Mechanical & Aerospace Engineering A path through The Default Library ======================================== How a machine, a bridge, an engine and an aircraft stand up, move, heat up and stay under control. Step 1. Calculus I: Slopes, Totals and the Bridge Between Them -------------------------------------------------------------- Why: Every rate and every total in engineering is a derivative or an integral. Your car's speedometer is a derivative. Its odometer is an integral. Everything a first calculus course teaches is a way of moving between the two. Look at a dashboard. The odometer says how far you have gone in total. The speedometer says how fast the total is changing right now. Press the accelerator and the speedometer moves first; the odometer follows, a little behind, like a patient accountant adding up what the speedometer reported. Nobody gave the speedometer a ruler or a clock to look at the whole trip. It reads one instant. Yet it reports a rate, which is a statement about change, from a single moment. How can a single moment say anything about change? That puzzle is what the course exists to answer, and the answer turned out to run most of modern engineering and computing. This is the second time the library has met calculus. Calculus: Derivatives, Integrals, and the Language of Motion (12/11) is the easier first read, with the tangent line and the rectangle sum. Here we go a step further: what the course trains you to see, how a computer gets it wrong, and where the idea is used afterwards. The course in one box Who takes it: first-year students in Mechanical & Aerospace Engineering and in Computer Science, and almost every other science or engineering degree. What it is for: turning "how much?" into "how fast?" and back, so that change can be calculated instead of guessed. Comes after: nothing in this series. Algebra and a little trigonometry are all it assumes. Leads to: Multivariable Calculus (34/02), Differential Equations (34/03), Probability & Statistics (34/05), Physics I: Mechanics (34/06), Thermodynamics (34/14), and from them nearly everything else on both majors. Honest line: you will not be able to differentiate a hard formula by hand after this. You will see why a flat spot on a curve is where the best answer hides, and why the area under a rate is a total. A limit is where a calculation settles The speedometer puzzle is this. Speed is distance divided by time, but in an instant no time passes and no distance is covered, so you seem to be dividing zero by zero. The way out is to stop asking what happens at zero and ask what a sequence of better and better answers is settling towards. Measure over one second, then a tenth, then a hundredth, and watch the average speed home in on a number. That number is the limit. speed now = limit of (distance change ÷ time change) as the time change shrinks to 0 A graduate does not carry the formal definition around, but carries the habit: if a quantity looks like 0 ÷ 0, ask what it is approaching. Take sin(x) ÷ x as x goes to zero, with x in radians. At x = 0.1 it is 0.99833; at 0.01, 0.99998; at 0.001, 0.9999998. It never "is" 1 at zero, since it is undefined there, but it settles on 1. That one limit is why the derivative of sine turns out to be cosine. Interactive · Shrink the gap until the computer gives up We estimate the slope of ex at x = 1 by (e1+h − e) ÷ h. The true slope is e = 2.718281828459045. Make the gap h smaller. Gap h = 10−k The error falls as the gap shrinks, as the limit promises, until about h = 10−8, where it bottoms out near 7 × 10−9. Shrink further and it gets worse: the two values being subtracted agree in almost all of their roughly 16 stored digits, so the subtraction wipes out the information. At 10−16 the answer is exactly 0. This is a computer-science lesson hiding in a calculus course: the limit is a mathematical idea, and a machine can only approach it so far. The derivative is a slope, and a slope is a rate Zoom in on any smooth curve and it straightens out. The steepness of that straight piece is the derivative. In plain words: how much the output moves per unit nudge of the input. If the curve is distance against time, the derivative is speed. If it is temperature against position, it is how sharply it warms as you walk. Rate and slope are one thing seen two ways. f′(x) = limit of [f(x+h) − f(x)] ÷ h as h → 0 The graduate's real asset is not computing derivatives but reading them. Where the curve is rising, the derivative is positive. Where it is flat, the derivative is zero. Where the curve is steepest, the derivative is at its peak. Try it: the lower graph is the derivative of the upper one, drawn point by point. Interactive · A curve and its slope, side by side x³ − 3x sin x bell: e^(−x²) Position x Check the three curves. Every time the top curve peaks or dips, the bottom one crosses zero. Every time the top is climbing, the bottom is above the axis. For x³ − 3x the slope is 3x² − 3, which is zero at x = −1 and x = 1: the hill top and the valley floor. The chain rule: rates multiply down a chain Rates often come in chains. A bicycle's pedals turn the chainring, the chain turns the rear wheel, the wheel moves the bike. If the wheel turns 3 times per pedal turn, and the bike moves 2 metres per wheel turn, the bike moves 6 metres per pedal turn. You multiply the ratios. That is the chain rule, and it is the most used rule in the course. d(outer)/d(input) = d(outer)/d(middle) × d(middle)/d(input) The classic worked case is a balloon. Its volume depends on its radius, and the radius depends on time. At radius r the surface has area 4πr², so one extra centimetre of radius adds a shell of volume about 4πr² cm³: that is dV/dr. Multiply by how fast the radius is growing and you have how fast the volume is growing. At r = 10 cm, growing 2 cm each second, volume grows at 4π × 100 × 2 ≈ 2,513 cm³ per second. Pull the sliders and confirm it. Interactive · Inflate a balloon Radius now Radius is growing at The picture shows why it works: the shaded shell is what one second adds, a skin of area 4πr² and thickness equal to the radius growth. The same rule runs inside every neural network, where the "chain" is dozens of layers long and the multiplication is called backpropagation (Machine Learning, 34/32; Multivariable Calculus, 34/02). Where the slope is zero, something is best At the top of a hill you stop going up and have not yet started going down. The slope is zero. So to find the biggest or smallest value of anything smooth (strongest, cheapest, lightest, fastest) look for where its derivative is zero. That is optimisation, and it is why engineers love the derivative. Take a 30 cm by 20 cm sheet of card. Cut a square of side x from each corner and fold up the sides to make an open box. Its volume is V = x(30 − 2x)(20 − 2x). Cut too little and the box is shallow. Cut too much and it is narrow. In between there is a best x, and the slope tells you where: at x ≈ 3.92 cm the volume peaks at about 1,056 cm³. Interactive · The best box Corner cut x Jump to the flat spot Notice what the slope does as you drag. To the left of the peak the slope is positive, so a bigger cut helps. To the right it is negative, so a smaller cut helps. At the peak it is zero and nothing nearby is better. Gradient descent, the algorithm that trains machine-learning models, is exactly this reasoning repeated: look at the slope, step downhill, repeat. The integral adds up a rate into a total The odometer is the speedometer added up over time. Draw speed against time; the area between the curve and the time axis is distance. That is the integral. Anywhere a rate is known and a total is wanted (litres per minute into a tank, watts into a battery, newtons over a distance) the answer is an area. Try it before you read on. A car accelerates steadily from 0 to 20 m/s in 10 seconds, cruises at 20 m/s for 30 seconds, then brakes steadily to a stop in 10 seconds. How far has it gone? Interactive · Predict, then reveal 600 m 800 m 1,000 m 1,200 m Choose an answer. The shape is two triangles and a rectangle: ½ × 10 × 20 + 30 × 20 + ½ × 10 × 20 = 100 + 600 + 100 = 800 m. The common wrong guess, 1,000 m, uses the top speed for all 50 seconds and forgets that the car spends time slower than that. The Fundamental Theorem: two jobs, one idea Here is the surprise that makes the course a single subject. Build a running total from a rate: the amount in the tank after time T. Then ask how fast that total is growing at T. The answer is just the rate at T. Accumulating and differentiating undo each other, the way the odometer and speedometer do. Newton and Leibniz, working separately in the late 1600s, each saw this. if A(T) = area under the rate up to T, then A′(T) = the rate at T Its practical power is that you can find a total without adding anything: find a function whose slope is your rate and subtract its values at the two ends. Below, water flows into a tank at a rate that wobbles: 4 + 3 sin(t/2) litres per minute. Drag the time and watch the right-hand graph, the tank's total, steepen exactly when the rate on the left is high. Interactive · The tank and its rate Time T The readout checks the theorem two ways: area by adding 600 thin slices, and the exact formula 4T + 6(1 − cos(T/2)). At T = 12 minutes both give 48.24 litres. The slope of the total at that moment is 4 + 3 sin 6 ≈ 3.16 litres per minute, the rate at that moment. Every smooth thing is nearly a straight line The last habit is the most useful in daily engineering. Close to any point, a smooth curve is almost its tangent line, so you can replace it with a straight line and be nearly right. The small-angle rule sin x ≈ x is why a pendulum swings at a steady rhythm. The rule √(1 + x) ≈ 1 + x/2 gives mental square roots: √1.1 ≈ 1.05, true value 1.0488. f(a + x) ≈ f(a) + f′(a) · x   (good when x is small) How small is small? The error shrinks faster than the step. Halve the step and the error falls about four times for √(1 + x) and ex, and about eight times for sine. Interactive · How good is the straight line? sin x ≈ x √(1+x) ≈ 1 + x/2 eˣ ≈ 1 + x Step x from the point What a graduate just sees A rate and a slope are the same thing, and "how fast?" always means a derivative. A total is an area under a rate, and adding up thin slices is a calculation, not a mystery. Rates multiply along a chain: gear ratios, units, layers of a network. The best value of a smooth quantity sits where its slope is zero. Differentiating and accumulating are inverse jobs, so a total can often be found without adding. Up close, everything smooth is a straight line, and engineers exploit that constantly. Subtracting two nearly equal numbers on a computer destroys precision, so a naive slope estimate gets worse when the gap gets too small. Where it shows up next Physics I (34/06) and Dynamics (34/12): velocity is the derivative of position, acceleration of velocity, and Newton's law ties force to the second one. Differential Equations (34/03): when a law is written as a rate, solving it means integrating, which is the whole next course. Thermodynamics (34/14) and Heat Transfer (34/16): heat added is an integral of power; temperature profiles are read from slopes. Probability & Statistics (34/05): the chance of a range of outcomes is the area under a density curve. Algorithms (34/24) and Machine Learning (34/32): growth rates of running time are compared with limits, and models are trained by following slopes downhill. If you want to go deeper 📘 Textbook: James Stewart, Daniel Clegg and Saleem Watson, Calculus: Early Transcendentals (9th ed., Cengage). The most-assigned text; best for worked examples and a huge problem bank. 📘 Textbook: Hass, Heil, Bogacki and Weir, Thomas' Calculus (15th ed., Pearson). Strong on applications, popular in engineering departments. ▶️ Gentler: Grant Sanderson's Essence of Calculus series on 3Blue1Brown, a visual tour of the same ideas. Also Steven Strogatz, Infinite Powers, for the story. 🌐 Free course: MIT OpenCourseWare 18.01SC Single Variable Calculus, with video lectures, notes and problems. Khan Academy also covers it. The takeaway Calculus I teaches two moves and one bridge. The derivative turns a total into a rate (a slope); the integral turns a rate into a total (an area); and the Fundamental Theorem says they undo each other. From that come the habits of mind: chain rates together, look for zero slope to find the best, and treat smooth things as lines up close. Our slope estimate for ex was best near a gap of 10−8 and useless at 10−16, a reminder that the ideas are exact and the machines are not. Try this: next time you are in a car or on a bike, watch the speedometer and ask what the area under its history means. Then on a hill, find the spot where the road is flat. That is a derivative equal to zero. Sources & Further Reading Sources. Every number in the interactives was computed in node: the forward-difference error of ex at x = 1 for h = 10−1 to 10−16 (best 6.6 × 10−9 at 10−8, exactly 0 at 10−16, in IEEE double precision); sin(x)/x at 0.1, 0.01, 0.001; the box V(x) = 4x³ − 100x² + 600x, zero slope at x = (50 − √700)/6 = 3.9237, V = 1,056.3 cm³; the balloon 4π × 10² × 2 = 2,513.27 cm³/s; the car, 800 m; the tank, A(12) = 48.239 L; small-step errors of the three approximations. Book editions checked against the publishers' pages (Cengage, Pearson) and the course numbers against MIT OpenCourseWare. 🌐 MIT OpenCourseWare 18.01SC Single Variable Calculus ▶️ 3Blue1Brown: Essence of Calculus 🌐 Khan Academy: Calculus 1 📘 Stewart, Clegg and Watson, Calculus: Early Transcendentals, 9th ed. (Cengage) 📘 Hass, Heil, Bogacki and Weir, Thomas' Calculus, 15th ed. (Pearson) 📘 Steven Strogatz, Infinite Powers, a popular history of calculus Step 2. Physics I: Mechanics — Three Ways to Keep the Books ----------------------------------------------------------- Why: Forces, energy and momentum: the three ways engineers account for motion. Why doubling your speed quadruples your stopping distance, why a skater spins up when she pulls her arms in, and the three questions an engineer asks before touching a calculator. A car doing 50 km/h stops in about 14 metres on a dry road. Most people guess that at 100 km/h it takes about 28. It takes about 56. The extra speed doesn't just cost twice the distance, it costs four times, and the same fact is why a crash at 100 km/h is not "twice as bad" as one at 50. Nobody needs a formula to be afraid of that, but a person who has done a first mechanics course sees it: the brakes have to get rid of the car's energy, energy grows with the square of speed, and the road can only take it away at a steady rate per metre. That one chain of thought is most of the course. Physics I is less a list of topics than a way of keeping accounts. There are three ledgers, force, energy and momentum, and a good physicist picks the one in which the problem is easiest. If you want an easier first read, the library has The Four Forces (13/01), Energy: The One Currency (13/02) and Orbits (13/11). This piece goes further: it makes you use the ledgers. The course in one box Who takes it: first-year students of Mechanical & Aerospace Engineering, and in many programmes Computer Science too, as the science requirement. What it's for: turning "what will this object do?" into something you can calculate. Comes after: Calculus I (34/01). Leads to: Physics II: Electricity & Magnetism (34/07), Statics (34/10), Dynamics (34/12), Materials Science & Manufacturing (34/13), Thermodynamics (34/14). Honest line: you won't be able to design a suspension after this. You will see why a car's tyres, not its engine, decide how fast it can take a corner. Newton's laws are one accounting system Isaac Newton set out his three laws in the Principia in 1687. Read as a bookkeeper would, they say: momentum (mass times velocity) is the thing being tracked; a force is a deposit or withdrawal; and every deposit has a matching withdrawal somewhere else. The first law says that with no net force the account doesn't change: things keep their velocity. The second says that a net force changes momentum at a steady rate, which for a fixed mass is the famous rule below. The third says forces come in pairs between two bodies: if you push a wall, the wall pushes you, and the pair never acts on the same body, which is why they don't cancel. Fnet = m · athe net force on a body is its mass times its acceleration Worked number: a 1,500 kg car getting from 0 to 100 km/h (27.8 m/s) in 8 seconds accelerates at 3.47 m/s², so the road must push it forward with about 5.2 kN. Two ice skaters of 50 kg and 70 kg who push apart from rest leave with equal and opposite momentum: if the heavier moves at 1.0 m/s the lighter goes at 1.4 m/s the other way. Total momentum stays at zero. Try the classic: you on a scale in an elevator. The scale doesn't read your weight, it reads the force pushing up on you. Interactive · The elevator scale (a 70 kg rider) Speeding up, going up Cruising Speeding up, going down Cable cut Elevator acceleration (up is positive) Accelerating upward at 2 m/s² the scale reads about 827 N, like 84 kg. At constant speed it reads your true weight, whether the elevator is moving or still. In free fall it reads zero. That is exactly the "weightless" astronaut of Orbits. Free-body diagrams, and friction as a force that only pushes as hard as it must The graduate's first reflex on any problem is to draw one object, alone, and every push on it: weight, the surface pushing back, friction, tension. Nothing else. That is a free-body diagram. Add the forces as vectors, split them along the directions that matter, and F = ma does the rest. Friction is the force beginners get wrong. Static friction is not a fixed number: it pushes exactly as hard as needed to stop sliding, up to a ceiling of μs times the normal force (how hard the surface pushes back). Only once the ceiling is passed does the block slide, and then friction drops to μk × normal force. f ≤ μs · Nfriction is at most a coefficient times how hard the surfaces press together On a ramp the weight splits into a part along the slope, mg sin θ, and a part pressing into it, mg cos θ. Sliding begins when the first beats μs times the second, which gives tan θ = μs. The mass cancels. Check that in the tool: change the mass and nothing about when it slips changes. (The coefficients are round textbook values; real surfaces vary a lot.) Interactive · A block on a ramp Wood on wood Rubber on concrete Steel on ice Ramp angle Mass Arrows are drawn as fractions of the weight. With μs = 0.5 the block holds until 26.6° and then slides. At 40° it accelerates down at about 4.05 m/s² (with μk = 0.3). This is why a pile of sand has a steepest slope, and why designers of ramps, screws and wedges care so much about one number. Energy: the balance that doesn't care about the path Force problems get painful when the path is complicated. Energy is the escape. The work a force does is force times distance moved along it, and the work–energy theorem says the net work on a body equals its change in kinetic energy. When only gravity or springs are involved, the total of kinetic plus potential energy is simply conserved, whatever the route. KE = ½ · m · v²speed counts squared; this is why speed is so costly A bobsled dropping 30 m ends at √(2 × 9.81 × 30) = 24.3 m/s (87 km/h) with no friction, however many bends the track has. Now back to the opening: the brakes do work μ × m × g × d against the car's ½mv², so the stopping distance is v² ÷ (2μg). Double v, quadruple d. Interactive · Brake, or hit Dry (μ 0.7) Wet (0.5) Snow (0.2) Ice (0.1) Speed when you brake Obstacle ahead Braking only, no reaction time. μ values are round, typical figures for tyres. The sharpest result: brake from 100 km/h with an obstacle 14 m away, the distance that stops a 50 km/h car on a dry road. You still hit it at about 87 km/h, with 75% of the energy you started with: 14 metres of hard braking has removed only a quarter of it. Momentum: what survives every crash Energy has a weakness: in a collision much of it turns to heat, sound and bent metal, so the energy ledger is no use. Momentum, though, is conserved in any collision, because the two forces are equal and opposite for the same short time (the third law again). The total before equals the total after, with direction as a sign. How bouncy the collision is gets one number, the restitution e: 1 for perfectly bouncy, 0 for stuck together. Momentum doesn't care about e. Kinetic energy does. Interactive · Two carts collide Equal masses, bouncy Stick together Light hits heavy Blue mass Blue velocity Gold mass Gold velocity Bounciness e Velocity to the right is positive; arrows show velocity. With equal masses and e = 1, the moving cart stops dead and the other leaves with its speed: the toy known as Newton's cradle. Stuck together, two 1,000 kg cars (one at 10 m/s, one parked) leave at 5 m/s and half the kinetic energy, 25 kJ, becomes damage. That lost half is the reason cars have crumple zones: they spend the energy deforming metal instead of people. Rotation: the same laws with new names Everything above has a spinning twin. Force becomes torque, force times the perpendicular distance from the pivot: 80 N on a 0.25 m wrench is 20 N·m, and a longer wrench gives more for the same push. Mass becomes moment of inertia I, which counts not just how much mass there is but how far from the axis it sits. Momentum becomes angular momentum L = I × ω. τ = I · α   ·   L = I · ωwith no outside torque, L is conserved Mass far from the axis counts for much more, because I grows with the square of distance. That's why a solid disc beats a hoop of the same mass and radius down a ramp: the hoop keeps all its mass at the rim, so it spends more of its energy on spinning (the disc's acceleration is two thirds of g sin θ, the hoop's one half, whatever the size or mass). And it's why a skater spins faster when she pulls her arms in. Interactive · The skater's weights Arms out Halfway Tucked in Weights' distance from the axis A model: body I = 0.9 kg·m², plus two 2 kg weights; start at one turn per second with arms out. From arms out to tucked in, I falls from 3.15 to 0.99 kg·m², so the spin rises from 1 to about 3.2 turns per second. The energy also rises, from 62 J to 198 J, and the difference is the work her muscles do pulling the weights inward. Circular motion and gravity: a force that points at the middle Going round a bend at constant speed is still accelerating, because the velocity keeps changing direction. The acceleration points to the centre and equals v²/r, so something must push sideways with force mv²/r. On a flat road that something is friction. No more friction, no more turn: the car continues in a straight line, which feels, from inside, like being thrown outward. vmax = √(μ · g · r)fastest speed a flat bend of radius r allows Notice the engine's power isn't in the formula. On a 50 m bend with μ = 0.5 (wet) the limit is 56 km/h; dry (0.7), 67 km/h; ice (0.1), 25 km/h. Interactive · Take the bend Dry (0.7) Wet (0.5) Ice (0.1) Speed Radius of the bend Replace friction with gravity and you get orbits. Set v²/r equal to GM/r² and the period T satisfies T² ∝ r³, Kepler's third law. Check it: the Space Station's lap of 92.6 minutes at 6,779 km from Earth's centre predicts, for the Moon at 384,400 km, a period of 27.5 days. The real month is 27.3. One law, from a bend in a road to the Moon. Which ledger? Choosing force, energy or momentum This is the real skill. A hard problem in one ledger is a one-liner in another. Three rough rules: if the question asks for a force or a time, use F = ma; if it asks for a speed between two positions, and the path is messy, use energy; if there's a collision or explosion, use momentum. If something spins, use angular momentum. Try five. Interactive · Pick the first tool Force Energy Momentum Angular momentum Next question What a graduate just sees Stopping distance goes with speed squared: double the speed, four times the distance. A scale reads the push on you, not your mass: it changes in an accelerating elevator. Static friction is only as large as it has to be, up to a limit that depends on μ and the pressing force. Energy ignores the path; use it when the route is complicated. Momentum survives every collision; kinetic energy rarely does. Mass far from the axis counts for much more in rotation, because distance is squared. A turn needs a sideways force; remove the grip and you go straight. Weightless means falling together, not "no gravity". Where it shows up next Statics (34/10) is F = ma with a = 0: the free-body diagram becomes the bridge or crane that stays put. Dynamics (34/12) and Vibrations (34/17) take rotation and the second law into machines and springs, usually as differential equations (Differential Equations, 34/03). Thermodynamics (34/14) is the energy ledger with heat added: the lost half of that crash is where it starts. Physics II (34/07) redoes the same accounting for charges, with Coulomb's force and voltage as energy per charge. In Programming Fundamentals (34/09), you'll turn F = ma into a simulation loop; in Computer Science game engines and robotics code, it's the physics step. If you want to go deeper 📘 Standard textbooks. Halliday, Resnick & Walker, Fundamentals of Physics (12th ed., Wiley, 2021), or Young & Freedman, University Physics (16th ed., Pearson); the usual first-year choices, with worked examples and thousands of problems. For a sharper, more mathematical route, Kleppner & Kolenkow, An Introduction to Mechanics (2nd ed., Cambridge). 📘 Gentler, free. The Feynman Lectures on Physics, Volume I, free online at feynmanlectures.caltech.edu; chapters 4 (energy) and 10 (momentum) are a good start. 🌐 Free course. MIT OpenCourseWare 8.01SC Classical Mechanics (Fall 2016): lectures, problem sets and solutions. The takeaway Mechanics is three ledgers kept over the same world. Force tells you what changes motion and when; energy lets you skip the path; momentum survives collisions; and each has a spinning twin. Doubling speed quadruples stopping distance because energy goes with v². The graduate's skill is choosing which ledger makes this problem easy. Try this: next time you're a passenger, estimate how far the car needs to stop at the speed you're doing, using "14 m at 50 km/h, four times that at 100". Then notice how far ahead the next car really is. Sources & Further Reading Sources. Newton's laws first published in Philosophiae Naturalis Principia Mathematica (1687). Textbook physics throughout. Friction coefficients are rounded from published tables, which vary widely between sources (for example Engineering Toolbox's "Friction coefficients for common materials", which gives rubber on dry concrete 0.6–0.85, on wet concrete 0.45–0.75). Every number in the interactives was computed in node and checked by hand: 14.05 m and 56.2 m braking distances at 50 and 100 km/h (μ = 0.7); 86.6 km/h residual speed; 5.21 kN; 24.3 m/s; skater I = 3.15 and 0.99 kg·m², 3.18 turns per second, 62 J and 198 J; 56.4, 66.7 and 25.2 km/h bend limits; Kepler scaling from the Space Station (92.6 min at 6,779 km) to the Moon (27.46 days from 384,400 km). 🌐 MIT OpenCourseWare: 8.01SC Classical Mechanics 📘 The Feynman Lectures on Physics, Vol. I (free online, feynmanlectures.caltech.edu) 📘 Kleppner & Kolenkow, An Introduction to Mechanics, 2nd ed. (Cambridge University Press) 🌐 Engineering Toolbox: friction coefficients Step 3. Linear Algebra: Machines That Move Space ------------------------------------------------ Why: Real structures have hundreds of unknowns; matrices solve them all at once. A grid of numbers is really a machine that moves all of space at once. Learn to see it, and circuits, bridges, search engines and AI all start to look alike. You pinch two fingers on a photo and it rotates and zooms smoothly under them. A game character turns to face you and every corner of its face swings round together. A search for "best pizza near me" ranks millions of pages in a fraction of a second. A bridge designer presses "solve" and a screen of colour shows how every beam is being squeezed. Underneath all four is the same object: a grid of numbers, and one habit of thought about what a grid of numbers does. A graduate of a linear algebra course no longer sees a table of numbers. They see a machine that grabs the whole plane, or the whole of space, and moves it. This piece builds that picture with seven things you can drag. You will not come out able to compute a determinant of a 5×5 by hand (nobody does that on purpose). You will come out seeing why a system of equations sometimes has no answer, why a bridge hums at particular pitches, and why the chips that run AI are built for one operation only. The course in one box Who takes it: nearly every engineering and computer science student, usually in the first or second year. It is the single most reused maths course in both majors. What it's for: handling many unknowns and many dimensions at once, with one set of tools. Comes after: only school algebra and a little geometry. (Calculus helps but is not required.) Leads to: Statics (34/10), Control Systems (34/20) and Machine Learning (34/32) list it as a prerequisite; Vibrations (34/17), Computer Architecture (34/25) and Multivariable Calculus (34/02) lean on it constantly. Honest line: you won't be able to run a finite element program after this piece. You will see why it spends nearly all its time solving one equation, Ax = b. If you want an easier first read, try the library's Vectors, Matrices and Linear Transformation (12/14); this one goes further into solving, eigenvectors, fitting and why hardware cares. A matrix moves all of space at once Start with the picture a graduate keeps. A 2×2 matrix is not four numbers. It is two arrows. The first column says where the arrow pointing one step right (call it i) ends up. The second column says where the arrow pointing one step up (j) ends up. Once you know where those two land, you know where everything lands, because every point is some amount of i plus some amount of j, and the machine is straight and even: grid lines stay straight, and evenly spaced lines stay evenly spaced. Ax = x1(column 1) + x2(column 2) Read that in words: the matrix times a point is "that many of the first column, plus that many of the second". Try it. The grey grid is the original plane; the blue grid is where the matrix sends it. Interactive · Move the plane Do nothing Rotate 30° Stretch Shear Mirror Column 1, right part (a) Column 2, right part (b) Column 1, up part (c) Column 2, up part (d) Without JavaScript: the rotate-30° matrix sends the arrow (1, 0) to (0.87, 0.5) and (0, 1) to (−0.5, 0.87). Every grid line turns 30° and the unit square keeps area 1. Notice what the shaded unit square does. In "Rotate" it keeps its size. In "Stretch" it doubles. In "Shear" it leans over but keeps its area. That area change is the next idea. The determinant is how much area gets scaled The determinant is one number that says how much the machine scales area. If the unit square becomes a shape with area 5, the determinant is 5. If it becomes a mirror image, the determinant is negative: area is kept (or scaled) but orientation is flipped. For a 2×2 matrix with columns (a, c) and (b, d) the rule is short: det = ad − bc Worked number: the matrix with columns (2, 1) and (1, 3) has det = 2×3 − 1×1 = 5, so it stretches every area in the plane by a factor of five. In three dimensions the determinant scales volume. The interesting case is zero. A determinant of zero means the machine flattens space: the plane is squashed onto a line (or a point), and the area becomes nothing. Squashing cannot be undone, because many different starting points land on the same spot. Drag the slider until the determinant reaches zero. Interactive · Squash the plane flat Tilt the second arrow: s Without JavaScript: the matrix with columns (1, 0.5) and (s, 1) has determinant 1 − 0.5s. At s = 2 the determinant is 0, the whole plane lands on one line, and the points (0, 0) and (2, −1) both land on the origin. The orange dot is a probe at (2, −1). At s = 2 it lands exactly on the origin, the same place as the point (0, 0). Two different inputs, one output: no machine could tell you which one you started from. That is the whole meaning of "this matrix has no inverse", and it is why a zero determinant is the alarm bell in every solver you will ever meet. Solving many equations at once: Ax = b Now use the machine backwards. Given the output b, which input x did the machine start from? That is the equation Ax = b, and it is the most common problem in applied mathematics. Every time you have several unknowns that are tied together by several straight-line relationships, you have one. A circuit is the cleanest example. Two loops share a resistor. Kirchhoff's voltage law says the voltage gained around each loop equals the voltage lost, which gives one equation per loop. With a 10 V battery, R1 = 2 Ω, R2 = 3 Ω (shared) and R3 = 4 Ω: 5I1 − 3I2 = 10−3I1 + 7I2 = 0 The determinant is 5×7 − 3×3 = 26, so there is exactly one answer: I1 = 2.69 A and I2 = 1.15 A. Change the resistors below and watch the matrix and the answer change together. Interactive · Solve a circuit with a matrix Battery V R1 (loop 1 only) R2 (shared) R3 (loop 2 only) Without JavaScript: with 10 V, 2 Ω, 3 Ω and 4 Ω the equations are 5I1 − 3I2 = 10 and −3I1 + 7I2 = 0, its determinant is 26, and the loop currents are 2.69 A and 1.15 A. The reason an engineer cares: nothing about this method knows it is a circuit. A truss is the same. At each joint of a bridge, the horizontal forces must add to zero and so must the vertical ones: two equations per joint. A simple triangle truss has three joints and so six equations, for three member forces and three support reactions, and the answer is one solve of Ax = b. A real structure has thousands of joints. Finite element programs routinely solve systems of many thousands of equations, and research codes have reported solving over a million unknowns at once. The method is the same one you just used; only the size has changed. (Statics, 34/10, is this idea applied to forces.) A zero determinant in a structure's equations is a warning: the structure may be able to move without any member pushing back. Eigenvectors: the directions a transformation only stretches Most arrows are turned by a matrix as well as stretched. But for many matrices there are special directions where the arrow comes out pointing the same way (or exactly backwards), only longer or shorter. Those are the eigenvectors, and the stretch factor along each is its eigenvalue. (Eigen is German for "own": the matrix's own directions.) Av = λv In words: feed the matrix the special arrow v and it hands back the same arrow multiplied by a plain number λ. For the matrix with columns (2, 1) and (1, 2) the special directions are 45° (stretched by 3) and 135° (stretched by 1). Every other arrow gets bent towards the stretchy one. Turn the dial and look for the moments the two arrows line up. Interactive · Find the directions that only stretch Stretch & mix Shear Rotation Two springs Direction of the input arrow Without JavaScript: for the matrix with columns (2, 1) and (1, 2), an arrow at 45° comes out at 45° three times longer, and an arrow at 135° comes out at 135° unchanged. A pure rotation has no such direction at all. Three things to notice. The shear has only one special direction (along the x-axis, where nothing happens at all). The rotation has none, because every arrow turns. And "Two springs" has two special directions with different stretch factors, 1 and 3; hold that thought. Eigenvectors at work: bridges that hum, pages that rank Vibration. Picture two equal masses on springs between two walls. Push them any way you like and the motion looks messy. But there are exactly two clean patterns: both swing together, or they swing in opposite directions. These are the modes, and they are the eigenvectors of the "two springs" matrix above, (1, 1) and (1, −1). The eigenvalues 1 and 3 fix how fast each mode vibrates: the squared frequency is the eigenvalue times k/m, so the opposite-motion mode vibrates √3 ≈ 1.73 times faster. Any messy motion is just a mix of the modes. That is how engineers find which pitches will shake a wing, a bridge deck or a drive shaft (Vibrations, 34/17). Ranking pages. Google's founders modelled a web surfer who follows a random link from each page, and now and then jumps to a random page instead. The long-run share of time spent on each page is its PageRank, and it is the eigenvector (with eigenvalue 1) of the matrix describing one step of that surfer. In their 1998 paper Brin and Page used a "damping" probability of 0.85 for following links. You can find the eigenvector without solving anything fancy: start with equal scores and apply the machine over and over. Press Step. Interactive · A four-page web finds its eigenvector Step onceRun 20 stepsReset Without JavaScript: A links to B and C, B links to C, C links to A, D links to C. After many steps with damping 0.85 the ranks settle to A 0.373, B 0.196, C 0.394, D 0.038. D, which nobody links to, keeps only the base share. After about twenty steps the numbers stop changing: the vector has become an eigenvector, because applying the machine again returns the same vector. Page C wins because two pages point at it and one of them (A) is itself well-ranked. Least squares: the best answer when there is no exact one Real data is never perfect. Six measurements that "should" lie on a straight line will not, so the equations Ax = b have more equations than unknowns and no exact solution. Do you give up? No. You choose the line whose total mistake is smallest. Measure each miss vertically, square it, and add up the squares. The least squares line makes that total as small as possible. In pictures: each squared miss is literally a square. Drag the line and watch the squares shrink and grow. Interactive · Make the squares as small as possible Slope Height at x = 0 Snap to the least-squares line Without JavaScript: for the six points (1, 1.2), (2, 2.9), (3, 3.1), (4, 4.8), (5, 5.2), (6, 7.1) the best line is y = 1.089x + 0.24, with total squared miss 0.798. The guess y = x gives 2.75. The graduate's picture is geometric: the right-hand side b is a point outside the flat sheet of everything the matrix can reach, and the best answer is the point on the sheet closest to it, found by dropping a perpendicular. That perpendicular condition gives the one equation to remember: ATA x̂ = ATb In words: multiply both sides by the matrix flipped on its side and you get a square system you can solve. The history is a priority fight: Legendre published the method in 1805, Gauss said he had used it since 1795 and published in 1809, and Gauss's use of it to predict where the asteroid Ceres would reappear in 1801 made it famous. Today it sits under every trend line, every calibration of a sensor, and the first model most machine-learning students ever train. Why GPUs and AI are matrix multiplication Multiplying two matrices is just many dot products: every cell of the answer is one row of the first matrix paired with one column of the second, multiplied and added. Click a cell below and see the row and column that make it. Interactive · Build a product cell by cell Tap a cell of C = A × B. Matrix size n (n × n times n × n) 1 billion multiplies per second 100 billion per second 10 trillion per second The speeds are round illustrative numbers, not any particular chip. Without JavaScript: multiplying two n×n matrices takes n×n×n multiplications. For n = 1,000 that is 1,000,000,000, and the 1,000,000 cells of the answer can all be computed independently. Two facts follow. First, the work is huge: n×n matrices need n³ multiplications, so 1,000×1,000 is a billion. Second, the cells are independent, so a thousand workers (or a million) can each take cells at the same time. That is exactly the shape of work a graphics chip was built for, because drawing a 3D scene means pushing every corner of every triangle through a matrix that rotates, moves and projects it. Modern neural networks turned out to be the same shape. A layer of a network is a matrix of learned numbers times a vector (or a matrix) of inputs. In their scaling-law paper, Kaplan and colleagues estimate that a transformer language model with N parameters needs roughly 2N arithmetic operations to process one token. GPT-3, with 175 billion parameters, therefore needs about 3.5×1011 operations for each token, nearly all of them inside matrix products. NVIDIA's Volta GPUs (2017) put dedicated "tensor core" circuits on the chip that do nothing but a small matrix multiply-and-add (4×4) per clock tick. When people say AI runs on GPUs, they mean it runs on hardware shaped like linear algebra. (Computer Architecture, 34/25, and Machine Learning, 34/32.) What a graduate just sees A matrix is a machine that moves all of space; its columns say where the basis arrows go. The determinant is the area (or volume) scale factor, and zero means the machine flattens space and cannot be undone. "Solve Ax = b" is the same question for a circuit, a truss, a fitted line and a game physics step. If there are as many independent equations as unknowns there is one answer; fewer, many answers; contradictory, none. Eigenvectors are the directions a machine only stretches; they simplify any problem the machine drives. A vibrating structure is a mix of modes, one per eigenvector; the eigenvalues are the pitches. When there is no exact answer, take the closest one: project onto what you can reach. Matrix multiplication is many independent dot products, which is why it parallelises and why hardware is built around it. Where it shows up next Statics (34/10) and Mechanics of Materials (34/11): joint and member forces are a linear system; stiffness matrices tie loads to deflections. Vibrations (34/17) and Dynamics (34/12): mode shapes and natural frequencies are eigenvectors and eigenvalues of mass and stiffness matrices. Control Systems (34/20): the eigenvalues of a system's matrix say whether a drone's wobble dies away or grows. Machine Learning (34/32): data is a matrix, a layer is a matrix, and training adjusts matrices; least squares is the model you meet first. Computer Architecture (34/25) and graphics: GPUs and tensor cores exist to multiply matrices fast. If you want to go deeper 📘 Gilbert Strang, Introduction to Linear Algebra (Wellesley-Cambridge Press; 6th edition, 2023). The standard first course for engineers; best for the four fundamental subspaces and for application. 📘 David C. Lay, Steven R. Lay and Judi J. McDonald, Linear Algebra and Its Applications (Pearson; 6th edition, 2021). Gentle, example-heavy, widely assigned. For proofs, Sheldon Axler's Linear Algebra Done Right (Springer; 4th edition, free to read online at linear.axler.net) is the elegant theoretical route. ▶️ 3Blue1Brown, Essence of Linear Algebra: the best visual series for learning alone; it builds exactly the "matrix as moving space" picture used here. 🌐 MIT OpenCourseWare 18.06, Linear Algebra (Gilbert Strang): full video lectures, free. The takeaway Linear algebra is the study of machines that move all of space at once. Read a matrix by its columns, read its determinant as the area scale, solve Ax = b when you need to run the machine backwards, look for the eigenvector directions when you want it to be simple, and project onto what you can reach when there is no exact answer. Try this: next time an app zooms, rotates or recolours a photo, picture the two column arrows moving. And for the fun of it, in the circuit widget raise R2 to 10: the shared resistor starts to decide what both loops do. Sources & Further Reading Sources. Strang, Introduction to Linear Algebra (6th ed., 2023) and MIT 18.06 for the standard course content. Kirchhoff loop equations and the spring-mass modes are textbook derivations; every number above was recomputed in code. Legendre's 1805 publication and Gauss's priority claim and Ceres prediction: Wikipedia, "Least squares"; MacTutor biography of Legendre. PageRank and the 0.85 damping factor: Brin and Page, "The Anatomy of a Large-Scale Hypertextual Web Search Engine" (1998). Volta tensor cores (4×4 matrix multiply-accumulate): Markidis et al., "NVIDIA Tensor Core Programmability, Performance & Precision" (2018). About 2N operations per token: Kaplan et al., "Scaling Laws for Neural Language Models" (2020); GPT-3 at 175 billion parameters: Brown et al. (2020). Million-unknown structural solves: NASA Technical Reports Server and OSTI conference papers on parallel sparse solvers. Go deeper. The books and courses above. Step 4. Programming Fundamentals: Instructions, State and Taking Things Apart ----------------------------------------------------------------------------- Why: Engineers model in code (MATLAB, Python) long before they build in metal. A computer can't guess what you meant. Everything in a first programming course is a way of saying it exactly enough, and of working out why you didn't. Tap "sort by price" and a list of a thousand items rearranges before you've lifted your finger. Somewhere, a person wrote down steps so exact that a machine with no judgment, no memory of yesterday and no idea what a "price" is could do it perfectly, a thousand times over. That is the whole trade. A programmer isn't someone who knows a language, they're someone who can state a task so precisely that nothing is left to guess. And since people are bad at being precise on the first try, the other half of the skill is finding out what you actually said, as opposed to what you meant. Mechanical and aerospace engineers use the same habits to simulate a wing before it's built; computer scientists use them to build every other subject on the degree. Python and MATLAB are the common first languages for engineers, and this piece uses Python-style code. The ideas carry to every language. The course in one box Who takes it: first-year students of Computer Science (usually their very first course) and Mechanical & Aerospace Engineering (often in MATLAB or Python). What it's for: telling a machine what to do exactly, and finding out when you didn't. Comes after: nothing; it has no prerequisites. Leads to: Data Structures (34/23), Computer Architecture (34/25) and Software Engineering (34/31) on the computing side; Differential Equations (34/03), Dynamics (34/12) and Control Systems (34/20) on the engineering side, all of which are solved on a computer in practice. Honest line: you won't be able to ship an app after this. You will be able to read ten lines of code, predict what they do, and make a physics model in a loop. A program is instructions plus state Strip away the syntax and a program is two things: a list of instructions, and the state, the current values of all its named boxes. Each instruction reads the state and changes it a little. Running the program is just that, one step after another, and the answer is whatever state is left (or printed) at the end. The graduate habit is to run the program in their head: keep a small table of variable values and update it line by line. Below is a tiny interpreter that does it visibly. Press Step. Then edit the program: change the 6 in the Collatz example, or write your own with =, +=, if/else, while, for … in range(…) and print. Interactive · Step through a program Add 1 to 5 Collatz steps Count down by 3 The program (edit me) StepRun to the endStart again A real mini-interpreter, written for this page. It stops after 400 steps, in case you write a loop that never ends. The first program adds 1 + 2 + 3 + 4 + 5 and prints 15. The Collatz one halves even numbers and triples-plus-one odd ones until it reaches 1: starting from 6 it takes 8 steps. Nobody knows if every starting number reaches 1, and yet the program is five lines long. That's what "simple instructions, deep behaviour" feels like. Variables and types: what a number is made of A variable is a name for a box in memory. A type says what kind of thing is in the box, and therefore what you may do with it: you can add two numbers, join two pieces of text, but adding "3" and 4 is an error or a surprise. Types exist because the machine stores everything as patterns of bits, and the pattern means nothing without the type. The box has a size too. A signed 16-bit integer holds −32,768 to 32,767, and nothing more. Decimals are stored as floating point, binary fractions of limited precision, so 0.1 can't be stored exactly and 0.1 + 0.2 gives 0.30000000000000004. On 4 June 1996 the first Ariane 5 rocket veered off course and broke up about 37 seconds after lift-off, a loss of more than US$370 million. The investigation traced it to the guidance software, reused from Ariane 4: a 64-bit floating-point value (the horizontal bias) was converted to a 16-bit signed integer, and on the faster rocket it didn't fit. The failure shut down both inertial units. A number too big for its box was enough. Interactive · Does it fit? 8 bits 16 bits 32 bits Number to store Ariane's code reportedly raised an error on overflow; some systems wrap silently instead, which is shown here. Conditionals and loops: choosing and repeating With state and instructions, only two more ideas are needed to compute anything: choose (if something is true, do this, else that) and repeat (while something is true, or for each value in a range). Together they let ten lines do work that would take ten million lines written out. The classic trap is in the edges. In Python's range(1, 10) the loop starts at 1 and stops before 10, so it runs 9 times, not 10. Off-by-one errors are the "fencepost" mistake: a fence of 10 sections needs 11 posts. Try the bounds yourself. Interactive · Edit the loop Add up i Add up i² Just count Start Stop (never reached) Step With the defaults, range(1, 11) runs 10 times and adds up to 55. Make the step negative and the start bigger than the stop to count down. Set the start above the stop with a positive step and the body never runs at all, which is perfectly legal and a common source of silent bugs. Functions and recursion: naming a piece, then calling it again A function packages some instructions under a name, takes inputs, and gives back an output. That is decomposition, the central skill of engineering of any kind: break a big problem into pieces small enough to understand and test alone, give each a name, then build on the names. The tracer above could be one function; so could braking distance. Recursion is a function that calls itself on a smaller version of its problem, until it reaches a case it can answer directly. The Fibonacci numbers (0, 1, 1, 2, 3, 5, 8, …, each the sum of the previous two) have a one-line recursive definition. Run it naively and the machine re-solves the same sub-problems again and again. Interactive · Count the calls Naive Remember answers (memo) Compute fib(n), n = Time it on this device Naively, fib(5) makes 15 calls, fib(20) makes 21,891, and fib(30), whose answer is just 832,040, makes 2,692,537. Remembering answers brings fib(30) down to 59 calls. Same answer, a different algorithm, and the course after this one (Algorithms, 34/24) is the study of exactly that difference. Data: lists and dictionaries Most programs aren't about single numbers but collections. A list is an ordered row of values, found by position (the third item). A dictionary maps a key to a value (a word to its count, a student ID to a name) and finds an entry by key without scanning. Choosing the right container is the first lesson of Data Structures (34/23). The difference is speed. To find a word in a list of n you may have to look at all n. A dictionary (usually built as a hash table) jumps almost straight to it, in roughly constant time on average, however large it is. Paste some text below and look a word up. Interactive · Count the words Textthe cat sat on the mat and the dog sat on the cat Look up a word Debugging is the scientific method Programs don't do what you meant; they do what you wrote. Finding the gap is science: look at the result, form a hypothesis ("the loop is skipping the last item"), make a prediction ("if I print the count here it will be one short"), run the experiment (print it), and revise. Guessing and changing things at random is the opposite, and it's how you add new bugs. The strongest move is bisection. If the code is a chain of steps and the result is wrong, check the value halfway along. If it's right, the fault is in the second half; if wrong, the first. Each check halves the suspects. In 1947 operators of Harvard's Mark II computer taped a moth found stuck in a relay into the logbook with the note "First actual case of bug being found" — a joke that only works because engineers already called faults "bugs". Interactive · Find the bug in as few probes as you can Start with x = 5. Slide to a step and probe the value there against what you'd get by hand. Look at x Probe hereBlame this stepNew bug Twelve suspect lines need at most four probes when you halve each time, because 2⁴ = 16 is at least 12. Checking from the top, one line at a time, can take twelve. Modelling: a physics simulation in a dozen lines Engineers model long before they build. Most physical laws give a rate of change (acceleration from forces, as in Physics I), and a computer can follow a rate forward in small steps: update the position using the velocity, update the velocity using the acceleration, repeat. That is the Euler method, and the loop is the whole simulation. x, y = 0.0, 0.0 vx, vy = 21.2, 21.2 # 30 m/s at 45 degrees dt = 0.1 # time step in seconds while y >= 0: x = x + vx * dt y = y + vy * dt vy = vy - 9.81 * dt # gravity changes vy only print(x) # about 93.9 metres Smaller time steps are more accurate and cost more. For the cannonball at 30 m/s and 45°, the exact range without air is 91.7 m. With a step of 0.5 s the simulation says 102.1 m, 0.1 s gives 93.9 m, and 0.001 s gives 91.8. Now add air drag, for which no tidy formula exists: the same loop still works, which is why engineers simulate. Interactive · A cannonball in code No air With air drag Launch speed Launch angle Time step Drag is modelled as a quadratic term with a baseball-like coefficient (0.007 per metre); a model, not a measurement. Readable code is a letter to a person A program is run by a machine once and read by people many times: the next engineer, the reviewer, you in six months. The well-known line from the preface of Structure and Interpretation of Computer Programs is: "Programs must be written for people to read, and only incidentally for machines to execute." The machine will accept either version below. Only one of them can be checked by a human. def f(a, b): return (a / 3.6) ** 2 / (2 * b * 9.81) def braking_distance(speed_kmh, friction): speed = speed_kmh / 3.6 # convert to metres per second return speed ** 2 / (2 * friction * 9.81) Both give 14.05 m for 50 km/h and a friction of 0.7, the braking distance of Physics I. Only the second tells you what it means, and so lets a reader spot a wrong constant. Good names, small functions and comments that say why are the cheapest debugging tools there are. People disagree on how much structure is enough: some teams favour heavy up-front design, others short cycles of build, test and revise; Software Engineering (34/31) covers both camps. What a graduate just sees A program is instructions plus state; to understand it, track the variables line by line. Every value has a type and a size, and a number too large for its box is a real way to lose a rocket. Loops run until their stop value, not through it: watch the edges. Break a problem into named functions you can test one at a time. A function that calls itself can be elegant and wildly wasteful; counting calls shows which. The right container (list or dictionary) can be the difference between instant and never. Debugging means hypotheses and experiments; halve the suspects each time. Simulation is a rate of change plus a loop, and smaller steps cost more for less error. Code is read far more often than it is written. Where it shows up next Data Structures (34/23) and Algorithms (34/24): the list-versus-dictionary and naive-versus-memo contrasts above, measured properly. Computer Architecture (34/25): the fixed-size boxes and overflow are the hardware's registers; instructions plus state is literally how a processor works. Software Engineering (34/31): decomposition, testing and readable code, scaled to teams and years. Differential Equations (34/03), Dynamics (34/12), Control Systems (34/20): the Euler loop is the starting point of numerical solvers, vehicle simulators and the code that runs a controller. Machine Learning (34/32): training is a loop that nudges numbers until the error falls. If you want to go deeper 📘 Standard textbook. John Guttag, Introduction to Computation and Programming Using Python (3rd ed., MIT Press, 2021), the text behind MIT's introductory course, written for engineers and scientists, with modelling and data. 📘 Gentler, free. Allen Downey, Think Python (3rd ed., Green Tea Press / O'Reilly), free to read online at greenteapress.com. For the deeper "why", Structure and Interpretation of Computer Programs by Abelson and Sussman is free online from MIT Press. 🌐 Free course. MIT OpenCourseWare 6.0001 Introduction to Computer Science and Programming in Python (Fall 2016), or Harvard's CS50x (starts in C, then Python and SQL) and CS50P (Python only), free on OpenCourseWare. The takeaway A program is instructions acting on state. Variables, types, choices, loops and functions are how you say it exactly; lists and dictionaries are how you hold data; debugging is hypothesis and experiment; and a simulation is a rate of change inside a loop. The graduate's real skill is reading code, predicting it, and taking a big problem apart. Try this: take any small thing you do by hand, like splitting a bill or converting recipe units, and write it as exact steps for a friend to follow without any judgment. Then watch where they get stuck: those are your bugs. Sources & Further Reading Sources. Ariane 5 flight 501 (4 June 1996; 64-bit floating-point to 16-bit signed integer conversion overflow; about 37 seconds; more than US$370 million): Wikipedia, "Ariane flight V88", which summarises the Inquiry Board's report. The Mark II moth (9 September 1947, logbook note "First actual case of bug being found", now at the Smithsonian) from the Computer History Museum and JSTOR Daily, "The bug in the computer bug story", which notes the term was already in use. The SICP quotation from the preface of Abelson & Sussman, Structure and Interpretation of Computer Programs (MIT Press, 1985). CS50x language order from cs50.harvard.edu/x; Think Python 3rd ed. from greenteapress.com. Computed and checked in node, including by running the page's own interpreter on its example programs: 15; 8 Collatz steps; −2; 40,000 wraps to −25,536 in 16 bits; fib call counts 15, 177, 21,891 and 2,692,537 (and 2n−1 with a memo); projectile ranges 91.74 m exact and 102.1, 93.9, 91.8 m for steps of 0.5, 0.1 and 0.001 s; 14.05 m braking distance. 🌐 MIT OpenCourseWare: 6.0001 🌐 Harvard CS50x and CS50P 📘 Think Python, 3rd edition (free online) 📘 Guttag, Introduction to Computation and Programming Using Python, 3rd ed. (MIT Press) 🌐 JSTOR Daily: the bug in the computer bug story Step 5. Engineering Drawing & CAD: The Contract Between Designer and Maker -------------------------------------------------------------------------- Why: A drawing is a contract: what you draw is what gets made. A drawing is not a picture of a part. It is a promise about what will be made, how exactly, and who is to blame when it isn't. Buy a shelf bracket, and the screw holes line up with the studs in your wall. Buy a replacement bearing for a bicycle, a car, a washing machine, and it fits a shaft that was made in a different factory, on a different continent, years earlier. Nobody phoned anyone. What made all that work is a sheet of paper, or these days a file, that says: make it exactly like this, to within this much. That sheet is an engineering drawing, and it is the quietest important idea in mechanical engineering. A graduate looks at one and reads it the way you read a sentence: the views, the numbers, the little symbols. The skill sounds dull until you see how many ways a part can come back wrong when a drawing is vague. The course in one box Who takes it: mechanical, aerospace and civil engineers, and anyone in product design, usually in year 1. What it's for: saying precisely what shape a part is, how exact it must be, and then turning that into a model that can be analysed and made. Comes after: nothing; it needs only geometry. Leads to: Machine Design (34/19), where every part you design is communicated by a drawing; it also supports Materials Science & Manufacturing (34/13), Mechanics of Materials (34/11) and the Capstone (34/22). Honest line: you won't learn a CAD package here (those change every decade); you will learn what every package is asking you to say. A drawing is a contract A drawing is not an illustration. It is the document the designer hands to the maker, and it is what the buyer pays against. If the maker builds exactly what is drawn and it doesn't work, the question "whose fault is that?" is settled by reading the drawing. If the maker builds something different and it fails, that is settled by the same document. This is why engineers insist on one rule: if it matters, it is on the drawing. Not in an email, not in a phone call, not "everyone knows". A change that isn't on the drawing doesn't exist, and a change that is on it has to be checked by someone who understands what it does. The sad proof is the Hyatt Regency walkways in Kansas City, which collapsed in 1981 and killed 114 people (some sources say 113). The contract drawings showed the hanger rods holding two walkways as single long rods. The steel fabricator's shop drawings changed that to two sets of shorter rods, which roughly doubled the load on the critical connection. The US National Bureau of Standards investigation found two problems: the original connection was already inadequate (it would have reached only about 60% of the capacity the building code expected), and the change made it far worse. The lesson isn't that someone was careless with a pencil. It is that a drawing carries a decision, and an unchecked change to a drawing is an unchecked change to a building. A drawing says what to make, how exactly, and who is responsible when it isn't. Three flat views tell you a solid A 3D object has to be described on flat paper. The method used for roughly two centuries is orthographic projection: imagine the object inside a glass box, and look straight at each face from far away. Projection lines are parallel, there is no perspective, and so a length on the drawing is a true length: parallel edges stay parallel, and a hole that is 12 mm across is drawn 12 mm across in the view where you look down its axis. Usually three views are enough: front, top and right side. The front and top share the same widths, the front and side share the same heights, and from any two you can find the third. Edges you can't see are drawn dashed. If the inside matters, you imagine slicing the part and drawing the cut face with hatching, a section view. Several parts together appear in an assembly drawing, often with an exploded view and a numbered parts list. There are two conventions for where the views go on the sheet. Third angle (used in the US and Canada) puts the top view above the front. First angle (used in most of Europe and Asia) puts it below. Same views, different layout; a small symbol in the title block says which one you are reading. Try it below: the three views come from a real 3D model, projected live, with hidden lines worked out by ray-casting. Interactive · Rotate the part, read its three views Hidden lines Third angle Turn Tilt The part is an L-bracket, 60 × 40 × 40 mm, with a 12 mm vertical hole in the base, a 10 mm horizontal hole through the upright and a notch in one corner. Solid lines are visible edges and dashed lines are hidden ones. Drag the picture to rotate it. Every dimension is a range: tolerance and stack-up No machine makes anything exactly 25.000 mm, and nothing needs to. What matters is how far off is still acceptable. A tolerance is that allowed range: 25 ± 0.1 mm means anything from 24.9 to 25.1 is good. Tight tolerances cost money (more careful machines, more inspection, more scrap), so the designer's job is to give each dimension the loosest tolerance that still works. The trouble is that tolerances add up. Put three parts in a housing: the gap left over is the housing size minus all three parts, so every ± feeds into the gap. Adding the extremes gives the worst case, which is safe but costly. But it is very unlikely that every part lands at its extreme in the same direction. If you treat each ± as three standard deviations of a bell curve, the combined spread is the square root of the sum of the squares, the statistical (root-sum-square) stack: worst case = Σ ti    statistical = √(Σ ti²) Worked number: a 100 mm housing holds parts of 30, 40 and 28 mm, leaving a 2 mm gap. With tolerances ±0.30, ±0.20, ±0.20 and ±0.15 mm, worst case is ±0.85 mm (gap 1.15 to 2.85 mm). The statistical range is ±0.44 mm. That 2:1 difference is why statistical stacks let designers loosen tolerances; it is also a bet that your manufacturing really is centred and random. Interactive · Tolerance stack-up calculator Housing (100 mm) Part 1 (30 mm) Part 2 (40 mm) Part 3 (28 mm) The statistical figure assumes each tolerance is three standard deviations of a normal distribution and the parts are independent. Check this against real inspection data before relying on it. One more everyday consequence: how you dimension decides how tolerances add. Dimension a row of holes as a chain (each from the last) and the errors accumulate along the chain. Dimension them all from one datum edge and each hole only inherits its own tolerance. Good drawings dimension from the surfaces that matter, and never make the machinist do arithmetic. Fits, and GD&T in plain words When a shaft goes into a hole, what matters is the gap between them. A fit is a standard pairing of tolerances chosen for the job. ISO 286 names them with letters and numbers: a capital letter for the hole, a lowercase one for the shaft, and a number for the tolerance grade. For a 25 mm size, H7 for the hole means 25.000 to 25.021 mm. Pair it with g6 and the shaft runs freely; with k6 you may get either a small gap or a small squeeze (a transition fit); with p6 you must press the shaft in (an interference fit), and the parts then grip by friction alone. Interactive · Pick a fit H7/g6 H7/h6 H7/k6 H7/p6 Deviations are for the 18 to 30 mm size band from ISO 286 tables (H7: 0 to +21 µm, g6: −20 to −7, h6: −13 to 0, k6: +2 to +15, p6: +22 to +35). A micrometre is a thousandth of a millimetre; a human hair is very roughly 50 to 100 of them. A ± size tolerance only controls the size at each point. It says nothing about whether the hole is straight, round, square to the face, or in the right place. Geometric dimensioning and tolerancing (GD&T) is a language of symbols for exactly that: flatness, perpendicularity, roundness, and above all position. Each rule reads like a short sentence: "this hole's axis must lie within a 0.2 mm circle, measured from datum surfaces A, B and C." The datums are the three flat faces that sit the part on a table, against one wall, and against another; they are how a maker and an inspector agree on where "measure from" is. A circular position zone also holds more good parts than a square ± box with the same corner reach: a ±0.1 mm square has area 0.040 mm², and the circle through its corners (0.283 mm across) 0.063 mm², about 57% more. In the US the rulebook is ASME Y14.5 (the 2018 edition, reaffirmed in 2024); ISO has its own set. Parametric CAD: you draw the intent, not the shape Computer-aided design began as an electronic drafting board. Ivan Sutherland's Sketchpad, in the early 1960s, let a person draw with a light pen and have the computer keep the lines connected. Modern CAD went further: since the late 1980s (PTC's Pro/ENGINEER shipped in 1988), mainstream CAD records how you built the shape. You draw a 2D sketch, add constraints (this line is horizontal, that circle is 12 mm, these two are equal), then extrude or cut it into a solid. Every step goes into a feature tree and every number is a parameter, so changing one rebuilds the part. The point is design intent: the rules you chose say what must stay true when something changes. Widen this plate: should the hole stay 30 mm from the wall, or stay in the middle? Both are legitimate, and they are different parts. Try it. Interactive · A sketch that keeps its promises Hole 30 mm from the left Hole stays centred Plate width How many constraints does a sketch need? Tap constraints on and off. The plate sketch is a rectangle (8 coordinates) and a circle (3 numbers) = 11 degrees of freedom. Each constraint removes some. From the model to simulation and the shop floor Once a part exists as a precise digital solid, two things become cheap. First, you can simulate it. The standard tool is the finite element method (FEA): chop the part into thousands of little elements, write the equations of stiffness for each one, and solve all of them together for how the part deforms and where the stress is. It is a numerical answer, and the discipline is to ask how wrong is it? The way to find out is to refine the mesh and see whether the answer settles. The bar below is a real, tiny finite-element model: a tapered aluminium bar, pulled with 10 kN at the thin end. Interactive · Refine the mesh Number of elements Bar: 100 mm long, cross-section tapering from 200 mm² at the wall to 50 mm² at the free end, E = 70 GPa, 10 kN axial load. Each element has linear displacement, so its stress is one constant number. The exact answers come from integrating the load over the taper. Look at the two errors. Movement settles quickly; stress, which depends on the slope of the movement, converges much more slowly, and the worst stress is usually where it matters most. A graduate never trusts a coloured stress plot with one mesh. Second, the same model feeds manufacturing. CAM (computer-aided manufacturing) turns the solid into toolpaths: a list of moves for a CNC machine (often G-code) to cut the part from stock. For 3D printing the model is sliced into layers and each layer into a toolpath. The drawing doesn't disappear: the tolerances, fits and datums are still what the inspector checks, even if the model carries them. What a graduate just sees A drawing is a contract: if it matters, it is on the drawing, and a change needs a check. Any object can be read from two or three orthographic views; hidden features are dashed. Every dimension is a range, and tolerances cost money. Tolerances stack: worst case is the sum, statistical is the root-sum-square, and datum dimensioning stops errors accumulating. A fit is a pairing of two tolerances, and "clearance, transition, interference" is a ladder from sliding to pressed. A size tolerance doesn't say the part is straight or in the right place; GD&T does. A CAD model is a record of intent, with sketch, constraints and a feature tree; what you constrain is what stays true. A simulation is a number with an error bar, and you check it by refining the mesh. Where it shows up next Machine Design (34/19): every shaft, gear and bolted joint you design ends up as a dimensioned drawing, with fits for the bearing seats. Materials Science & Manufacturing (34/13): the process (casting, machining, printing) sets the tolerances you can realistically ask for. Mechanics of Materials (34/11): FEA is the numerical cousin of the beam and stress formulas. Systems Engineering & the Capstone (34/22): interface drawings are how subsystems built by different teams agree to fit. In the real world: product design, a car's body panels fitting with an even gap, an aircraft's thousands of parts matching across suppliers. If you want to go deeper 📘 Standard texts: Giesecke and co-authors, Technical Drawing with Engineering Graphics (Pearson, 15th edition; a 16th is listed), the usual first-year text. Gene Cogorno, Geometric Dimensioning and Tolerancing for Mechanical Design (McGraw Hill, 3rd edition), for GD&T. The standard itself is ASME Y14.5-2018. 📘 Gentler: the free MIT OpenCourseWare short course How to CAD Almost Anything (RES.16-002), a hands-on introduction to parametric modelling through reverse engineering. 🌐 Free course: NPTEL: Engineering Graphics and Design (IIT Delhi, Profs. Naresh Datla and S. R. Kale). The takeaway A drawing is the contract between a designer and a maker: views to say the shape, dimensions and tolerances to say how exactly, fits and GD&T to say how parts meet. Tolerances add up, so you stack them deliberately. CAD records your intent so changes rebuild correctly, and simulation and manufacturing both read the same model, which is why you check each by asking how wrong it could be. Try this: pick any small object with a few parts (a hinge, a jar lid) and sketch its front, top and side views. Then guess which two dimensions matter most for it to work, and what a ±1 mm error on each would do. Sources & Further Reading Sources. Hyatt Regency collapse: US National Bureau of Standards, NBSIR 82-2465 (1982) and its NIST-hosted summaries; the Linda Hall Library notes (114 deaths; NIST pages and UPI report 113). Third/first angle: ISO 128 and ASME Y14.3 as summarised by Xometry and Peachpit. ISO 286 fits for 18 to 30 mm: ISO 286-2 values as reproduced by cadforum.cz (H7, g6, h6, k6, p6). ASME Y14.5-2018 (R2024): ASME and ANSI webstore listings. Pro/ENGINEER commercial shipments from January 1988: PTC history summaries. Sketchpad (Sutherland, early 1960s): PTC/NovEdge history summaries. Numbers computed in node: the three views and hidden lines by ray-casting; stack-up (worst case 0.85, statistical 0.439); the p6 and g6 limits; position-zone areas (0.040 and 0.063 mm²); and the tapered-bar finite-element model checked against its exact integral (exact tip movement 0.13203 mm). 🌐 NPTEL: Engineering Graphics and Design 🌐 MIT OpenCourseWare: How to CAD Almost Anything 📘 Giesecke et al., Technical Drawing with Engineering Graphics 📘 Cogorno, Geometric Dimensioning and Tolerancing for Mechanical Design Step 6. Multivariable Calculus: Calculus on a Landscape ------------------------------------------------------- Why: Stress, flow and heat vary across a part, so the maths has to vary in 3D too. A raindrop on a hillside picks the steepest way down without thinking. The arrow it follows is the same one that trains every neural network and drives heat through a metal plate. Stand on a hillside with a map. Height depends on two things at once: how far east you are and how far north. Single-variable calculus handles one input at a time. The hillside needs a new question: of all the directions I could step, which one rises fastest? Almost nothing real depends on one number. Temperature in an engine block depends on position in three dimensions. The error of a language model depends on billions of adjustable weights. Air pressure depends on place and time. This course takes the slope-and-area ideas of Calculus I (34/01) and lets them live in more than one dimension. Surprisingly little new is needed: one new arrow, the gradient, and one new accounting rule, the idea that what flows out of a region is what is made inside it. The course in one box Who takes it: second-year Mechanical & Aerospace Engineering students; Computer Science students meet it in year 4 when machine learning needs it. What it is for: calculating change and totals when many things vary at once. Comes after: Calculus I (34/01). Leads to: Physics II: Electricity & Magnetism (34/07), Fluid Mechanics (34/15), Machine Learning (34/32), and through them Heat Transfer (34/16) and Propulsion (34/21). Honest line: you will not be able to grind through a surface integral after this. You will see why a gradient points uphill, why training an AI is "just" rolling downhill, and what the big theorems say in plain words. Two inputs make a landscape, and a partial derivative is one slope A function of two inputs is a surface: height above each point of a map. Engineers draw it the way hikers do, with contour lines: closely spaced lines mean steep ground, widely spaced lines mean gentle ground. To ask "how steep?" you must say in which direction. Walk due east and note the slope: that is the partial derivative with respect to x, written ∂f/∂x. Hold east still and walk north: that is ∂f/∂y. The recipe is simple: treat every other variable as frozen and differentiate as before. ∂f/∂x = slope of f as only x changes, with y held still The landscape below has a tall hill near (1, 1) and a lower one near (−1.2, −0.8). Drag the point on the map (or use the sliders). The cyan arrow shows the slope eastward and the green arrow the slope northward; a leftward or downward arrow means the ground falls in that direction. Interactive · Walk the map East–west position x North–south position y At the starting point (−0.2, 0.3) the height is 0.684, the eastward slope +0.621 and the northward slope +0.226: the ground rises both ways, more steeply towards the east. The gradient points straight uphill Put the two partial slopes together as one arrow, east component first: that is the gradient, written ∇f. It has a beautiful property: it points in the direction of steepest ascent, and its length is how steep that is. Step in any other direction and the slope is the gradient's length times the cosine of the angle between your step and the arrow. Straight along it, cosine is 1: the most you can get. Sideways to it, cosine is 0: you walk along a contour and stay level. slope in direction u = ∇f · u = |∇f| cos(angle) Two consequences follow. The gradient is always perpendicular to the contour line through that point, which is why water runs across contours, not along them. And the same arrow governs heat: in Fourier's law, heat flows down the temperature gradient, from hot to cold, at a rate proportional to its steepness. Many diffusion laws have this shape: flow follows the negative gradient. Interactive · Which way is uphill? Direction you step (0° = east, 90° = north) Turn to the gradient Turn the pink direction until the slope is largest: it lines up with the orange arrow. Turn it 90° from the arrow and the slope is zero. At the very top of a hill the gradient shrinks to nothing, which is the multivariable version of "zero slope means a peak" from Calculus I. Rolling downhill is how a network learns Now flip the sign. A machine-learning model has adjustable numbers (weights), and a loss measures how wrong it is on the training data. The loss is a landscape over the weights, with one dimension per weight. To learn, take a small step against the gradient, so downhill, then recompute and repeat. This is gradient descent, which Augustin-Louis Cauchy proposed in 1847 for solving equations, long before computers. The chain rule from Calculus I is what lets the gradient be computed layer by layer, in the method called backpropagation, published by Rumelhart, Hinton and Williams in Nature in 1986. A model like GPT-3 (2020) has 175 billion weights, so its landscape has 175 billion dimensions, yet the rule is the same arrow. new position = old position − (step size) × ∇(loss) The step size matters, and you can feel why. In the bowl below, one direction is twelve times steeper than the other. Too small a step crawls; just right is fast; too large and each step overshoots the valley floor by more than it started and the loss explodes. For this bowl, steps above 1/6 (about 0.167) diverge. Interactive · Learning rate Too small Good Zig-zag Too big Step size (learning rate) Steps taken Real training uses refinements (momentum, adaptive step sizes, random mini-batches), but all of them are ways of coping with exactly this picture: a long narrow valley where the steepest direction is not the direction of the destination. Double and triple integrals add up a landscape's worth Integrals extend the same way. A single integral adds thin slices along a line. A double integral adds tiny patches over an area: if each patch has a density (kilograms per square metre), the sum is the plate's mass. A triple integral does the same over a solid, with density in kilograms per cubic metre. The recipe is slice, multiply, add, repeated once per dimension. mass = sum over all pieces of (density × piece size)  =  ∬ ρ dA The same sums also give the centroid, the balance point: weight each position by its piece's mass and divide by the total. Engineers use it constantly: the centre of mass of a rocket, a car or an aircraft decides whether it is stable. Try a plate 2 m long and 1 m wide whose density rises along the length. Interactive · Where does the plate balance? Density rises along the length: a (kg per m² per m) Density rises across the width: b The density here is 1 + ax + by kg/m². With a = 3 and b = 0, the plate's mass is 2 + 2×3 = 8 kg and it balances at x = 1.25 m, shifted from the middle (1 m) towards the heavy end. A vector field puts an arrow on every point Wind on a weather map, water in a pipe, the pull of a magnet: each attaches an arrow (a direction and strength) to every point in space. That is a vector field. Two numbers computed from the field's partial derivatives tell you most of what it is doing locally. Divergence: is the field spreading out from here, or converging? A tap under water is a source; a drain is a sink. A positive divergence means a small blob of fluid is expanding. Curl: is the field spinning? Put a tiny paddle wheel in the flow. If it turns, the curl is not zero, and the sign tells you which way. divergence = ∂vx/∂x + ∂vy/∂y    curl = ∂vy/∂x − ∂vx/∂y Pick a field and let a round blob of fluid (grey dashes show where it started) be carried along. You will see that a source inflates the blob without turning it, a whirlpool turns it without changing its area, and a saddle stretches it one way and squeezes it the other while keeping its area. Interactive · Drift a blob through a field Source Whirlpool Shear Saddle Time elapsed Divergence and curl are the two words in Maxwell's equations of electromagnetism (Physics II, 34/07) and in the equations of fluid flow (Fluid Mechanics, 34/15). Spreading and spinning are the two things a field can do locally, besides moving along as a whole. What flows out of a region is what is made inside Here is the idea behind the three big theorems, in the language of a bathtub. If water pours out through the boundary of a region, then some taps inside must be producing it (or the level inside must be falling). Add up every tiny source and sink inside and you get the net flow out through the boundary. Nothing more is needed. net flow out through the boundary = total divergence inside That is the divergence theorem, the three-dimensional version of the statement. Green's theorem is the flat version (the name honours George Green, whose Essay of 1828 contained the related surface-to-volume formula), and Stokes's theorem relates the spinning of a field inside a surface to the flow around its edge. All three say the same kind of thing: a sum over a boundary equals a sum of local changes inside. It is why one can measure a rotating machine from outside and why conservation laws can be written either as "this region's total changes by what crosses its walls" (integral form) or as an equation about each point (differential form). Interactive · Count the flow two ways Source Cubic Source in, sink out Steady wind Circle radius Circle centre, left or right The first number adds up the field's push through the circle itself. The second never looks at the circle's edge: it adds the divergence in every patch inside. They agree for every field, radius and position. For the plain source (x, y), the divergence is 2 everywhere, so a unit circle encloses 2 × π = 6.283 units of production, and exactly 6.283 flows out through its rim. What a graduate just sees A slope needs a direction; a partial derivative is the slope with every other input held still. The gradient points straight uphill, its length is the steepness, and it is perpendicular to the contour. "Downhill" is the whole idea behind training a model: step against the gradient, and mind the step size. Heat, and many other things, flow down a gradient from high to low. A double or triple integral is slice, multiply, add, once per dimension; mass and balance point come from it. A field can only spread (divergence) or spin (curl), locally. What leaves a region through its wall is exactly what its insides produce. Where it shows up next Physics II: Electricity & Magnetism (34/07): Maxwell's equations are statements about the divergence and curl of the electric and magnetic fields. Fluid Mechanics (34/15): conservation of mass is a divergence equation, and the divergence theorem turns "what crosses the walls of this control volume" into "what happens inside it". Heat Transfer (34/16): conduction is heat flowing down the temperature gradient; the heat equation is the gradient and divergence together. Machine Learning (34/32): training is gradient descent on a loss landscape with millions or billions of dimensions. Mechanics of Materials (34/11) and Dynamics (34/12): centroids and moments of inertia are integrals of exactly the kind you added up above. If you want to go deeper 📘 Textbook: Stewart, Clegg and Watson, Calculus: Early Transcendentals (9th ed.), chapters 14 to 16: the standard multivariable text, rich in examples. 📘 Textbook: Jerrold Marsden and Anthony Tromba, Vector Calculus (6th ed., W. H. Freeman): the classic for the geometry of divergence, curl and the big theorems. ▶️ Gentler: Khan Academy's Multivariable Calculus course, which has videos with a strong visual feel, and Grant Sanderson's multivariable lessons. 🌐 Free course: MIT OpenCourseWare 18.02SC Multivariable Calculus, with lectures, notes and exercises. The takeaway With many inputs, a slope needs a direction. The partial derivatives are the slopes along the axes; the gradient joins them into one arrow that points uphill and measures how steep it is. Walk against it and you are doing gradient descent, which trains models with billions of weights and also describes how heat flows. Integrals extend to areas and volumes, giving mass and balance points, and for flows the whole subject rests on one bookkeeping rule: what leaves a region through its boundary is what its interior produces. Try this: on any hill, a park slope or a street, find the line along which a ball would roll: that is the negative gradient. Then notice that the ball's path crosses the contours at right angles. Sources & Further Reading Sources. Gradient descent: A.-L. Cauchy, "Méthode générale pour la résolution des systèmes d'équations simultanées", Comptes rendus 25 (1847), pp. 536–538; Claude Lemaréchal, "Cauchy and the gradient method", Documenta Mathematica, Extra Volume ISMP (2012). Backpropagation: Rumelhart, Hinton and Williams, Nature 323 (1986), pp. 533–536. GPT-3's 175 billion parameters: Brown et al., "Language Models are Few-Shot Learners" (2020, arXiv 2005.14165). Green's Essay of 1828 containing the theorem named after him: MacTutor biography and the essay itself. Computed in node: the landscape f = 2e−((x−1)²+(y−1)²)/1.5 + 1.2e−((x+1.2)²+(y+0.8)²)/1.0 and its analytic gradient (checked against finite differences, agreement to 3 × 10−10); gradient descent on ½(x² + 12y²), stable for step sizes below 2/12; plate mass and centroid by 3,200-cell sums against the exact formulas; blob area against ediv·t; circle flux against the divergence integral for four fields (agreement to 4 decimals). 🌐 MIT OpenCourseWare 18.02SC Multivariable Calculus 🌐 Khan Academy: Multivariable calculus 🌐 Documenta Mathematica, ISMP volume, which contains Lemaréchal's short history of gradient descent 📘 Stewart, Clegg and Watson, Calculus: Early Transcendentals, 9th ed. (Cengage) 📘 Marsden and Tromba, Vector Calculus, 6th ed. Step 7. Differential Equations: Laws Written as Rates ----------------------------------------------------- Why: Almost every physical law an engineer uses is written as a differential equation. A cup of coffee cools, a capacitor charges, a car bounces over a bump, a fish stock collapses. Each follows a rule that says how fast things change, not where they will be. Solving it is the whole course. Pour coffee and watch it cool. It loses heat fast at first, when it is far hotter than the room, and ever more slowly as it approaches room temperature. Nobody wrote down the temperature at every minute in advance. All you need is a rule: the hotter the cup is above the room, the faster it cools. That sentence, turned into an equation, is a differential equation, and it predicts the whole curve. This is the secret structure of physics. Nature rarely says "the position will be this". It says "the rate of change is that, given the state right now". Newton's second law is such a rule. So are the equations of electrical circuits, chemical reactions, populations, heat, and the flow of air over a wing. A differential equations course teaches how to read these rules and how to turn them into predictions, either exactly or step by step on a computer. The course in one box Who takes it: second-year Mechanical & Aerospace Engineering students. It is not on the Computer Science route in this series, but CS students meet the same ideas whenever they simulate, animate or train anything that evolves in time. What it is for: predicting how a system evolves when you only know the rule for its rate of change. Comes after: Calculus I (34/01). Leads to: Dynamics (34/12), Fluid Mechanics (34/15), Heat Transfer (34/16), Vibrations (34/17), Control Systems (34/20). Honest line: you will not learn the solution tricks that fill the textbook. You will see the three behaviours that recur everywhere (settling, oscillating, tipping) and why a simulation can lie to you. An equation about a rate hands you a direction at every point An ordinary differential equation, an ODE, is an equation that mentions a quantity and how fast it is changing. The simplest useful kind says the rate depends on how far the quantity is from a target: dy/dt = k (target − y) Read it aloud: the further I am from the target, the faster I move towards it. Coffee towards the room's temperature, a capacitor towards the supply voltage, a medicine level towards zero. The picture is a slope field: at every point of the plane the equation fixes the direction a solution must go through, drawn as a tiny segment. A solution is any curve that follows those segments the whole way, like a leaf carried along a stream. Different starting points give different curves, all flowing toward the target. Interactive · Follow the arrows Pull strength k Target value Starting value Notice that the segments get flatter as you approach the dashed line: the rate shrinks with the distance. That is why the curve approaches the target but never quite arrives. The exact solution is the target plus the starting gap times a decaying exponential, e−kt. The time constant: how fast a first-order system forgets The rule dy/dt = k(target − y) has one number that sums it up: the time constant, τ = 1/k. In one τ the system covers about 63.2% of the remaining gap; after about 4.6 τ it has covered 99%. Engineers read response speed from that single number. fraction of the gap covered after time t = 1 − e−t/τ In an electrical circuit, a resistor R in series with a capacitor C charges towards the supply with τ = R × C. A 10 kΩ resistor and a 100 µF capacitor give τ = 10,000 Ω × 0.0001 F = 1 second: after 1 s the capacitor holds 63.2% of the supply (3.16 V of 5 V), after 4.6 s, 99%. A cooling coffee follows the same curve; only the story and the units change. Interactive · Charge a capacitor Resistance R Capacitance C Notice what the sliders do: double either one and the curve stretches twice as wide, but keeps exactly the same shape. The same shape is why a thermometer takes a while to read a fever, and why a control engineer (Control Systems, 34/20) asks first: "what is the time constant?" Solving it step by step: Euler's method Most differential equations have no neat formula. The fall-back is to walk along the slope field in small steps. At the current point read the slope, move a short time h in that direction, and repeat. This is Euler's method, set out by Leonhard Euler in his Institutionum calculi integralis (1768), and it is the ancestor of every physics engine, weather model and circuit simulator. ynext = y + h × (the rate at y) Test it on a rule we can also solve exactly, dy/dt = −y, starting from 1. Each Euler step multiplies the value by (1 − h). Small steps follow the true curve e−t closely. But raise h and something stranger happens. Interactive · Euler versus the exact answer Step size h At h = 1 the first step lands exactly on zero. Beyond 1 the steps overshoot, and past h = 2 each step makes the number bigger in size, so the "answer" explodes even though the real one decays. That is numerical instability: a stepping rule has its own stability limit, and a simulation that ignores it will happily print nonsense. At h = 0.5 Euler gets 0.000244 at t = 6, where the true value is 0.00248: right shape, ten times too small. Smaller steps, or smarter methods (the Runge–Kutta family), fix that. Second order: the spring, the mass and the damper Newton's second law says force equals mass times acceleration, and acceleration is the rate of change of the rate of change of position. So a mass on a spring obeys a second-order equation: the second derivative of position x depends on position and velocity. m x″ + c x′ + k x = 0 The three terms are inertia (mass m), a damper that resists velocity (c) and a spring that pulls back with force proportional to displacement (k). Two numbers decide everything: the natural frequency, ω = √(k/m), and the damping ratio, ζ = c ÷ (2√(km)). ζ below 1, underdamped: it rings, swinging past rest and fading. ζ = 1, critically damped: the fastest return without overshoot. ζ above 1, overdamped: no swing, but a sluggish crawl home. Engineers tune for it. A door closer is aimed near critical damping so the door shuts without banging or bouncing. A shock absorber is a compromise: enough damping to stop the bouncing, not so much that the wheel can't follow the road. Interactive · Tune the damper No damping Light Critical Heavy Mass m Spring stiffness k Damping c With m = 1 kg and k = 25 N/m the natural frequency is 5 rad/s, or about 0.8 Hz. Setting c = 10 N·s/m gives ζ = 1, critical. With c = 2, ζ = 0.2 and the first swing past rest reaches 52.7% of the starting displacement on the other side. This same equation, with different symbols, describes the RLC circuit, a building in an earthquake and a robot arm: that is the next courses (Vibrations, 34/17, and Control Systems, 34/20). Equilibria: where things settle, and where they tip Often you do not need the whole curve, only the places where change stops: the equilibria, where the rate equals zero. Two questions follow. Where are they? And is each one stable (nudge it and it comes back) or unstable (nudge it and it runs away)? The answer is the slope of the rate curve at that point, from Calculus I: negative slope means stable, positive means unstable. A toy model shows why this matters. Fish regrow according to dy/dt = y(1 − y/100) per year (a standard logistic model), and a fleet takes a fixed catch H every year. For H = 20 there are two equilibria, at about 27.6 and 72.4 fish: the upper is stable, the lower unstable. Start just above 27.6 and the stock recovers to 72.4. Start just below it and the stock collapses to zero in a few years. Raise the catch above 25, the most the stock can regrow, and no equilibrium remains at all. Interactive · A fishery on the edge Catch 20, start 30 Catch 20, start 25 Catch 26 Fixed catch H per year Starting population The lesson generalises: a system can look healthy while it sits near an unstable tipping point, and a slow change in a parameter (here the catch) can make the stable state vanish suddenly. Bridges, power grids, aircraft and ecosystems are all analysed this way. The model is a classroom toy, not a description of any real fishery. Why physics is written in differential equations Why do laws come as rates? Because change is local. What a system does next depends only on what is happening here, now: the forces, the gradients, the current state. Newton's second law (second order in position), Maxwell's equations for electric and magnetic fields, the equations of heat flow and the equations of fluid motion all have this shape. Give the rule and the starting conditions and the future follows. Two kinds appear. An ordinary equation (ODE) has one changing variable, usually time: a pendulum, a circuit, a rocket's trajectory. A partial differential equation (PDE) has several, such as time and position: the temperature along a rod, the pressure in the air around a wing. The Navier–Stokes equations of fluid flow (Fluid Mechanics, 34/15) are PDEs so hard that the Clay Mathematics Institute named a question about them one of its seven Millennium Prize problems in 2000. Most real ones are solved on computers using the stepping idea above. A glimpse of PDEs: the heat equation Joseph Fourier wrote the heat equation in his Théorie analytique de la chaleur (1822). It says: the temperature at a point changes at a rate proportional to how much its surroundings, on average, are hotter or cooler than it is, which is the curvature of the temperature curve. ∂u/∂t = α ∂²u/∂x² A peak has negative curvature, so it cools; a dip has positive curvature, so it warms. Heat flows from hot to cold and the profile smooths out. The rod below is insulated at both ends, so the average temperature, the total heat, never changes, only its distribution. Fourier's great discovery (see the library's piece on the Fourier transform, 12/12) is that smooth wiggles of different sharpness die away at different rates: a wiggle five times sharper dies 25 times faster, because it decays like e−(n²π²)t. Interactive · Heat spreading along a rod Hot spot Half hot Gentle wave Sharp wave Time Time step, as a fraction of the stability limit Try the two waves at the same time. The gentle wave 1 + 0.8 cos(πx) has faded to 0.298 high at t = 0.1, matching the exact 0.8e−π²t. The sharp wave 1 + 0.8 cos(5πx) is gone almost at once, 25 times faster in the exponent. Then drag the time-step slider above 0.5 and watch the simulation explode: Euler's lesson again, now in a PDE. Real heat solvers (Heat Transfer, 34/16) must respect the same limit or use methods built to avoid it. What a graduate just sees A law about a rate is a direction at every point; a solution is a curve that follows those directions. First-order systems approach their target by a time constant: 63% in one τ, 99% in about 4.6 τ. Cooling, charging and decay are the same curve with different labels. A step-by-step simulation has a stability limit: too big a step creates growth that is not in the physics. Mass, spring and damper give three behaviours: ring, crawl, or the sweet spot at ζ = 1. Equilibria matter, and slope tells stable from unstable. Systems tip. Sharp features smooth out first in diffusion, and total heat is conserved. Where it shows up next Dynamics (34/12): every motion problem becomes an equation of the form above. Vibrations (34/17): the damped spring is the building block of everything that shakes, from engines to aircraft wings. Control Systems (34/20): a thermostat, cruise control or autopilot is a differential equation you shape with feedback; time constants and damping ratios are its vocabulary. Heat Transfer (34/16) and Fluid Mechanics (34/15): the heat equation and Navier–Stokes are PDEs; the stepping and stability ideas carry straight over. Computer science: physics engines in games, climate and weather models, and circuit simulators are loops of Euler-like steps; neural ODEs even treat a network as a flow. If you want to go deeper 📘 Textbook: William Boyce, Richard DiPrima and Douglas Meade, Elementary Differential Equations and Boundary Value Problems (12th ed., Wiley): the classic course text. 📘 Textbook: Steven Strogatz, Nonlinear Dynamics and Chaos (3rd ed., CRC Press): an unusually readable treatment of equilibria, stability and tipping. ▶️ Gentler: 3Blue1Brown's differential equations series, and Strogatz's Infinite Powers for the story of why calculus describes nature. 🌐 Free course: MIT OpenCourseWare 18.03SC Differential Equations, with video lectures, notes and problem sets. The takeaway Physics speaks in rates. A differential equation gives a direction at every point and a solution follows it. First-order systems approach a target with a time constant (63% in one τ). Second-order systems, from springs and masses, ring, crawl or sit at the sweet spot of critical damping. Equilibria are where change stops and slope tells stable from unstable. When no formula exists, you step along on a computer, but only small enough steps tell the truth: past a limit, a simulation blows up even though nature does not. Try this: take a hot drink and a thermometer (or just your hand) and note how long it takes to lose half its excess heat over the room. It will take about the same time again to lose half of what is left. That constant "half-life" is the time constant in disguise. Sources & Further Reading Sources. Euler's method: Institutionum calculi integralis, vol. 1 (1768); Encyclopedia of Mathematics, "Euler method". Newton's law of cooling originates in Newton's anonymous 1701 paper "Scala graduum caloris"; the modern differential form is not in the original (Wikipedia, "Newton's law of cooling", and history-of-science references). Fourier's Théorie analytique de la chaleur (1822). Millennium Prize problems announced by the Clay Mathematics Institute in 2000, Navier–Stokes among them. Computed in node: RC charging (τ = RC = 1 s for 10 kΩ and 100 µF, 63.2% at τ, 99.0% at 4.605τ); Euler at h = 0.5 gives 2.44 × 10−4 at t = 6 against the exact 2.48 × 10−3; spring closed-form solutions for the three regimes checked against a step-by-step integration (agreement to 5 decimals) and the first-swing overshoot e−πζ/√(1−ζ²) = 52.7% for ζ = 0.2; harvested logistic equilibria 50(1 ± √(1 − H/25)) = 27.64 and 72.36 for H = 20, with collapse after 4.3 years from a start of 25; the explicit heat scheme (100 cells, stable below 0.5) against the exact decay 0.8e−π²t = 0.2982 (numerical 0.2981) at t = 0.1. 🌐 MIT OpenCourseWare 18.03SC Differential Equations ▶️ 3Blue1Brown: differential equations 📘 Boyce, DiPrima and Meade, Elementary Differential Equations and Boundary Value Problems, 12th ed. (Wiley) 📘 Steven Strogatz, Nonlinear Dynamics and Chaos, 3rd ed. (CRC Press) 🌐 Newton's law of cooling (with the history of the law) Step 8. Statics: Why Things Stand Still --------------------------------------- Why: Nothing that stands still has unbalanced forces; that one rule designs bridges. Every bridge, shelf and crane is a bargain between forces that cancel exactly. Learn to see the bargain and you can read a structure at a glance. A shelf sags a little under its books, and then it stops. It does not creep lower all night, and it does not suddenly give up. Somewhere inside it, a quiet bargain has been struck: the wall pushes up on the bracket with exactly as much force as the books push down. If the books were heavier by one paperback, the bargain would be re-struck with a slightly bigger push and a slightly bigger sag. That bargain is the whole of statics. A bridge, a crane, a chair, a bone, the wing spar of an airliner on the ground: each is a collection of pushes and pulls that cancel exactly. When engineers say a structure is "in equilibrium," they are saying something stronger than "it isn't moving." They mean every piece of it, however small, has forces on it that add to zero. Once you can see that cancellation, a surprising amount of the built world becomes legible. You can look at a crane and tell which arm is being stretched and which squashed. You can see why a tall narrow thing falls over before it slides, and why a seesaw does not care how big the children are, only where they sit. The course in one box Who takes it: mechanical, aerospace and civil engineers, usually in the second year, straight after first-year physics. What it is for: working out the forces inside and on a thing that is standing still, so you know what each part has to carry. Comes after: Physics I: Mechanics (34/06) and Linear Algebra (34/04). Leads to: Mechanics of Materials (34/11), which asks how a part responds to those forces, and from there Machine Design (34/19). What this piece will and won't give you: you won't be able to size a real bridge member after this. You will see why a truss is made of triangles, and why a tall crate tips before a squat one. An easier first read on the same ground: Everyday Physics: The Answers Hiding in Plain Sight (Physics 5) and The Four Forces That Run Everything (Physics 1). Nothing is accelerating, so everything adds to zero Newton's second law says that a net force makes a body accelerate. Statics is the special case where nothing accelerates, so the net force is zero. That is one sentence, and an entire first course grows from it, because it can be applied to any piece of a structure you care to isolate. The habit that makes it work is the free-body diagram. Draw a closed outline around the thing you are studying (the whole bridge, one beam, one bolt) and cut every connection to the outside world. At each cut, draw the force that connection supplied: gravity, a push from a support, a pull from a cable. If you have drawn them all, the arrows must balance. If they don't, you forgot one. Most of a statics course is learning to draw that outline honestly. Two kinds of balance are needed for something that can turn as well as slide. The forces must add to zero, and the turning effects (moments, next section) must add to zero. In symbols: ΣF = 0    and    ΣM = 0 Read it as: add up every force, add up every turning effect, and both totals come out as nothing. In a flat problem that is three equations: horizontal forces, vertical forces, and moments. Three equations can find three unknowns, which is why a simple beam with one pin support and one roller support (three unknown reactions) can be solved on the back of an envelope. Supports are described by what they refuse to let the beam do. A roller blocks only up-and-down motion, so it supplies one force. A pin blocks sliding in both directions, so it supplies two. A fixed wall blocks sliding and turning, so it supplies two forces and a moment. The forces a support supplies are called reactions, and they are exactly as large as they need to be. Interactive · Where does the load go? A beam is held up at both ends. Drag the red load along it, or use the slider. Position of the load Size of the load Put 1,000 N on a 10 m beam 6 m from the left end. The right support carries 1,000 × 6 ÷ 10 = 600 N and the left carries 400 N. The nearer support always carries more. Slide the load to the middle and they split it equally; slide it over a support and that support takes everything. A force's turning power is force times distance Push a door near its hinge and it barely moves. Push at the handle and it swings. The force is the same; the distance from the hinge is not. The turning effect of a force about a point is the force multiplied by the shortest distance from the point to the force's line of action. Engineers call it a moment (physicists say torque). M = F × d    (newton-metres) This is why a long spanner loosens a stubborn nut: 100 N pushed 0.2 m from the nut gives 20 N·m, and the same push 0.4 m out gives 40 N·m. And it is why the second balance condition is so useful. If you add up moments about a point where an unknown force acts, that force has zero distance and drops out of the equation. Choosing the pivot cleverly is half the craft. A seesaw is the cleanest case. It balances when the turning effect on one side equals the turning effect on the other. The weights matter, and the distances matter, but only their product counts. Interactive · The seesaw Left mass Left distance from pivot Right mass Right distance from pivot Start with 30 kg on the left at 2 m and 20 kg on the right at 2 m. The left side wins. Now move the lighter child out to 3 m: 30 × 2 = 60 and 20 × 3 = 60, and the seesaw sits level. (The reading shows newton-metres, which is the same product times 9.81.) Gravity cancels from both sides, so the balance holds on the Moon too. A truss is a machine for sorting pushes from pulls Take a rectangle of four sticks pinned at the corners and it folds flat with a finger. Add one diagonal and it cannot. A triangle is the only polygon that is rigid with pinned corners, so bridges, roof frames, cranes and pylons are built of them. In an ideal truss every member is pinned at both ends and loads act only at the joints, so each member carries a force along its own length and nothing else: either a pull (tension) or a push (compression). To find which, use the method of joints. Isolate one joint, draw the arrows of the members meeting there, and make them balance. Start at a joint with only two unknown members, solve it, and move to the next. For a real truss with dozens of joints this becomes a big set of linear equations, which is exactly what Linear Algebra (34/04) is for. The tool below does it live: seven member forces and three support reactions, ten equations solved at once. Interactive · Which members pull, which push? A 6 m span truss. Red members are in tension, blue in compression; thicker means more force. Load at bottom middle Load at top left Load at top right Load Truss depth With 10 kN at the bottom middle and a depth of 3 m, each support carries 5 kN. The top member is squashed (compression, 5 kN), the two bottom members are stretched (2.5 kN of tension each), and each of the four sloping members carries 5.59 kN, alternating between push and pull. Now squash the truss to 1.5 m deep: the top member's force doubles to 10 kN. A shallow truss works its members much harder than a deep one, for the same reason a thin plank sags more than a deep beam. You will meet that reason again in Mechanics of Materials (34/11). Switch the load to a top joint and watch members change sex. A member that was pulling can start pushing. Engineers care, because tension and compression fail differently: a rod in tension can be thin, but a long member in compression may bow sideways long before it is crushed. Friction pushes back only as hard as it has to Friction is the most misread force in school physics. It is not a fixed number. If you push a heavy crate gently and it does not move, friction is pushing back with exactly your push, no more. Push harder and friction rises to match, until it hits a ceiling. Past the ceiling, the crate slides. For dry solid surfaces, the ceiling is roughly proportional to how hard the surfaces are pressed together (the normal force), and, to a good approximation, does not depend on the contact area. This is the Amontons–Coulomb law: Ffriction ≤ μ × N The coefficient μ (mu) depends on the pair of materials. A sliding object usually feels a slightly smaller friction than the ceiling that held it still, so there are two coefficients, static and kinetic. The numbers below are made up for illustration (0.5 static, 0.4 kinetic), not the values for any particular pair of materials. Interactive · Push the crate Mass of crate Your push A 50 kg crate presses on the floor with 490.5 N. At μ = 0.5, friction can resist up to 245 N. Push with 100 N and friction pushes back with exactly 100 N and nothing happens. Push with 300 N and it breaks free; kinetic friction is then 0.4 × 490.5 = 196 N, leaving 104 N to accelerate 50 kg at about 2.1 m/s². That sudden drop is also why things "stick-slip": a squeaking door or a violin bow. The centre of gravity decides what tips A body can be treated, for balance, as if all its weight acted at a single point: the centre of gravity. For a uniform object this is its centroid, the geometric centre of its shape. A rectangle's centroid is at the middle; a triangle's is one third of the way up from the base. For awkward shapes, engineers find the centroid by splitting them into simple pieces and averaging their positions, weighted by area. The rule for tipping is then simple and a little startling: an object stays up as long as the vertical line through its centre of gravity falls inside its base of support. The moment that line passes outside the edge it is resting on, gravity's turning effect about that edge has nothing to oppose it, and the object goes over. That gives a clean comparison between tipping and sliding on a tilting surface. A block tips when the tilt reaches the angle whose tangent is its width divided by its height. It slides when the tilt reaches the angle whose tangent is μ. Whichever threshold comes first wins. Interactive · Tilt the ramp Ramp angle Block height : width Friction coefficient μ A crate 1 m tall and 0.5 m wide tips when tan θ reaches 0.5, which is 26.6°. With μ = 0.4 it would slide at 21.8° first. Raise μ to 0.8 (rough rubber matting, say) and it slides only at 38.7°, so now it tips at 26.6° instead. Same crate, different outcome, and the reason is entirely in the comparison of two numbers. A low, wide object, or one with a heavy bottom, has a larger tipping angle. That is why lorries are loaded heavy-side-down and why a laden roof rack makes a car feel twitchy. What a graduate just sees Anything standing still has forces that add to zero, and turning effects that add to zero. If a piece of the picture does not balance, a force is missing. Before solving anything, they cut the thing out of the world and draw what each cut supplies. A support exerts exactly the reaction needed, and the nearer support carries more of a load. A long lever arm is a force multiplier; the pivot you pick for adding up moments can make an unknown vanish. Triangles are rigid. Every member of a truss is secretly either pulling or pushing along its length, and the pushed ones are the ones to worry about. A deep truss works its members less than a shallow one for the same load. Friction is a limit, not a number: it matches your push until the limit, then lets go. Tall and narrow tips, low and wide slides: compare the width-to-height ratio with μ. Where it shows up next Mechanics of Materials (34/11): statics gives the force in each member or cross-section; that course asks how big a bar must be so that force doesn't break or bend it. Dynamics (34/12): everything here is Newton's second law with the acceleration set to zero. Let it be non-zero and the same free-body diagram describes a moving machine. Machine Design (34/19): every shaft, gear tooth and bolt is sized from the reactions found by statics. Linear Algebra (34/04): real structures have hundreds of joints, and the equations are solved as matrices, just as the truss tool does. In the world: crane booms, bridge trusses, aircraft wing ribs, scaffolding, bike frames, the load rating on a ladder, the rollover rating of a vehicle. If you want to go deeper 📘 Standard textbook: R. C. Hibbeler, Engineering Mechanics: Statics (15th edition, Pearson). The most widely assigned text, with a great many worked problems. 📘 Standard textbook: J. L. Meriam and L. G. Kraige, Engineering Mechanics: Statics (Wiley). Another common choice, careful on free-body diagrams. 📘 Gentler: J. E. Gordon, Structures: Or Why Things Don't Fall Down (Penguin 1978; Da Capo reprint). No equations needed to follow it; written by a materials scientist who loves bridges, boats and bones. 🌐 Free book: Engineering Statics: Open and Interactive by Daniel W. Baker and William Haynes, a free online textbook with interactive figures. 🌐 Free course: MIT OpenCourseWare 2.001 Mechanics & Materials I, which treats statics as the first step into materials. The takeaway If a thing is standing still, every force on it, and every turning effect, adds to zero. Draw the thing alone, draw what each cut supplies, and the unknown pushes and pulls fall out of three equations. That is how a beam shares its load between supports, how a truss sorts its members into pushers and pullers, and why a tall crate tips while a squat one slides. Try this: this week, look at a bookshelf, a bridge or a bike stand and ask three questions. Where are the supports? Which parts are being pulled and which pushed? What would make it tip? Sources & Further Reading Sources. The equilibrium equations, support types, method of joints, Amontons–Coulomb friction, centroids and tipping are standard textbook statics (see the books below). Every number in the interactives was computed in node: beam reactions (RA = W(L−x)/L), seesaw moments, the ten-equation truss solve (checked by hand for the 3 m deep case: chords 5 kN and 2.5 kN, diagonals 5/sin 63.4° = 5.59 kN), friction (N = 50 × 9.81 = 490.5 N; a 300 N push gives 2.08 m/s²) and tipping (arctan 0.5 = 26.6°; arctan 0.4 = 21.8°; arctan 0.8 = 38.7°). The friction coefficients 0.5 and 0.4 are illustrative. The truss is the ideal, pin-jointed model, as in textbook problems; real joints are stiffer. Go deeper. Hibbeler's Statics, 15th edition, is listed by Pearson (ISBN 9780137514663). Gordon's Structures was first published by Penguin in 1978. 🌐 Engineering Statics: Open and Interactive 🌐 MIT OpenCourseWare 2.001 Mechanics & Materials I 📘 J. E. Gordon, Structures: Or Why Things Don't Fall Down Step 9. Mechanics of Materials: Why Things Bend, Buckle and Break ----------------------------------------------------------------- Why: Why beams bend, columns buckle and parts crack, and how to size them so they don't. A bridge cable, a bicycle frame and a paperclip are all springs with limits. Learn the few ideas that say how far they stretch, and what finally breaks them. Bend a paperclip once and it holds its new shape. Bend it back and forth a dozen times and it snaps in your fingers, even though you never once pulled as hard as it can be pulled. A ruler squeezed end to end skips the squashing and bows sideways with a sudden bang. A diving board flexes under your weight and springs back, but a bit more weight would leave it permanently bent. None of that is mysterious once you have the picture a mechanical engineer carries. Every solid is a very stiff spring. Every load makes it stretch or squash or twist by an amount you can predict, and the way it fails depends on what kind of load, how often, and where the stress piles up. Statics told you how big the forces are. This course is about what they do to the stuff. The course in one box Who takes it: mechanical, aerospace, civil and materials engineers, usually in the second year (sometimes called Strength of Materials or Solid Mechanics). What it is for: predicting how much a part deforms and when it will fail, so you can choose its shape and size. Comes after: Statics (34/10), and through it Physics I: Mechanics (34/06) and Linear Algebra (34/04). Leads to: Machine Design (34/19), which turns it into shafts, bolts and gears. It also underlies Materials Science & Manufacturing (34/13) and Vibrations (34/17). What this piece will and won't give you: you won't be able to certify a beam. You will see why a beam is shaped like an I, why a long thin column fails differently from a short fat one, and why a small hole can matter more than a big load. Easier first read: Everyday Physics: The Answers Hiding in Plain Sight (Physics 5). Read Statics: Why Things Stand Still (34/10) first. Stress and strain: divide out the size A thick rod and a thin rod made of the same steel and pulled with the same force behave very differently, and a long rod stretches more than a short one. To compare materials rather than particular objects, engineers divide the size out. Stress is force per unit area: how hard the material is working. Strain is stretch per unit length: how much it has deformed, as a fraction. A rod that grows from 1,000 mm to 1,001 mm has strain 0.001, or 0.1%. σ = F ÷ A     ε = ΔL ÷ L For small stretches, stress and strain are proportional. Robert Hooke published the idea in 1678 as ut tensio, sic vis, "as the extension, so the force". The constant of proportionality for a material is its Young's modulus E, named after Thomas Young. It is the material's stiffness, independent of shape: σ = E × ε Steel's E is about 200 GPa, aluminium's about 69 GPa, and pine's roughly 10 GPa (values vary by alloy and species). Read it as: for the same stress, aluminium stretches about three times as much as steel. Try it. Interactive · Pull a rod Steel Aluminium 6061-T6 Pulling force Rod diameter With 10 kN on a 10 mm steel rod 1 m long: area 78.5 mm², stress 127 MPa, strain 0.000637, stretch 0.64 mm. Halve the diameter and the stress quadruples, because the area falls with the square of the diameter. Swap to aluminium and the stretch nearly triples (200 ÷ 69) while the stress stays put. That is why stiffness-critical parts (not just strength-critical ones) are chosen by E. The stress–strain curve is a material's personality Pull a bar in a test machine, record stress against strain, and you get a curve that is a fingerprint of the material. It has a few landmarks worth knowing by heart. First comes a steep straight line: the elastic range. Let go and the bar returns exactly to its original length. The slope is E. At the yield point the line gives way: atoms begin to slip past each other, and release no longer restores the old shape, so some stretch is permanent. In a ductile metal like structural steel the curve then rises further (the metal work-hardens) to the ultimate strength, the highest stress it ever carries. After that the bar thins in one place (necking) and finally breaks. A brittle material such as glass or cast iron stays elastic almost to the end and snaps with little or no warning. Engineers prefer ductile materials for structures because ductility gives notice: a steel beam sags visibly and absorbs energy long before it breaks. The yield point, not the ultimate strength, is usually the number a designer works from. Interactive · Stretch it until it gives Illustrative curves: a steel-like metal (E = 200 GPa, yield 250 MPa, ultimate 400 MPa) and a glass-like brittle solid (E = 70 GPa, snaps at 50 MPa). They are idealised in shape, not measured. Ductile, steel-like Brittle, glass-like Whole curve Zoom in on the first 0.4% Stretch the bar to The elastic range of the steel-like metal ends at strain 0.00125, since 250 MPa ÷ 200,000 MPa = 0.00125. That is just 1.25 mm on a metre of bar, a sliver of the total 25% it can stretch before breaking. Use the zoom button to see that sliver. Nearly everything a structure does in service happens inside it, which is why it is designed to stay there. Beams: stiffness grows with depth cubed When a beam sags, its top edge is squashed and its bottom edge is stretched. Somewhere between them is a layer that does neither, the neutral axis. Material near the middle is barely working. Material at the top and bottom is working hardest and is also the farthest from the middle, which gives it the most leverage against bending. The shape's resistance to bending is captured in one number, the second moment of area I. For a rectangle of width b and depth d, I = b·d³ ÷ 12. The cube is the whole story. For a beam resting on two supports with a load in the middle: sag = P L³ ÷ (48 E I) Double the depth of a beam, keeping the width, and its stiffness rises 2³ = 8 times. Compare a plank on edge with the same plank laid flat: the edge-up plank is twice as deep (a factor of 8) but half as wide (a factor of ½), so it is 4 times stiffer. Doubling the span makes the sag 2³ = 8 times bigger. Depth and span, not weight, rule the stiffness. That also explains the I-beam. If material near the middle isn't doing much, move it to the flanges and leave a thin web to hold them apart. The shape gives up some stiffness for a lot less steel. Interactive · Shape the beam Steel Aluminium Pine Solid rectangle I-section Lay it flat Load at the middle A solid steel 50 × 100 mm beam on edge, 2 m span, 5 kN at the middle, sags 1.0 mm. Lay it flat and it sags 4 mm. Make the same outline an I-section (flanges and web each a tenth of the size) and it uses 28% of the steel but keeps 54% of the stiffness, sagging 1.85 mm. The beam is shaped like an I because that puts the steel where the bending lives. Buckling: slender columns bow before they crush You would expect a column to fail when the compression exceeds the material's strength. A short, stout one does. A long thin one never gets there. Press on the ends of a plastic ruler and at some load it suddenly bows out sideways, far below what it could carry if it stayed straight. That is buckling: an instability, not a strength failure. Leonhard Euler worked out the load at which a perfectly straight slender column bows, in the 1750s: Pcr = π² E I ÷ (K L)² The load depends on stiffness (E I), not strength, and falls with the square of the length: double the length and the load a column can take falls to a quarter. The factor K depends on how the ends are held: 1 for pinned ends, 0.5 if both are clamped (four times stronger), 2 for a flagpole fixed at the bottom only (a quarter as strong). A column's crushing load, on the other hand, is just yield stress times area and doesn't care how long it is. Interactive · Crush or bow? Pinned both ends Clamped both ends Clamped and pinned Flagpole Rod diameter Length A 20 mm steel rod, 1 m long and pinned at both ends, buckles at 15.5 kN, though it would take 78.5 kN to crush it. At 2 m it buckles at only 3.9 kN. At 0.2 m the buckling load is far above the crushing load, so it crushes instead. The formula is for ideal columns; real ones have imperfections and do worse, and real designs use empirical corrections with generous margins. In structures, compression members are the ones engineers watch for exactly this reason, and why a long pushed member is deeper or braced. Torsion: a twisted shaft is a bar in shear Stretching is not the only way to load a bar. Push the top of a deck of cards sideways and the cards slide over each other: that is shear, a sliding stress (force parallel to the surface, divided by the area). A bolt holding two plates that try to slide past each other is in shear. The shear modulus G plays the role for sliding that E plays for stretching; for steel it is about 80 GPa. Twist a shaft and every thin slice slides a little relative to its neighbour, and the sliding is biggest at the surface. The surface stress is torque times radius divided by the shaft's polar second moment J (for a solid round shaft J = πD⁴ ÷ 32): τ = T r ÷ J     twist = T L ÷ (G J) Because the centre of a shaft barely strains, you can bore it out and keep most of its torsional stiffness. That is why drive shafts, bicycle cranks and aircraft tubes are hollow. Interactive · Twist a shaft Outer diameter Hollowness Torque A solid 20 mm steel shaft carrying 100 N·m has a surface shear stress of 63.7 MPa and twists 4.6° over a metre. Bore it out to 16 mm (80% of the outer diameter) and it keeps 59% of the twisting stiffness with 36% of the steel. The surface stress rises to 108 MPa, so you do pay something, which is why designers also check it. The yield strength in shear is roughly 0.58 times the tensile yield (about 144 MPa for 250 MPa steel). Holes, cracks and repetition: why parts break below their strength Stress flows through a part like water through a pipe. Where the pipe narrows or bends, the flow bunches up. A hole in a plate in tension makes the stress lines crowd around it, so the stress at the hole's edge is higher than the average, by a stress concentration factor K. For a small round hole in a wide plate, K = 3. For an elliptical hole whose width across the pull is a times its height, K = 1 + 2a. A sharper, longer notch across the pull means a bigger K, and a crack is the sharpest notch of all. Interactive · A hole in a loaded plate Average stress in the plate Shape of the hole At 80 MPa average stress (well under yield) a round hole brings the edge to 240 MPa, a factor of safety of about 1.04. At 100 MPa it goes to 300 MPa and the metal there yields. A 5 : 1 slot puts K at 11. The factor of safety is just capacity divided by what you expect to load, with a deliberate margin for the unknowns. Aircraft designers typically work with slimmer margins than builders of bridges, because every kilogram of margin is paid for in fuel, so they have less room for a surprise like a concentration they missed. Repetition makes it worse. A load applied again and again, far below the yield stress, can start a tiny crack at a stress concentration and grow it a little each cycle until what remains cannot carry the load. That is fatigue, and it is the paperclip's story at engineering scale. The de Havilland Comet, an early jet airliner, suffered two mid-air break-ups over the Mediterranean in 1954 (10 January off Elba, 35 people killed; 8 April near Naples, 21 killed). At Farnborough, investigators sealed a whole fuselage in a water tank and pressurised it again and again to imitate flights. It burst open after 3,057 pressure cycles (1,221 real flights plus 1,836 simulated), from a crack at the corner of the forward escape-hatch window. On the Elba wreckage the fatigue crack began at a rivet hole by the opening for an aerial. The story is often told as "square windows"; sharp corners were stress concentrators, but they were one of several in that airframe. The lesson that stayed: repeated pressurisation plus a stress concentration can break even a strong skin, so pressurised airframes must be designed and tested for fatigue. What a graduate just sees Strength is stress (force over area), not force. Halving the diameter quadruples the stress. Every solid is a spring, and Young's modulus is its stiffness: steel stretches a third as much as aluminium for the same stress. Most of the working life of a structure is spent in the first fraction of a percent of strain, deliberately. Ductile means warning; brittle means none. A deep beam is stiff because stiffness grows with the cube of the depth. Put material where it is farthest from the middle. A long thin thing in compression is a buckling problem, not a crushing problem, and doubling its length cuts its capacity to a quarter. Hollow tubes are nearly as stiff in twist as solid bars at a fraction of the weight. Sharp corners, holes and cracks multiply stress; round them. Repetition finds them. Where it shows up next Statics (34/10) feeds it: the reaction or member force found there becomes the F and T here. Machine Design (34/19): sizing a shaft, a bolt or a gear tooth is these formulas with factors of safety and fatigue data added. Materials Science & Manufacturing (34/13): why steel yields where it does, why glass is brittle, and how heat treatment moves the curve you played with. Vibrations (34/17): a beam is a spring, and its stiffness sets the pitch at which it rings. In the world: aircraft skins and wing spars, bridge girders, bicycle frames, tall chimneys, engine crankshafts, the reason window corners on ships and planes are rounded. If you want to go deeper 📘 Standard textbook: R. C. Hibbeler, Mechanics of Materials (Pearson). Widely assigned, clear on method, with a great many worked problems. 📘 Standard textbook: F. P. Beer, E. R. Johnston Jr., J. T. DeWolf and D. F. Mazurek, Mechanics of Materials (McGraw Hill). The other common choice. 📘 Standard textbook: J. M. Gere and B. J. Goodno, Mechanics of Materials (Cengage). Known for patient explanations. 📘 Gentler: J. E. Gordon, Structures: Or Why Things Don't Fall Down, and its companion The New Science of Strong Materials. Written for non-engineers, and mostly equation-free. 🌐 Free course: MIT OpenCourseWare 2.001 Mechanics & Materials I, with lecture notes and problem sets. The takeaway Stress is force over area, strain is stretch over length, and Young's modulus is the exchange rate between them. A beam's stiffness rises with the cube of its depth, so it is shaped like an I. A slender column bows rather than crushes, at a load that falls with the square of its length. A twisted shaft is a bar in shear, and a hollow one is almost as good as a solid one. And every hole, corner and crack concentrates stress, which repeated loads then turn into fatigue. Try this: bend a paperclip back and forth and count the bends until it breaks. Then look at any bridge, crane or bicycle and find where the metal is placed far from the middle, and where the corners are rounded. Sources & Further Reading Sources. The formulas (σ = F/A, ε = ΔL/L, Hooke's law, I = bd³/12, centre-load sag PL³/48EI, Euler's Pcr = π²EI/(KL)², τ = Tr/J, twist TL/GJ, the Inglis ellipse K = 1 + 2a/b) are standard textbook results. Hooke published the anagram's solution ut tensio, sic vis in 1678 (Wikipedia, "Hooke's law"). Typical E values: Wikipedia, "Young's modulus" (steel about 200 GPa, aluminium about 68–69 GPa, wood roughly 9–12 GPa). Euler's column formula dates from 1757 (Wikipedia, "Buckling"). Comet: BOAC Flight 781, 10 January 1954, 35 dead; South African Airways Flight 201, 8 April 1954, 21 dead; water-tank test fuselage G-ALYU failed on 24 June 1954 after 3,057 cycles (1,221 actual flights plus 1,836 simulated), from Wikipedia, "De Havilland Comet"; the fatigue origin on G-ALYP at a rivet hole by the aerial (ADF) window, and the square-window caveat, from Aerossurance, "Common Comet Misconceptions". Every number in the interactives was computed in node (steel rod: 127.3 MPa, 0.637 mm; beam: 1.00, 4.00 and 1.85 mm; column: 15.5, 3.88 and 78.5 kN; shaft: 63.7 and 107.8 MPa, 4.56°; notch: K = 3, 11). The two stress–strain curves are idealised and the G, yield and ultimate values are round textbook figures. Go deeper. Gordon's two books were first published by Penguin in 1978 (Structures) and earlier (The New Science of Strong Materials). 🌐 MIT OpenCourseWare 2.001 Mechanics & Materials I 🌐 Aerossurance: Common Comet Misconceptions 📘 J. E. Gordon, Structures: Or Why Things Don't Fall Down Step 10. Dynamics: Why a Hoop Loses the Race -------------------------------------------- Why: Moving things: rotation, momentum, and why a spinning wheel resists turning. Statics freezes the world to find the forces. Dynamics lets it move, and asks where it goes next: why a gyroscope turns sideways, and why the full can wins. Roll a full soup can and an empty one down the same slope, side by side, and the full one wins. Hold a spinning bicycle wheel by its axle and try to tilt it: it twists in your hands in a direction you didn't push. Stand on a bathroom scale in a lift as it starts to rise and the number jumps up, though you haven't gained an ounce. These look like three unrelated tricks. They are one idea: when things accelerate, the forces on them no longer add to zero, and the leftover is exactly mass times acceleration. Statics freezes the world to find forces. Dynamics lets it move, and asks where it goes next. The course in one box Who takes it: mechanical and aerospace engineers (and many physics and civil students), usually in the second year. What it is for: predicting how forces change the motion of bodies, from a thrown ball to a spinning turbine to a linkage in an engine. Comes after: Physics I: Mechanics (34/06) and Differential Equations (34/03). Leads to: Vibrations (34/17), and through it Control Systems (34/20). Its ideas also run through Machine Design (34/19) and Propulsion (34/21). What this piece will and won't give you: you won't be able to simulate a robot arm after this. You will see why a hoop loses a race to a disc, why a gyroscope turns sideways, and how an engine's piston is a spinning crank in disguise. Easier first reads: Energy: The One Currency of the Universe (Physics 2), Orbits: Why the Moon Doesn't Fall (or Rather, Keeps Falling) (Physics 11) and Calculus: The Language of Change (12/11). Kinematics: position, velocity, acceleration, and who is measuring Kinematics describes motion without asking what causes it. There are three quantities and they are one story told three times. Position says where. Velocity is how fast position changes. Acceleration is how fast velocity changes. Each is the derivative of the one before it; run the chain backwards with integrals and acceleration gives you velocity gives you position. The most useful trick is to split motion into independent directions. Throw a ball: gravity pulls down only, so the sideways speed never changes and the vertical speed changes by 9.81 m/s every second. The two parts do not interfere. Combine them and you get a parabola. Interactive · Throw something Launch speed Launch angle Time in flight At 20 m/s and 45° (no air resistance), the ball travels 40.8 m, peaks at 10.2 m and stays up 2.88 s. Angles 30° and 60° give the same range, 35.3 m, because they trade height for distance in matching amounts. Real balls feel air drag, which breaks that symmetry; this is the idealised textbook case. Motion is also relative to whoever measures it. An aeroplane's engines fix its speed through the air; the wind carries the air along; the ground sees the sum. Velocities add as vectors. Interactive · A plane in a crosswind Airspeed Heading (0 = north, 90 = east) Wind speed Wind blowing towards Aim to track due north Point north at 200 km/h with a 50 km/h wind blowing towards the east: the ground sees 206 km/h along a track 14° east of north. To actually go north, the pilot must point about 14.5° into the wind (heading 345.5°), and ground speed drops to 194 km/h. Pilots call that the crab angle. Newton's second law: the free-body diagram, now with an acceleration Draw the same free-body diagram you would for a statics problem. The difference is the right-hand side. The forces no longer add to zero; they add to mass times acceleration: ΣF = m a Read it as: the net force is what is left over after the pushes and pulls cancel, and it makes the body speed up in its own direction by that leftover divided by its mass. A lift is the cleanest example, because the "scale" is a force meter: it reports the push of the floor on your feet. Interactive · The lift and the scale Lift acceleration (up is positive) For a 70 kg person, the floor pushes with N = m(g + a). Accelerating upward at 2 m/s² gives 827 N, a scale reading of 84 kg. Accelerating downward at 2 m/s² gives 547 N, or 56 kg. Only the acceleration matters: moving at a steady speed, up or down, reads your true 70 kg. Slide it to −9.8 and the floor pushes with nothing. That is the same "weightless, not gravity-less" free fall as in the Orbits piece. Energy and momentum: two shortcuts that skip the forces Solving every motion by integrating F = ma in time is possible but slow. Two integrals of Newton's law do the work for you. The work–energy method integrates over distance: the work done on a body equals its change in kinetic energy, ½mv². It is the tool when you care where something ends up and not how long it took. A car braking at constant friction needs a distance of v² ÷ (2μg); double the speed and the stopping distance quadruples. At 100 km/h (27.8 m/s) and a tyre grip of 0.7 (a made-up but plausible number), that is about 56 m. The impulse–momentum method integrates over time: a force acting for a time (the impulse) equals the change in momentum, mv. It is the tool for impacts, where the forces are huge, brief and unknown, yet the totals are knowable. When two bodies collide with no outside force, their total momentum is unchanged, whatever happens inside. Energy is not so loyal: collisions waste some as heat, sound and dents. Δ(½mv²) = work     F·Δt = Δ(mv) Interactive · Crash two carts Cart 1 moves at 10 m/s and hits cart 2, which is at rest. Mass of cart 1 Mass of cart 2 How bouncy Two 10 kg carts that stick together after the hit leave at 5 m/s: momentum is 100 kg·m/s before and after, but energy falls from 500 J to 250 J. A perfect bounce passes cart 1's whole motion to cart 2 and keeps all 500 J. Whatever the bounciness, the momentum line never moves. This is how crash investigators estimate speeds from wreckage, and how a rocket works: throw mass one way and the rest goes the other. Rotation: mass measured by how far it sits from the axis Everything above has a spinning twin. Force becomes torque, mass becomes moment of inertia I, acceleration becomes angular acceleration. Newton's law for a spinning body is torque equals I times angular acceleration. The one new idea is that "mass" now depends on where the mass sits: I = Σ m r² Read it as: every bit of mass counts by its distance from the axis, squared. A kilogram at the rim is far harder to spin up than a kilogram at the centre. Figure skaters use this: pulling their arms in shrinks r and the spin speeds up. It is also why a hoop trails a disc. Rolling downhill, the energy released from height is shared between moving forward and spinning. The more of the mass that sits at the rim, the more energy goes into spin, so the less is left for speed. For a body of radius R the acceleration down a slope is g sin θ ÷ (1 + k), where k = I ÷ (mR²) is 1 for a hoop, ½ for a solid disc, ⅖ for a solid ball and ⅔ for a hollow ball. The mass and the size cancel out. Interactive · The rolling race Slope angle Time since release On a 20° slope 2 m long, the solid ball takes 1.29 s, the disc 1.34 s, the hollow ball 1.41 s and the hoop 1.54 s. A heavier hoop, or a bigger one, finishes at the same moment as a lighter, smaller hoop; only the shape of the mass distribution counts. The full soup can wins for a related reason: a can of thin liquid is not rigid, so the liquid barely joins the spin and less of the mass is rotating, while an empty can is a thin shell with its mass out at the rim, like the hoop. (A frozen can would behave more like a solid cylinder.) Gyroscopes: why a spinning thing turns sideways when pushed A spinning object carries angular momentum L = I × ω, which points along its axis. A torque changes L in the direction of the torque, just as a force changes momentum in the direction of the force. If the torque is sideways to the axis, L is not made bigger or smaller; it is swung round. The axis turns at right angles to the push. That swinging is called precession, and for a fast-spinning wheel it goes at: Ω = m g r ÷ (I ω) Read it as: the gravity torque (m g r) divided by the angular momentum (I ω). Faster spin means slower precession. A wheel held by one end of its axle with gravity trying to tip it does not fall; it circles slowly like a hand on a clock. Interactive · The wheel on a pivot A 2 kg wheel with its mass mostly at the rim (I = 0.18 kg·m²), supported at one end of its axle. The sliders set the spin and how far the wheel's centre sits from the pivot. Spin Wheel centre from pivot Time elapsed At 300 rpm (31.4 rad/s) the precession is 0.69 rad/s, about 40° per second, so one lap in 9 s. At 600 rpm the lap takes 18 s. The same physics keeps a spinning top upright and lets spacecraft use spinning wheels to turn without thrusters. It is also one of several effects at work in a bicycle, though it is not the whole story: researchers have built a bicycle that balances itself without any gyroscopic effect. Linkages: turning one motion into another A mechanism is a chain of rigid bars joined by pivots, built to turn one motion into a different one. The simplest has four bars, and it appears in windscreen wipers, car suspensions and bicycle rear shocks. The most famous is the slider-crank: a crank turning in a circle, a connecting rod, and a piston sliding in a line. Run it one way and it is a petrol engine; run it the other and it is a pump. Kinematics (earlier in this piece) gives the piston's position from the crank angle by geometry, and its speed by differentiating. The piston's motion is not a pure sine wave: it moves faster in one half of its stroke than the other, and its speed peaks before mid-stroke. Interactive · A piston and crank Crank angle Connecting rod length With a 40 mm crank (an 80 mm stroke), a 130 mm rod and the crank turning at 3,000 rpm, the piston's speed reaches about 13.2 m/s at roughly 74° of crank angle, against an average of 8 m/s. The piston is going nowhere at the two ends of its stroke, and its acceleration there, about 5,200 m/s² or roughly 530 times gravity, is the reason engine parts are so carefully balanced. (These dimensions are illustrative.) What a graduate just sees Position, velocity and acceleration are one quantity differentiated or integrated; split a motion into independent directions and each one is easy. Velocity is always relative to something. Add the vectors. If the forces don't cancel, something accelerates, and ΣF = ma says by how much. Weight on a scale is a force reading, not a mass. Use energy when you want to know where; use momentum when you want to know what an impact did. Momentum is conserved in a collision even when energy isn't. Mass far from the axis is hard to spin, so shape beats weight in a rolling race. Push on a spinning thing and it moves at right angles to the push. A linkage is a piece of geometry; its speeds are the derivative of its positions. Where it shows up next Vibrations (34/17): a mass on a spring is ΣF = ma with a spring force, a differential equation that rings. Resonance comes from it. Control Systems (34/20): to make a drone or robot arm hold a position, you first need its equation of motion. Machine Design (34/19): flywheels, cams, gears and crankshafts are sized with inertia and balance in mind. Propulsion (34/21): a rocket is conservation of momentum applied to a stream of exhaust. In the world: car crash tests, aircraft navigation in wind, satellites that point using spinning wheels, robot arms, roller coasters, every engine. If you want to go deeper 📘 Standard textbook: J. L. Meriam and L. G. Kraige, Engineering Mechanics: Dynamics (Wiley). Known for clear explanations and sketches. 📘 Standard textbook: R. C. Hibbeler, Engineering Mechanics: Dynamics (Pearson). The other widely assigned choice, with a great many worked problems. 📘 Gentler: The Feynman Lectures on Physics, Volume I, which treats rotation and the gyroscope with no more mathematics than this piece uses; free to read online from Caltech. 🌐 Free course: MIT OpenCourseWare 2.003SC Engineering Dynamics (Fall 2011), with lectures, recitations and problem sets. The takeaway Dynamics is the same free-body diagram as statics with one change: the forces need not cancel, and the leftover is mass times acceleration. Energy and momentum are two integrals of that law that skip the details. Rotation repeats the story with moment of inertia, which is why a hoop trails a disc and a figure skater speeds up by pulling in her arms. A spinning body answers a push by turning at right angles, and a linkage turns one motion into another by geometry. Try this: race a full and an empty tin can, or a bangle and a disc, down a tilted tray. Then ask yourself what the scale in the lift is actually measuring. Sources & Further Reading Sources. The equations (kinematics, ΣF = ma, work–energy, impulse–momentum, I = Σmr², rolling acceleration g sin θ ÷ (1 + k), precession Ω = mgr ÷ (Iω), the slider-crank position formula) are standard textbook dynamics. Every number in the interactives was computed in node: projectile (range 40.77 m, apex 10.19 m, flight 2.88 s at 20 m/s and 45°; 35.31 m at 30° and 60°), wind triangle (206.2 km/h, track 14.0°; crab angle 14.48°, ground speed 193.6 km/h), lift (827 N and 547 N), collisions (500 J to 250 J), rolling race (1.29, 1.34, 1.41 and 1.54 s on a 2 m, 20° slope), gyroscope (0.694 rad/s, period 9.05 s at 300 rpm) and slider-crank (peak 13.15 m/s at 74° for a 40 mm crank, 130 mm rod, 3,000 rpm). The tyre-grip value 0.7 and the engine dimensions are illustrative. That a bicycle can balance without gyroscopic effects: Kooijman et al., Science 332 (2011), "A bicycle can be self-stable without gyroscopic or caster effects". Go deeper. MIT OCW 2.003SC lists Hibbeler's Engineering Mechanics: Dynamics and Williams's Fundamentals of Applied Dynamics as suggested readings. 🌐 MIT OpenCourseWare 2.003SC Engineering Dynamics 🌐 The Feynman Lectures on Physics, Volume I (online) 📘 R. C. Hibbeler, Engineering Mechanics: Dynamics Step 11. Materials Science & Manufacturing: Why Things Are Made of What They Are -------------------------------------------------------------------------------- Why: Steel, aluminium, polymers, composites: why each part is made of what it's made of. A paperclip bends, a mug shatters, a rubber band springs back. Same push, three different answers, and the whole course is the reason why. Take a steel paperclip and bend it open. It stays bent. Bend that same kink back and forth, and it gets stubborn at the bend, then warm, then it snaps, usually after a handful of bends. Now try the same with a ceramic mug handle: it doesn't bend at all, it just breaks. Try it with a rubber band: it stretches a long way and comes back every time. Three materials, three completely different answers to the same push. A materials scientist doesn't see three mysteries. They see three arrangements of atoms, and one question that explains the lot: what happens to the atoms when you load them? That question, plus a second one (how was this thing made, and what did making it do to the atoms?), is the whole course. It decides why a bicycle frame is aluminium, why a surgeon's scalpel is hard and a car spring is not, why an aeroplane wing is turning from metal to carbon fibre, and why two parts with the same shape and the same alloy can behave differently. The course in one box Who takes it: mechanical, aerospace, civil and materials engineers, usually in year 2, beside Mechanics of Materials. What it's for: choosing what a part is made of, and from that, how to make it. Comes after: Physics I: Mechanics (34/06). Leads to: Machine Design (34/19), which needs it to pick shaft steels and bolt grades; it also feeds Mechanics of Materials (34/11), Propulsion (34/21) and the Capstone (34/22). Honest line: you won't be able to pick a steel grade from a handbook after this; you will see why a blacksmith plunges a blade into water, and why a bridge is not made of glass. Atoms are springs: where stiffness comes from A solid is atoms held at a comfortable spacing by bonds. Push two atoms closer than that spacing and they repel strongly. Pull them apart and they attract. At the right distance, they sit in an energy valley. Near the bottom of any smooth valley, the shape is almost a parabola, which is exactly the shape of a spring. So, for small stretches, every bond is a spring, and a solid is a huge three-dimensional mattress of them. Engineers measure how stiff the mattress is by stress (force per area) against strain (fractional stretch). Their ratio is Young's modulus, E: stress = E × strain    (σ = E ε) In words: the stiffer the bonds and the tighter the packing, the more stress it takes to stretch the material by the same fraction. Diamond sits at roughly 1,000 GPa. Steel is about 200 GPa, aluminium about 70, nylon about 3, and rubber 0.01 to 0.1, because rubber's long molecular chains are coiled like tangled string, so stretching it mostly uncoils the chains rather than stretching bonds. Worked number: pull a steel bar 1 m long with a 10 mm × 10 mm section using 10 kN. Stress = 10,000 N ÷ 100 mm² = 100 MPa. Strain = 100 MPa ÷ 200,000 MPa = 0.0005, so the bar grows 0.5 mm. The atomic spacing, about 0.25 nm, has changed by about 0.0001 nm. Now pull with a gentler 1 kN on the same section: steel stretches 0.05 mm, aluminium 0.14 mm, nylon 3.4 mm. Stiffness is a property of the bonds. Heat-treating a steel makes it far stronger, and does almost nothing to its stiffness. That last point is the surprise that a first-year loses and a graduate keeps: a soft mild steel and a hardened spring steel have almost the same E. A thin spring steel wire does not sag less than a mild steel one. It simply springs back from a much bigger load. Four words engineers keep apart: stiff, strong, tough, hard In everyday speech they blur together. In engineering they are different questions, and a material can say yes to one and no to another. Stiff: how little it stretches elastically under load (E). Strong: how much stress it carries before it yields (starts to stretch permanently) or breaks. Tough: how much energy it absorbs before it breaks, which is the area under its stress-strain curve. Hard: how well its surface resists being dented or scratched. Glass is stiff and, in a clean sheet, decently strong, but not tough: it breaks with no warning because cracks run through it. A rubber band is not stiff or strong, but is tough. Mild steel is the compromise that built the modern world: stiff, strong enough, and above all tough, because it yields, warns you by bending, and soaks up energy. Try the pull test below. Interactive · Pull test Mild steel Hardened spring steel Aluminium 6061-T6 Nylon 66 Glass Pull until it breaks (left to right) Simplified curves built from typical handbook values (modulus, yield, tensile strength, elongation); real batches vary. "Energy to fracture" is the area under the whole curve. The hardened spring steel row is illustrative. Each plot is scaled to its own material, so compare the table, not the picture. Compare steel and glass in the table. Their stiffness is within a factor of three, and yet steel absorbs roughly 4,500 times more energy per volume before it fails. Hardened spring steel has the same stiffness again, over five times the yield stress, and about the same area, but loses the long warning ductility gives. Strength, stiffness and toughness really are three separate dials. (Cracks have their own measure, fracture toughness; the area under the curve is the version for an uncracked bar.) Why metals are soft: crystals, grains and dislocations Metals are crystals: atoms stacked in neat repeating layers. A simple calculation of how much shear it takes to slide one whole layer over the next gives a stress of roughly one-tenth of the shear modulus G. For steel, G is about 80 GPa, so theory promises something like 8 GPa. Real mild steel yields at about 0.25 GPa, some thirty times lower, and for many pure metals the gap is hundreds of times. Something is wrong with the picture. In 1934, three researchers (Taylor, Orowan and Polanyi, working separately) gave the answer: a real crystal is not perfect. It contains dislocations, lines where an extra half-plane of atoms is wedged in. Slip doesn't need a whole layer to slide at once. It needs the dislocation to creep along, shifting a few atoms at a time, the way a ruck in a carpet moves far more easily than dragging the whole carpet. So everything a metallurgist does to make a metal stronger is a way of getting in the way of dislocations: Smaller grains. A metal part is made of many small crystals (grains) with misaligned layers. A dislocation can't cross the boundary easily, so more boundaries mean more strength. Alloying. Foreign atoms snag dislocations. Work hardening. Deforming a metal makes more dislocations that tangle each other. That's your paperclip getting stubborn at the bend. Precipitates. Tiny hard particles in the way (how aircraft aluminium is strengthened). The grain-size effect has a famous simple rule, the Hall–Petch relation: yield strength = σ0 + k ÷ √d Yield strength is a base strength plus a term that grows as the grain diameter d shrinks, with the square root making it gentler than you'd guess. For one measured low-carbon steel, σ₀ ≈ 100 MPa and k ≈ 600 MPa·µm½. Then grains of 25 µm give about 220 MPa; 10 µm give about 290; 3 µm give about 450. Interactive · Shrink the grains Average grain diameter The grains are a random pattern at the chosen average size, not a micrograph. Strength uses the Hall–Petch fit for one low-carbon steel; other metals have other constants, and the rule fails at nanometre grain sizes. Heat treatment: cooking steel Steel is iron with a little carbon (up to about 2%). Heat it red-hot and its atoms rearrange into a form that dissolves the carbon. Cool it slowly and the carbon has time to leave, leaving soft, tough mixtures of tiny crystals. Quench it (plunge it into water or oil) and the carbon has no time to escape. It is trapped, and the lattice is forced into a strained, distorted shape called martensite. That shape is extremely hard, and dislocations barely move. It is also brittle, and often has internal stresses big enough to crack the part. The fix is tempering: reheat the quenched steel to somewhere between about 150 and 600 °C. The trapped carbon can move a little, the strain relaxes, and you trade hardness for toughness. The higher the temper, the softer and tougher. A blacksmith reads the temperature from the colour of the thin oxide film on the bright steel: faint yellow, then straw, brown, purple, blue. Choose a tool below and see the colour it is tempered to. Interactive · Temper it Razor Rock drill Cold chisel Screwdriver Spring Tempering temperature Colours and uses are the traditional blacksmith's chart for carbon steel; the exact temperatures depend on the steel and how it was heated. The hardness and toughness trend is shown in words, because real numbers depend on the alloy. Tools that need an edge, such as razors and drills, are tempered low: hard, a little brittle. Things that must flex or take shocks are tempered higher. Notice what is not changing: the composition. Same iron, same carbon, different heat history, a different part. The same logic, with a different mechanism, is how aircraft aluminium is "aged" to full strength. Choosing a material: the Ashby trade-off Engineers rarely want "the strongest" or "the stiffest". They want the lightest part that is stiff enough, or the cheapest that is strong enough. The materials scientist Michael Ashby turned this into a method: plot every material on a chart of two properties (log scales, because properties span many powers of ten), and use a performance index to rank them for a given job. For a part that must stay below a given deflection at the lowest weight, the index depends on the shape of the job. For a tie rod or cable in tension it is E ÷ ρ (stiffness over density). For a beam whose cross-section you may scale, it is √E ÷ ρ. For a flat panel, ∛E ÷ ρ. Why the roots? Because in a beam you can use a stiffer material or make it deeper, and depth is worth a lot (it counts by the cube). Light, bulky materials win. Interactive · Which material makes the lightest part? Tie rod Beam Panel Anchor the guide line on: Mass at equal stiffness is 100% for steel; smaller is lighter. Data are typical values (carbon fibre depends strongly on layup; the "all fibres aligned" figure applies only along the fibres). Stiffness-limited design only: strength, toughness, cost and fatigue change the ranking. Two things stand out. First, for a tie rod, steel, aluminium, titanium and magnesium all score about the same (the metals are stiff in proportion to their weight). Aluminium wins only when the part can be made bulkier, in a beam or a panel, where a beam of aluminium is about 41% lighter than a steel one of equal stiffness. That is the real reason aeroplanes are aluminium: skin and spars are thin-walled, bulky shapes. Second, carbon fibre composite beats everything metallic, but only along its fibres; that's why a composite wing is built by laying plies in chosen directions. Boeing has said that about half of the 787's primary structure by weight is composite. Corrosion: a battery you didn't mean to build Rust is not "iron getting old". It is an electrochemical cell. Where iron meets water and oxygen, some patches give up electrons and dissolve (the anode), and the electrons run through the metal to other patches where oxygen picks them up (the cathode). Any water film is the electrolyte. The same thing happens on a larger scale when two different metals touch while wet. The more active one becomes the anode and corrodes faster than it would alone; the nobler one is protected. Galvanising uses this deliberately: zinc sacrifices itself to protect steel, and ships carry blocks of zinc or aluminium for the same job. Aluminium and stainless steel resist corrosion because of a thin, tough, self-healing oxide film (passivation) that blocks the reaction. Area matters too. The cathode reaction needs surface to run on, so a small anode with a big cathode concentrates the attack on a small area. Steel rivets in a big copper sheet is the bad case; a copper rivet in a steel plate, the forgiving one. Interactive · Who corrodes? Metal one Metal two Surface area, metal one : metal two Order follows the standard seawater galvanic series (most active first). It is a guide, not a prediction: real rates depend on the electrolyte, temperature and surface films. The area rule is a first-order approximation. How you make it changes what you made A drawing says "steel, this shape". It doesn't say how the steel was formed, and that changes the part: Casting (pour liquid metal into a mould) makes complex shapes cheaply, but the metal can trap gas pores and shrinks as it freezes, so cast parts tend to be weaker than wrought ones. Forging (hammer or press hot metal into shape) squeezes out pores and makes the grains follow the shape, like grain in wood. Crankshafts and connecting rods are often forged. Machining (cutting from solid stock) is accurate and keeps the stock's properties, but wastes material and leaves tool marks that can start fatigue cracks. Additive manufacturing (3D printing, building in layers) needs no tooling and can make internal channels no other process can, but parts are often weaker across layers than along them, and surfaces are rough. Injection moulding (molten polymer injected into a steel mould) makes millions of identical parts cheaply, after paying for the mould. Plastic chains line up in the flow direction, and where two flows meet they weld weakly. Cost shapes the choice as much as physics. Each process has a one-off cost (tooling, setup) and a cost per part. Divide the first over the quantity and add the second, and each process wins in its own range. Interactive · Which process, at what quantity? How many parts? The dollar figures are invented round numbers to show the shape of the trade-off, not quotes. Real tooling costs vary hugely (a simple mould can cost thousands, a car-door die millions), and the processes make parts in different materials with different properties. What a graduate just sees Stiffness belongs to the bonds, strength belongs to the microstructure: heat treatment changes the second and barely touches the first. Stiff, strong, tough and hard are four different questions, and brittle failure comes with no warning. Metals are weaker than their atoms promise because of dislocations, and every strengthening trick is a way to block them. A bent paperclip is harder at the bend: work hardening, then fatigue. Same alloy, different heat treatment or process, different part. "Lightest for a given stiffness" has an answer that depends on the shape of the job, and bulky, light materials win in bending. Two metals touching in water make a battery; the active one pays. Tooling cost against per-part cost decides how things are made. Where it shows up next Mechanics of Materials (34/11) uses E and yield strength as the inputs for sizing beams and shafts. Machine Design (34/19) is where you choose the steel for a gear or shaft and the grade of a bolt, using the heat-treatment and strength trade-offs from here. Propulsion (34/21): turbine blades live in gas hotter than their own melting point, using single-crystal alloys and cooling channels. Materials set the engine's temperature limit. Engineering Drawing & CAD (34/18): the drawing's material note and surface finish are how this course reaches the shop floor. In the real world: an aircraft's choice of aluminium or carbon, a car's crumple zone, a hip implant (titanium), the case of your phone. If you want to go deeper 📘 Standard texts: William Callister and David Rethwisch, Materials Science and Engineering: An Introduction (Wiley, 10th edition 2020), the most widely assigned survey. Michael Ashby, Materials Selection in Mechanical Design (Butterworth-Heinemann), for the property-chart method used above. For manufacturing, Kalpakjian and Schmid, Manufacturing Engineering and Technology (Pearson). 📘 Gentler: Mark Miodownik, Stuff Matters (2013), a prize-winning tour of everyday materials. J. E. Gordon, The New Science of Strong Materials, a classic with no equations required. 🌐 Free: MIT OpenCourseWare 3.091, Introduction to Solid-State Chemistry (bonding and crystals), and the University of Cambridge's DoITPoMS teaching packages, with micrographs and simulations. The takeaway Atoms are springs, so stiffness comes from the bonds and hardly changes with processing. Strength, hardness and toughness come from the microstructure (grains, dislocations, precipitates, trapped carbon), which heat treatment and manufacturing control. To choose a material, ask which property limits the part and use the right index, because the best material for a rod, a beam and a panel differ. And wherever metals meet water, expect a battery. Try this: bend a paperclip once, then bend the kink back and forth until it snaps, counting the bends. Feel the kink warm up. Then find two different metals touching in your home (a steel screw in a brass or aluminium fitting) and look for white powder or rust at the join. Sources & Further Reading Sources. Young's moduli: Wikipedia, "Young's modulus" (approximate values for relative comparison); aluminium and nylon properties from typical data sheets (MakeItFrom). Ideal shear strength about G/10 and the 1934 dislocation papers of Taylor, Orowan and Polanyi: standard textbook account (e.g. Wikipedia, "Dislocation"). Hall–Petch constants for low-carbon steel (σ₀ ≈ 100 MPa, k ≈ 600 MPa·µm½): Gotoh et al., Kanazawa University repository record on ECAP low-carbon steel. Tempering colours and uses: Wikipedia, "Tempering (metallurgy)". Performance indices √E/ρ, E/ρ, ∛E/ρ: Ashby, Materials Selection in Mechanical Design, summarised in NPTEL and Ansys teaching notes. Galvanic series: Wikipedia, "Galvanic series". Boeing 787 composite share: SAMPE/NASAMPE summary of Boeing figures. Numbers computed in node: bar stretch (0.5 mm), comparison stretches, energies to fracture, mass ratios, Hall–Petch values, and process switch points. 🌐 MIT OpenCourseWare: 3.091 Introduction to Solid-State Chemistry 🌐 DoITPoMS, University of Cambridge 📘 Mark Miodownik, Stuff Matters (2013) 📘 Callister and Rethwisch, Materials Science and Engineering: An Introduction Step 12. Physics II: Electricity & Magnetism — Charge, Fields and Light ----------------------------------------------------------------------- Why: Motors, sensors and actuators run on electromagnetism. A rubbed balloon beats the whole Earth's gravity, electrons crawl through your wiring slower than a snail, and light turns out to be a field vibrating. The one course that explains your phone, your motor and your Wi-Fi. Rub a balloon on your hair and stick it to the wall. It hangs there. Gravity, the pull of an entire planet, is trying to drop it, and a few billion stray electrons on its surface are winning. Between a proton and an electron the electric attraction is about 2.3 × 10³⁹ times stronger than their gravitational attraction. Gravity only seems strong because the Earth is enormous and its charge is balanced out. The second surprise is quieter. When you flip a switch, the light comes on at once, yet the electrons in the copper behind the wall drift at a fraction of a millimetre per second. What travels at nearly light speed is not the electrons. It is a field. Physics II is the course where those two sentences stop being puzzling. The library has an easier first read, How Electricity Reaches Your Wall (13/04), about the grid and Ohm's law. This piece goes further: into fields, capacitors, induction, and Maxwell's discovery that light is made of the same stuff. The course in one box Who takes it: second-term first-year or second-year students of Mechanical & Aerospace Engineering and, in many programmes, Computer Science. What it's for: predicting what charges and currents do, from a battery to a radio wave. Comes after: Physics I: Mechanics (34/06) and Multivariable Calculus (34/02). Leads to: the motors, sensors and actuators of Mechanical & Aerospace Engineering, then Control Systems (34/20); for Computer Science, Computer Architecture (34/25) and Computer Networks (34/27), which are electricity at tiny and large scales. Honest line: you won't design a power supply after this. You will know why a thin wire gets hot and a thick one doesn't, and why a transformer hums. Charge and the electric field Matter carries electric charge, positive or negative. Like charges push apart and opposite charges pull together, with a force that falls with the square of the distance, the same shape as gravity. A charge doesn't need to "reach" another: it fills the space around it with a field, and any other charge placed there feels it. E = F / q = k · Q / r²the field is the force a small test charge would feel, per unit of its charge (k ≈ 8.99 × 10⁹ N·m²/C²) Worked number: 1 nC (a billionth of a coulomb) makes a field of 900 V/m at 10 cm. Drag the gold probe below. Put a positive and a negative charge side by side and the field curves from one to the other. Interactive · Drag the probe through the field A positive and a negative Two positives One alone Left charge Right charge Charges are 30 cm apart; the grid shows field direction. Equal and opposite charges: the voltage along the midline between them is exactly 0. Voltage: energy per charge Lift a book and it gains energy; the same book on a higher shelf has more. Voltage is the electrical version of height: the energy a charge gains per coulomb moved. A 1.5 V battery gives every coulomb that crosses it 1.5 joules, whatever the battery's size. V = energy ÷ charge  ·  1 volt = 1 joule per coulomban electron falling through V volts gains eV of energy That makes electrons easy to launch. Accelerate one through 100 V and it leaves at about 5.9 × 10⁶ m/s, two per cent of the speed of light. At 10,000 V its speed is 0.195 c, and the ordinary formula (½mv²) overestimates it by about 1.5%. The old cathode-ray TV worked this way, aiming such electrons at a screen. Interactive · An electron falls through a battery Voltage it falls through Computed with the relativistic energy formula, and compared with the ordinary one. Current and resistance: the slow river Current is charge flowing past a point each second (1 ampere = 1 coulomb per second). Push a voltage across a wire and a current flows; the ratio is the resistance. For most metals it's constant, which is Ohm's law, and a wire's resistance grows with its length and shrinks with its thickness. Resistance turns electrical energy into heat at a rate P = I²R. V = I · R  ·  R = ρ · L / A  ·  P = I²Rρ is the material's resistivity: for copper about 1.68 × 10⁻⁸ Ω·m Now the surprise. The electrons' average drift speed is I ÷ (n·A·e), with n about 8.5 × 10²⁸ free electrons per cubic metre of copper. Ten amps through a 2.5 mm² house wire gives a drift of 0.29 mm/s. An electron takes nearly an hour to cover one metre. The lamp lights at once because the wire is already full of electrons and the field pushes them all together, like water in a pipe that is already full. Interactive · A copper wire under load Current Wire cross-section Wire length With those defaults the wire wastes 13 W as heat. Halve the thickness and the heat doubles. That is why extension cords are rated by thickness, and why power lines run at very high voltage: a given power needs a smaller current, and heat goes as current squared. Circuits: two rules Every circuit, however tangled, obeys two bookkeeping rules from Gustav Kirchhoff. The junction rule: current flowing into a junction equals current flowing out, because charge doesn't pile up. The loop rule: going around any closed loop, the voltage rises and drops add to zero, because energy is conserved. They are conservation of charge and of energy, again. From them follow the two shapes. Resistors in series share one current, so their resistances add. Resistors in parallel share one voltage, so their currents add, and the total resistance gets smaller as you add branches. That's why plugging more appliances into one socket pulls more current from the wall, not less. Interactive · Three resistors, a 12 V battery All in series All in parallel R1, then R2 ∥ R3 Battery R1 R2 R3 With 100, 200 and 300 Ω: in series the battery supplies 20 mA; in parallel the total resistance falls to 54.5 Ω and the current rises to 220 mA, eleven times as much. Capacitors and the RC clock A capacitor is two plates close together that store charge, and with it energy in the field between them. Its capacity C is how much charge it holds per volt. Connect one through a resistor to a battery and the plates fill, but ever more slowly, because the more it holds the less voltage is left to push more in. V(t) = V₀ (1 − e−t/RC)  ·  τ = R · Cthe time constant τ sets the pace; the exponential is the solution of Differential Equations (34/03) After one τ the capacitor is 63% full, after three τ 95%, after five 99%. A 10 kΩ resistor with a 100 µF capacitor makes τ = 1 second. Camera flashes, phone touch screens, power-supply smoothing and the blink of a turn signal all use this clock. Interactive · Charge it, discharge it Charging Discharging Resistor Capacitor Time elapsed A 5 V supply. Magnetic fields: the push on moving charge A current makes a magnetic field that circles the wire. Ten amps at 5 cm makes about 40 µT, the same size as Earth's own field (about 25 to 65 µT at the surface). A magnetic field pushes only on moving charge, and sideways: perpendicular to both the motion and the field. A sideways push with constant speed is the circular motion of Physics I. F = q · v · B  ⇒  r = m · v / (q · B)the radius is bigger for faster or heavier particles, smaller for stronger fields A wire carrying current in a field gets the same push: that is a motor. And a particle's lap time, 2πm/(qB), doesn't depend on its speed, which is how cyclotron accelerators work. Interactive · A charge in a magnetic field Electron Proton Speed Field Induction: change a field, get a voltage In 1831 Michael Faraday showed that a changing magnetic field through a loop of wire drives a voltage around it. A steady field does nothing; only change counts. Faster change, more loops, and a stronger field all give a bigger voltage. ε = − N · dΦ/dtvoltage = number of turns × how fast the magnetic flux changes; the minus sign says the induced current opposes the change This one rule explains three machines. Spin a coil in a field and you have a generator: the voltage rises and falls as a sine wave, peak N·B·A·ω. Run the same machine backwards and you have a motor. Wind two coils on one iron core and an alternating current in the first induces an alternating voltage in the second, with output voltage in the ratio of the turns: a transformer. It works only with alternating current, since only change induces, and it is why the grid uses AC and why a charger hums. Interactive · Generator, then transformer Turns in the coil Magnet field Spin speed Transformer input Output coil (input has 1,000) Coil area 10 cm × 10 cm. Ideal transformer, no losses. At the defaults the generator gives 31.4 V peak at 10 Hz, and the transformer steps 240 V down to 12 V: 20 times fewer turns, 20 times the current. Maxwell: four rules and the speed of light In the 1860s James Clerk Maxwell gathered everything known into four equations. In words: (1) electric charges make electric fields that spread outward; (2) there are no isolated magnetic charges, so magnetic field lines never end; (3) a changing magnetic field makes a circulating electric field (Faraday); (4) an electric current, and, Maxwell's own addition, a changing electric field, make a circulating magnetic field. Put (3) and (4) together and something remarkable happens: a changing electric field makes a changing magnetic field, which makes a changing electric field, and the disturbance can sustain itself and travel. Maxwell calculated its speed from two measured constants of electricity and magnetism, 1/√(μ₀ε₀), and got the measured speed of light, 299,792,458 m/s. In his 1865 paper he concluded that light is such a wave. Heinrich Hertz produced and detected radio waves in the late 1880s, and the rest of the spectrum followed. c = 1 / √(μ₀ ε₀) = 299,792,458 m/sthe speed of every electromagnetic wave in empty space Radio, microwaves, infrared, visible light and X-rays differ only in frequency, and the wavelength is c ÷ f. A 100 MHz FM station has a wavelength of 3 m; Wi-Fi at 2.4 GHz, 12.5 cm; red light at 700 nm, 430 THz. This also explains antenna sizes: a quarter wavelength works well, so an FM antenna is about 75 cm. Interactive · Slide along the spectrum Mains 60 Hz AM radio FM radio Wi‑Fi Red light Frequency (log scale) What a graduate just sees Electric force is absurdly strong; things look neutral because charges balance. A voltage is energy per charge; a current is charge per second; power is their product. A thin wire heats up because P = I²R, and that's why the grid uses high voltage. Electrons drift slowly; the signal is the field, and it moves near light speed. Series adds resistance, parallel adds current. Any capacitor and resistor make a clock, and 63% in one τ is the number to remember. Magnetism acts only on moving charge, and always sideways. Only change induces: no change in the field, no voltage. So transformers need AC. Light, radio and X-rays are the same wave at different frequencies. Where it shows up next Motors, generators and sensors in Mechanical & Aerospace Engineering are the F = qvB and induction sections above; Control Systems (34/20) then closes the loop around them. Vibrations (34/17) and Differential Equations (34/03): the RC curve is the same equation as a cooling cup of coffee, and an inductor with a capacitor oscillates like a mass on a spring. Computer Architecture (34/25): a logic gate is a switch, and the delay of a wire is an RC time constant, which sets how fast a chip can be clocked. Computer Networks (34/27): Wi-Fi and cellular are Maxwell's waves with data on them; the wavelength rule sets antenna size. Thermodynamics (34/14) picks up the I²R heat; a chip's heat is why laptops have fans. If you want to go deeper 📘 Standard textbooks. Halliday, Resnick & Walker, Fundamentals of Physics (12th ed.) or Young & Freedman, University Physics (16th ed.), the usual engineering-year texts. Purcell & Morin, Electricity and Magnetism (3rd ed., Cambridge, SI units) is the elegant one for stronger students. 📘 Gentler, free. The Feynman Lectures on Physics, Volume II, free at feynmanlectures.caltech.edu. 🌐 Free course. MIT OpenCourseWare 8.02 Physics II: Electricity and Magnetism (Spring 2007), with lecture notes and problem sets. The takeaway Charges make fields; a voltage is energy per charge; a current is charge per second, resisted into heat. Two rules, conservation of charge and of energy, solve any circuit. A changing magnetic field makes a voltage, which gives generators, motors and transformers, and Maxwell showed that a changing electric field and a changing magnetic field can sustain each other as a wave travelling at the speed of light. That wave is light. Try this: look at the charger of a laptop or phone. Find the voltage and amperes printed on it, multiply them, and you have the power it can deliver. Then feel how warm the cable and the brick get, and work out which part is I²R. Sources & Further Reading Sources. Constants from the NIST CODATA 2022 listing (elementary charge 1.602 176 634 × 10⁻¹⁹ C and speed of light 299 792 458 m/s are exact; electron and proton masses; G; μ₀; ε₀). Copper's resistivity (1.68 × 10⁻⁸ Ω·m at 20 °C) from the tabulation in Wikipedia's "Electrical resistivity and conductivity"; free-electron density 8.5 × 10²⁸ m⁻³ and the drift-velocity worked example from Wikipedia's "Drift velocity". Earth's field 25–65 µT: Wikipedia's "Earth's magnetic field". Faraday's 1831 demonstration: Wikipedia's "Faraday's law of induction" (Joseph Henry independently, 1832). Maxwell's paper read to the Royal Society 8 December 1864 and published 1865: Wikipedia's "A Dynamical Theory of the Electromagnetic Field". Hertz produced and detected radio waves in 1886 and published his evidence in 1887–88: Wikipedia's "Heinrich Hertz". Computed in node and checked: electric to gravitational force ratio 2.27 × 10³⁹; drift speed 0.294 mm/s and 13.4 W; electron speeds (5.93 × 10⁶ m/s at 100 V; 0.195 c at 10 kV); 54.5 Ω and 220 mA; 63.2%, 95.0%, 99.3% RC fractions; electron orbit radius 5.7 cm and period 35.7 ns at 10⁷ m/s and 1 mT; 31.4 V peak; 1/√(μ₀ε₀) = 299,792,458 m/s to 8 digits. 🌐 MIT OpenCourseWare: 8.02 Physics II, Spring 2007 📘 Purcell & Morin, Electricity and Magnetism, 3rd ed. (Cambridge University Press, 2013) 📘 The Feynman Lectures on Physics, Vol. II (free online, feynmanlectures.caltech.edu) 🌐 NIST: fundamental physical constants (CODATA 2022) Step 13. Thermodynamics: Why Every Engine Pays a Heat Tax --------------------------------------------------------- Why: Energy is conserved and always degrades; every engine lives inside those two laws. Energy is never lost, but it always degrades. Every engine, fridge and jet lives between those two laws, and two temperatures are enough to say how well it could ever do. Pump up a bicycle tyre and the barrel of the pump gets warm. Nobody heated it. You only pushed. Leave a phone charging under a pillow and it gets hot, because every bit of electrical energy that is not stored in the battery has to go somewhere, and "somewhere" is heat. Stand next to a power station and you will see a cooling tower pouring white steam into the sky: a machine built to make electricity that spends most of its effort throwing heat away on purpose. A mechanical or aerospace engineer looks at all three and sees the same two rules. Energy is never lost, only moved around. And when it is moved, some of it always ends up in a form that is hard to use again. Those two rules, written carefully, are thermodynamics. Every engine, every refrigerator and every jet is a way of living inside them. The course in one box Who takes itMechanical, aerospace, chemical and nuclear engineers, usually in year 3. Physicists and chemists meet a different version of it. What it is forPredicting how much work you can get out of heat, or how much work it takes to move heat, before you build anything. Comes afterCalculus I (34/01) and Physics I: Mechanics (34/06). Leads toFluid Mechanics (34/15), Heat Transfer (34/16) and Propulsion (34/21). What this piece gives youYou will not be able to size a power plant. You will see why no engine can ever be fully efficient, and you will be able to estimate the best any engine could do from two temperatures. Easier first reads in the library: Energy: The One Currency of the Universe (13/02) and Entropy: Why You Can't Unstir Your Coffee (13/12). This one goes further and puts numbers on the engines. Draw the box first: systems, properties and the ideal gas Every thermodynamics problem starts with a drawing: a dashed line around the thing you care about. That is the system. Everything outside is the surroundings. If nothing crosses the line, it is a closed system, like gas sealed in a cylinder. If matter flows through, it is a control volume, like a jet engine: air in at the front, hot gas out at the back. Most of the later subjects, from fluids to propulsion, are control-volume thinking. Draw the box, then count what crosses it. Inside the box we describe the stuff by properties: pressure, temperature, volume, internal energy. A property depends only on the state the stuff is in, not on how it got there. For air at ordinary conditions, the properties are tied together by the ideal gas law: p V = n R Tpressure × volume = amount of gas × 8.314 × absolute temperature (kelvin) A worked number: one litre of air at 100 kPa and 300 K is about 0.040 mole, roughly 1.2 grams. Now squeeze it to a tenth of the volume. What happens depends on how fast. Squeeze slowly and the walls carry the heat away, so the temperature stays 300 K and the pressure rises tenfold. Squeeze fast and the heat has nowhere to go, so the work you put in becomes temperature. Try both. Interactive · Squeeze a litre of air Slowly (heat escapes) Fast (no time for heat to escape) Compression ratio Air as an ideal gas with γ = 1.4, starting at 100 kPa, 300 K, 1 L. Slow: p₂ = p₁r, T₂ = T₁. Fast: p₂ = p₁r1.4, T₂ = T₁r0.4. At 10 to 1 and fast, the air reaches about 754 K, near 480 °C, and about 2.5 MPa. No spark, just squeezing. That is the whole idea of a diesel engine: compress air so hard that it is hot enough to light fuel sprayed into it. Real diesels typically compress about 15 to 20 to 1. The zeroth and third laws: the bookends The laws are numbered in the order people needed them, not the order they make sense in. The zeroth law was named last, after the first two already held their numbers. It says: if A is in thermal equilibrium with C, and B is in thermal equilibrium with C, then A and B are in equilibrium with each other. It sounds trivial. It is what makes a thermometer legitimate. C is the thermometer: it can compare two things that never touch. At the other end, the third law says you cannot cool anything all the way to absolute zero (0 K, −273.15 °C) in a finite number of steps. Each step removes less heat than the one before. It also says a perfect crystal at 0 K has zero entropy, which gives entropy a true zero to measure from. You will rarely compute with it. Cryogenic engineers live with it. The first law: energy is a bank balance Think of the energy inside your box as a bank balance. Two things can change it. Heat (Q) is energy that crosses the line because of a temperature difference. Work (W) is energy that crosses the line because of a force moving something, like a piston. There is no other way in or out of a closed system. So: ΔU = Q − Wchange in stored energy = heat added − work done by the gas Sign conventions differ between textbooks, so watch the first page of yours. Here, heat in is positive and work done by the gas is positive. The bank-balance picture also explains why "heat" and "work" are not things you hold. You cannot own heat; you can only watch it cross the line. Only the balance, the internal energy, is a property of the gas. Worked number: air has a specific heat at constant volume of about 0.718 kJ per kilogram per kelvin, so putting 50 kJ of heat into 1 kg in a sealed rigid can (no work) raises it by 50 ÷ 0.718 ≈ 70 K. Now let the same heat do something else. Interactive · The energy ledger Pump it, no heat escapes Heat a sealed can Heat it and let it push a piston Expand fast, no heat in Heat added to 1 kg of air Work done by the air ΔU = Q − W and ΔU = m·cv·ΔT with cv = 0.718 kJ/kg·K, starting at 300 K. Negative work means the surroundings did work on the gas. Notice what the third preset shows: 50 kJ of heat goes in, 50 kJ of work comes out, and the temperature does not change. That is the dream of an engine, and it only works for one expansion. An engine has to come back to its start and repeat. That is where the next law bites. The second law: heat only rolls downhill, and entropy counts the cost The first law allows a cup of coffee to cool by pushing its heat into the room, and it equally allows the room to push heat back into the coffee and warm it. Both conserve energy. Only one ever happens. The second law is the rule that picks the direction. Clausius's version: heat does not flow by itself from a colder body to a hotter one. Kelvin's version: no cyclic machine can turn heat from a single temperature entirely into work. They are the same law in different clothes. Engineers make this countable with entropy. When heat Q enters a body at absolute temperature T, its entropy changes by Q ÷ T. The same joule of heat adds more entropy when it enters something cold than when it leaves something hot, because dividing by a smaller T gives a bigger number. So heat that falls from hot to cold creates entropy overall, and heat that climbs from cold to hot would destroy it. The second law says total entropy never decreases. ΔS = Q ÷ Tentropy change of a body = heat in ÷ its absolute temperature · total ΔS ≥ 0 Interactive · Which way can the heat go? Hot → cold Cold → hot (try it) Hot body Cold body Heat moved At 400 K and 300 K, moving 1,000 J from hot to cold gives −2.5 J/K for the hot body and +3.33 J/K for the cold one: a net +0.83 J/K. Run it backwards and the total is −0.83 J/K, which never happens by itself. A refrigerator does move heat uphill, but only by being paid in work, and its exhaust heat creates more entropy than it removed. Energy is conserved. Entropy is not: it is made every time anything happens, and it can never be un-made. Carnot: the ceiling over every engine Now combine the two laws. A heat engine takes heat from a hot source, turns some into work, and must dump the rest into a cold sink. It has to dump some: if it took all its heat from the hot source and rejected none, the entropy of the world would fall. The least heat it can reject is the amount that keeps total entropy from decreasing. Working that out gives the highest efficiency any engine can have, discovered by the French engineer Sadi Carnot in 1824 (before the first law was even clear): ηmax = 1 − Tcold ÷ Thottemperatures in kelvin. This is the ceiling, not what real engines achieve. It depends only on the two temperatures, not on the fluid, the pistons or the cleverness. Two consequences stay with every engineer. To get more out, run hotter or reject colder; that is why turbine blades are cast to survive white-hot gas. And no engine reaches 100%, because the cold sink is never at absolute zero. Interactive · The Carnot ceiling Steam plant: 540 °C, river at 20 °C Geothermal: 150 °C Ocean thermal: warm 25 °C, deep 5 °C Hot side Cold side The steam plant's ceiling is 63.9%. Real steam plants fall well short of it, commonly around a third to 40% of the fuel's heat becoming electricity. The best gas-and-steam combined-cycle plants now reach about 64%: Siemens Energy reported 64.18% for the Keadby 2 plant in the UK, verified in May 2024. The ocean-thermal case shows the other side. A warm-water-over-cold-water engine in the tropics has a ceiling under 7%, which is why it is hard to make economical, even before friction. Cycles that run the world An engine must return to where it started, so that it can go round again. That loop of states is a cycle, and the four you will meet in every course have simple names for what the gas does. Otto (petrol engine): compress air and fuel, spark, expand, exhaust. Diesel: compress air alone until hot, inject fuel which ignites by itself, expand, exhaust. Brayton (gas turbines and jets): compress air continuously, burn fuel in a flowing stream, expand through a turbine. Rankine (steam and most of the world's electricity): pump water to high pressure, boil it, expand the steam through a turbine, condense it. For the ideal, air-standard versions of the first three, the efficiency depends on one ratio. For Otto it is 1 − r−0.4, where r is the compression ratio. At 10 to 1 that is 60%, well above what a real engine delivers, because real engines have friction, heat loss and incomplete combustion. The shape is the lesson: a higher ratio gives a higher efficiency, with diminishing returns. Petrol engines are limited by "knock" (fuel igniting early), which is part of why diesels, which compress only air, can use higher ratios. Interactive · Ideal cycle efficiency Piston engines: compression ratio Diesel: cutoff ratio (how long fuel burns) Gas turbine: pressure ratio Ideal air-standard cycles with γ = 1.4. Real engines come out lower. The Brayton cycle is the one to remember for later: a jet engine is a Brayton cycle laid out in a tube, with a compressor, a combustor and a turbine in a line. At a pressure ratio of 30 the ideal efficiency is 62%; at 10 it is 48%. That is why modern engines push their pressure ratios so high, and it is exactly the story told in Propulsion (34/21). Run it backwards: refrigerators, heat pumps and COP Run an engine in reverse, put work in, and it pumps heat from cold to hot. That is a refrigerator (the useful part is the heat taken from the cold side) or a heat pump (the useful part is the heat delivered to the hot side). Nothing breaks the second law, because you are paying work to climb the hill. For these machines "efficiency" is the wrong word, since the answer can be more than 1. Engineers use the coefficient of performance, COP: useful heat moved per unit of work paid. COPheating, ideal = Thot ÷ (Thot − Tcold)temperatures in kelvin. The smaller the lift, the bigger the COP. A resistance heater turns 1 kWh of electricity into 1 kWh of heat. A heat pump delivering heat at 35 °C from a 5 °C outdoors has an ideal COP of 10.3: it moves ten times more heat than the work it uses, because it is mostly moving heat, not making it. Real machines fall short of the ideal. The slider's 45% is an assumption for illustration, not a measurement; set your own. Interactive · A heat pump in winter Outdoor temperature Temperature of the delivered heat Fraction of the ideal that the machine achieves Two things to see. A colder day lowers the COP. And so does asking for hotter delivery: a heat pump feeding low-temperature floor heating at 35 °C beats one feeding old radiators at 55 °C, for the same weather. Everything is about the lift between the two temperatures. What a graduate just sees A system is a dashed line: first decide what is inside and count what crosses it. Compressing a gas fast heats it; expanding it fast cools it. Work and heat are two ways of moving the same energy. "Heat" is energy in transit. Things have internal energy, not heat. Energy is always conserved, so the question is never "where did it go" but "what form is it in now, and is it still useful". Efficiency has a ceiling set by two temperatures: 1 − Tcold/Thot. Running hotter is how engines improve. Every real process makes entropy. Friction, mixing and heat falling down a gradient all spend usable energy. A power plant's cooling tower is not waste by sloppiness. It is the second law's bill. A heat pump or a fridge looks like it breaks the rules, but it is moving heat, not creating it, and it is paying for it. Where it shows up next Fluid Mechanics (34/15): the control volume and the steady-flow energy equation become the frame for every pipe, pump and nozzle. Heat Transfer (34/16): thermodynamics says heat can flow from hot to cold; heat transfer says how fast through a wall, a fin or a vacuum. Propulsion (34/21): a jet is a Brayton cycle in a tube, and a rocket is a heat engine whose "work" is kinetic energy in the exhaust. In the world: power stations (Rankine), cars and aircraft (Otto, Diesel, Brayton), air conditioners, freezers, heat pumps in homes, and the energy accounting for any battery or fuel debate. If you want to go deeper 📘 Textbook. Yunus Çengel, Michael Boles and Mehmet Kanoğlu, Thermodynamics: An Engineering Approach (McGraw Hill, 10th edition). Widely assigned and full of worked examples and property tables. 📘 Textbook. Moran, Shapiro, Boettner and Bailey, Fundamentals of Engineering Thermodynamics (Wiley). The other standard choice, more careful on the first-principles setup. 📘 Gentler. Peter Atkins, Four Laws That Drive the Universe (Oxford, 2007; reissued in 2010 as The Laws of Thermodynamics: A Very Short Introduction). Almost no maths. 🌐 Free course. MIT OpenCourseWare 16.050 Thermal Energy (Fall 2002): basic concepts, the second law, propulsion and power cycles, and heat transfer in one aerospace course. Also the thermodynamics half of 16.01 Unified Engineering. The takeaway Draw a box around what you care about and count what crosses it. The first law says the energy balances; the second says some of it must always end up as low-grade heat, and entropy is how that is counted. The result is a ceiling, 1 − Tcold/Thot, that no engine can beat, and a trick, the heat pump, that beats 100% only because it moves heat rather than makes it. Try this: this week, feel for heat as the bill. Touch the back of a fridge, a phone charger, a laptop. Each one is a machine paying the second law, and the warmth is what it cost. Sources & Further Reading Sources. The laws, the ideal gas, air-standard cycles (Otto, Diesel, Brayton) and heat pump COP are standard textbook thermodynamics (Çengel & Boles; Moran & Shapiro). Air cv = 0.718 kJ/kg·K and γ = 1.4 are standard room-temperature values from the textbook property tables. Carnot's 1824 Reflections on the Motive Power of Fire is the standard origin of the ceiling. Combined-cycle record: Siemens Energy, Keadby 2 plant, 64.18% verified May 2024 (as reported by Guinness World Records and trade press; gross, net and ISO-corrected figures are not strictly comparable). Course contents: MIT OCW 16.050 and 16.01 pages. Every number an interactive shows was computed in node: 1 L at 100 kPa, 300 K = 0.0401 mol; 10:1 adiabatic gives 2,512 kPa and 753.6 K; 10:1 isothermal work = p₁V₁ ln 10 = 230 J, adiabatic work = 378 J; Carnot 540/20 °C = 63.9%, 150/20 °C = 30.7%, 25/5 °C = 6.7%; Otto at 10 = 60.2%; Brayton at pressure ratio 30 = 62.2%, at 10 = 48.2%; heat pump 35 °C from 5 °C = 10.27 ideal. 🌐 MIT OCW 16.050 Thermal Energy 🌐 MIT OCW 16.01 Unified Engineering: Thermodynamics and Propulsion 📘 Çengel, Boles & Kanoğlu, Thermodynamics: An Engineering Approach 📘 Atkins, Four Laws That Drive the Universe Step 14. Fluid Mechanics: Why Air and Water Push Back ----------------------------------------------------- Why: Air and water push, drag and lift; the Navier–Stokes equations are the rulebook. Water presses, air lifts, and the equations that govern both are worth a million dollars to anyone who can prove they behave. What a graduate still sees in every flow. Stand at the bottom of a swimming pool's deep end and your ears tell you something about the water above. Watch a heavy airliner climb off a runway. Tilt your hand out of a moving car's window and feel it rise. Squeeze a garden hose and watch the jet go further. All of these are one subject: a fluid, which is just something that flows, pushing on things. The strange part is that the full rulebook, the Navier–Stokes equations, has been known for about 180 years, fits on one line, and nobody can prove that it always behaves. Engineers use it anyway, every day. A fluid mechanics graduate learns which simple pictures work (and when), and which ones are traps. The course in one box Who takes it: mechanical, aerospace, civil and chemical engineers, usually in year 3. What it's for: predicting the forces, flows and losses when air or water moves past, through or around something. Comes after: Multivariable Calculus (34/02), Differential Equations (34/03), Thermodynamics (34/14). Leads to: Heat Transfer (34/16), Propulsion (34/21), and everything that flies or pumps. It also meets Vibrations (34/17) in wind-driven failures and Control Systems (34/20) in flight control. Honest line: you won't be able to size a pump or design a wing after this. You will see why a dam is thick at the bottom, why a wing needs speed, and why the Navier–Stokes problem is worth a million dollars. Pressure: the weight of everything above you Pressure is force per area. In a still fluid it comes from one thing: the weight of the column of fluid above. Dive down and you carry more water, so pressure rises in a straight line with depth. It pushes equally in every direction, which is why a diver's ears feel it from both sides. p = ρ · g · hpressure (extra, above the air) = density × gravity × depth For water that is about 9.8 kilopascals per metre, so 10 m down you carry about one extra atmosphere. Now the dam. The pressure grows with depth, so the push on the wall is a triangle, tiny at the top and largest at the bottom. Add up the triangle and the total push per metre of wall is ½·ρ·g·H². Double the water depth and the push quadruples. That is why a dam is thick at the bottom, and why a reservoir twice as deep is a very different structure, not a slightly bigger one. Interactive · The dam Water depth behind the wall At 10 m deep: base pressure 98 kPa, total push 491 kN per metre of wall. Fresh water, 1,000 kg/m³, g = 9.81 m/s². Try 10 m, then 20 m: the push goes from 491 to 1,962 kN per metre. Four times, not two. Buoyancy: the weight of the water you pushed aside The pressure under an object is bigger than the pressure on top, because it is deeper. That difference is a net upward push. Archimedes' principle says exactly how big: the weight of the fluid the object displaced. A block floats when it can displace its own weight before it is fully under. So it floats low or high depending on one ratio, its density against the fluid's. A steel ship is the proof that shape matters: steel is eight times denser than water, but a hull encloses so much air that the average density of the whole ship is below water's. An iceberg is ice (about 917 kg/m³) in seawater (about 1,025): 917 ÷ 1,025 = 0.895, so about 89% of it is below the surface. Interactive · Float or sink Fresh waterSeawaterMercury Density of the block Set to ice (917) A 600 kg/m³ block floats with 60% below the surface. Rough values: fresh water 1,000, seawater 1,025, mercury 13,534 kg/m³. Try a 7,800 block (steel) in mercury: it floats. Continuity and Bernoulli: faster flow, lower pressure Mass can't vanish. If water flows through a pipe that narrows, the same amount must pass each cross-section every second, so it must speed up. That is the continuity equation: area times speed stays constant. Squeeze the hose end with your thumb and the jet speeds up by exactly the area ratio. A₁v₁ = A₂v₂area × speed = the same all along the pipe Now the clever step. Speeding water up takes energy, and in smooth flow that energy comes from pressure. Daniel Bernoulli's relation is just energy conservation along a streamline: pressure plus the kinetic energy per volume plus the height term stays constant. p + ½ρv² + ρgh = constantpressure + motion energy + height energy, per cubic metre So where the water speeds up, the pressure drops. Worked example: a 10 cm pipe narrows to 5 cm. The area ratio is 4, so water at 2 m/s becomes 8 m/s, and the pressure falls by ½·1,000·(8² − 2²) = 30,000 Pa = 30 kPa. A carburettor, a perfume spray and a Venturi flow meter all run on this. Where it breaks. Bernoulli assumes steady flow, no friction, and no change in density. Friction in a long pipe steals pressure. Air above roughly Mach 0.3 (about a third of the speed of sound) starts to compress. And if the pressure falls far enough, water boils at room temperature (cavitation), which wrecks propellers and pumps. Interactive · The Venturi Speed in the wide pipe (10 cm) Throat diameter Pipe 10 cm to 5 cm at 2 m/s: 8 m/s in the throat, pressure down by 30 kPa. Water, 1,000 kg/m³, ideal flow, 150 kPa (absolute) before the throat. Squeeze the throat hard enough and the pressure falls to the boiling pressure of water. Lift: why "equal transit time" is wrong You may have been told that air going over the curved top of a wing has further to go, so it must go faster to "catch up" with air going underneath, and Bernoulli does the rest. The last step is right: the pressure on top is lower. The first steps are wrong. There is no rule that two neighbouring bits of air must meet again at the back edge, and they don't: in measurements and simulations the air over the top gets there first. Worse, the "longer path" story predicts far too little lift, and it can't explain a flat plate, a symmetric wing or a plane flying upside down, all of which fly. NASA Glenn's education pages call it "one of the most widely circulated, incorrect explanations". The better picture: a wing, tilted into the wind (the angle of attack), turns the air downward. Pushing air down pushes the wing up, which is Newton's third law. The pressure difference between top and bottom is how that push reaches the wing's skin, so Bernoulli and Newton are two bookkeepings of one event, not rivals. Why does the air go faster on top? Because the wing sets up a circulation of flow around itself, fixed by the sharp trailing edge, not by any race. L = ½ · ρ · v² · S · CLlift = dynamic pressure × wing area × lift coefficient For small angles the coefficient CL grows in proportion to the angle, until the flow can no longer follow the top of the wing. Then it separates and lift collapses: a stall. Worked case: a 1,100 kg light plane with 16 m² of wing at 50 m/s (97 knots) needs CL = 0.44, about 5° above its zero-lift angle. Slow to 30 m/s and the same plane needs 14°, almost at the stall. This is why landing speeds are what they are. Interactive · Hold the plane up Airspeed Angle of attack (from zero lift) Find the level-flight angle At 50 m/s and 5.1° the lift of 10.8 kN balances the weight of 10.8 kN. The stall is at about 28 m/s. A model, not a real aircraft: wing area 16 m², aspect ratio 7.5, sea-level air, lift slope about 5 per radian (thin-airfoil theory with a finite-wing correction), stall assumed at 16°. Viscosity and the Reynolds number: sticky against sluggish Viscosity is a fluid's internal stickiness: how hard layers resist sliding over each other. Honey is far more viscous than water; water is about 55 times more viscous than air. Whenever a fluid moves, two things compete. Inertia is the tendency of the moving fluid to keep going, and viscosity is the tendency to smooth the motion out. The ratio of the two is one number. Re = ρ · v · L / μReynolds number = density × speed × size, divided by viscosity Osborne Reynolds showed in 1883, by injecting a thread of dye into water flowing in a glass pipe, that this one number decides how a flow behaves. Low Re: the dye thread stays a straight line (laminar flow). High Re: it breaks into chaotic swirls (turbulent flow). In a pipe the change happens roughly between Re 2,000 and 4,000; it is a range, not a switch. The same number explains why a swimming bacterium can't coast. For a 2 micrometre bacterium swimming at 30 micrometres a second, Re is about 6 × 10⁻⁵, so the water behaves like thick syrup and it stops the instant it stops pushing (Purcell's classic 1977 essay "Life at low Reynolds number"). A human swimmer is near 4 million. Interactive · What is the Reynolds number of...? BacteriumSlow water in a 2 cm pipeSame pipe, 10× fasterWalking personThrown baseballSwimmer WaterAir Size L Speed v Water at 0.1 m/s in a 2 cm pipe: Re ≈ 2,000, right at the edge of laminar flow. Ten times faster is about 20,000: turbulent. Water 20 °C (998 kg/m³, 1.0 mPa·s); air 20 °C (1.2 kg/m³, 0.018 mPa·s). The labels follow pipe flow; around other bodies the thresholds differ. Boundary layers and drag: the thin skin that decides everything At the surface of any solid, fluid sticks to it (the no-slip condition). So speed rises from zero at the wall to the full stream speed over a thin skin called the boundary layer. On a flat plate, 0.3 m from the front edge in a 10 m/s breeze, that skin is only about 3 mm thick. Almost all the friction, and a lot of the drama, lives there. Drag comes in two kinds. Friction drag is the skin rubbing. Pressure drag appears when the boundary layer peels off the body behind its widest point (separation), leaving a low-pressure wake. Streamlining means keeping the flow attached so the wake stays thin. The whole story fits one formula. F = ½ · ρ · v² · Cd · Adrag = dynamic pressure × drag coefficient × frontal area Because it goes with the square of speed, the power to overcome it (force times speed) goes with the cube. A car with Cd = 0.30 and 2.2 m² of frontal area needs about 8.5 kW to push air aside at 100 km/h and 18.7 kW at 130 km/h: 2.2 times the power for 30% more speed. A golf ball shows the other trick. Its dimples trip the boundary layer into turbulence early; a turbulent layer clings to the ball longer, the wake shrinks, and drag falls (measured drops of more than a factor of two are reported for dimpled over smooth spheres at the same Re). A rougher surface can mean less drag. Interactive · Drag, force and power Flat plate face-on (1.2)Sphere (0.47)Car (0.30)Streamlined teardrop (0.05) Speed Car at 100 km/h: drag about 307 N, about 8.5 kW. At 30% more speed: 1.69× the force, 2.20× the power. Frontal area 2.2 m², air 1.204 kg/m³. The coefficients are typical round values and vary with the exact shape and with Re. Navier–Stokes: Newton's second law for a blob of fluid Take a tiny parcel of fluid and apply F = ma. That's it. Claude-Louis Navier wrote a version with viscosity in 1822, and George Stokes re-derived it on firmer ground in 1845. For water or slow air (incompressible flow), in words and then in symbols: ρ (∂u/∂t + u·∇u) = −∇p + μ∇²u + ρgmass × acceleration = pressure push + viscous smearing + gravity ρ(∂u/∂t + u·∇u): mass times acceleration. The parcel accelerates for two reasons: the flow at its spot changes in time (∂u/∂t), and it is carried to a place where the flow is different (u·∇u). −∇p: the parcel is pushed from high pressure toward low. μ∇²u: viscosity averages the velocity with the neighbours, so fast fluid drags slow fluid along. It smooths and dissipates, turning motion into heat. ρg: gravity (or any body force). Plus a second rule, ∇·u = 0, which is continuity. The troublemaker is u·∇u: the velocity appears multiplied by its own slope. This nonlinear term lets big eddies feed small ones, and it is why turbulence is so hard. Its job shows up already in a one-dimensional cousin, Burgers' equation, below. Try it: the nonlinear term steepens a smooth wave into a front, and viscosity decides how sharp the front can get. Interactive · Steepening against smoothing Viscosity ν Time Burgers' equation (fast parts catch slow parts, viscosity smooths). Without viscosity the front would become infinitely sharp at t ≈ 0.16. Computed live (finite differences, 400 points) and checked against the exact solution: errors under 1%. It is a 1-D toy, with no pressure and no turbulence. The million-dollar question. In 2000 the Clay Mathematics Institute named Navier–Stokes as one of seven Millennium Prize Problems, with $1,000,000 for each. The question is not whether the equations are useful (they are, endlessly). It is whether, in three dimensions, smooth starting flows always stay smooth, or whether a solution can ever "blow up" into infinite velocities in finite time. Nobody knows. As of this writing no proof has been accepted by the Clay Institute; claims appear regularly and are checked slowly. Dimensional analysis, wind tunnels and CFD You can't solve Navier–Stokes by hand for a car. So engineers do two things. First, test a small model. The trick, from dimensional analysis, is that two flows behave alike when their dimensionless numbers match, mostly the Reynolds number. A 1:10 model car must therefore run ten times faster in air, which would be 300 m/s, almost Mach 0.9: not the same physics at all. The fix is to change the fluid: water has about 15 times less kinematic viscosity than air, so the model only needs about 20 m/s underwater. Interactive · The scale model Air tunnelWater tunnel Model scale Real car at 30 m/s; a 1:10 model in water must run at 19.9 m/s to match the Reynolds number. Kinematic viscosity: air 1.516 × 10⁻⁵ m²/s, water 1.004 × 10⁻⁶ m²/s (20 °C). Speed of sound: air 343 m/s, water 1,480 m/s. Second, computational fluid dynamics (CFD): slice space into millions of cells and march the equations forward on a computer. Resolving every eddy (direct numerical simulation) costs roughly the cube of the Reynolds number, which is why aircraft and cars are simulated with turbulence models that average the small eddies away. CFD is numerical analysis and parallel computing as much as fluid mechanics, and it is where Computer Science meets this course. Modern work also trains machine-learning models on CFD results (Machine Learning, 34/32) to get approximate answers fast. In practice teams use CFD, tunnel tests and flight tests together, because each checks the others. What a graduate just sees Pressure is weight above you, so deep water pushes hard: depth, not volume, loads a dam. Squeeze a flow and it speeds up and its pressure drops. Check whether the "ideal" assumption still holds. A wing works by turning air downwards. "Equal transit time" is a myth. Lift and drag grow with speed squared; the power to overcome drag grows with speed cubed. Before anything else, ask for the Reynolds number: it tells you whether viscosity or inertia is in charge. Thin boundary layers carry the friction, and separation of the layer causes most drag. A rougher surface can have less drag (golf ball), and slower water can match faster air (model testing). The Navier–Stokes equations are Newton's second law with pressure, viscosity and gravity, and nobody can yet prove they always stay smooth. Where it shows up next Heat Transfer (34/16): a moving fluid carries heat. The boundary layer you met here is also the thermal layer that decides how fast a radiator or a chip cools. Propulsion (34/21): jets and rockets are pipes, nozzles and compressors where Bernoulli, compressibility and thermodynamics meet. Vibrations (34/17) and Control Systems (34/20): wind makes structures flutter, and a flight controller exists to keep a wing at the right angle of attack. The real world: aircraft and car design, wind turbines, pumps, pipelines, blood-flow models, weather forecasting. Computer Science: CFD codes are large parallel programs, and fluid solvers drive the water in films and games. If you want to go deeper 📘 Textbook: Frank M. White, Fluid Mechanics (McGraw-Hill): a standard undergraduate text, strong on worked examples. 📘 Textbook: Kundu, Cohen and Dowling, Fluid Mechanics (Academic Press; 7th edition, 2024): broader, with more on the mathematics and CFD. 📘 Gentler: Steven Vogel, Life in Moving Fluids (Princeton University Press): fluid mechanics through animals and plants, nearly no equations. ▶️ Free course: MIT OpenCourseWare 2.06 Fluid Dynamics (Spring 2013), an undergraduate course with problem sets and exams. 🌐 Easier first read in this library: Flying Machines (29/05) and Chaos Theory and the Butterfly Effect (12/07). The takeaway A fluid is a pile of tiny parcels obeying F = ma with three forces: pressure, viscosity and gravity. Pressure is the weight above, so dams are thick at the bottom. Continuity and Bernoulli say squeezed flow speeds up and loses pressure. Lift comes from turning air downward, not from equal transit times. The Reynolds number says whether a flow is smooth or turbulent, and drag grows with the square of speed. The full equations are the best rulebook we have, and whether they always behave is still an open million-dollar question. Try this: next time you drive or ride, notice how much harder the engine works at motorway speed than at 50 km/h. That is the cube law. Then hold a spoon's back under a thin tap stream and feel the water pulled around the curve. Sources & Further Reading Sources. Hydrostatics, continuity, Bernoulli, Navier–Stokes and dimensional analysis are standard textbook material (White; Kundu et al.). Navier 1822 and Stokes 1845: the 2024 review "200 Years of the Navier–Stokes Equation" (arXiv 2401.13669). Millennium Prize: Clay Mathematics Institute (claymath.org), official problem by C. Fefferman. Equal-transit myth and the downward-turning explanation: NASA Glenn Research Center, "Incorrect Lift Theory" and Tom Benson's Beginner's Guide to Aeronautics. Reynolds's dye experiment (1883) and the pipe transition range of about 2,000–4,000. Low-Re swimming: E. M. Purcell, "Life at low Reynolds number", American Journal of Physics 45 (1977). Golf-ball drag reduction: Balaras et al., APS Division of Fluid Dynamics 2016. Computed in node: pressure, force, Venturi, lift, Reynolds numbers, drag power and scale-model speeds; the Burgers solver was checked against the exact Cole–Hopf solution. 🌐 MIT OpenCourseWare 2.06 Fluid Dynamics 🌐 Clay Mathematics Institute: Navier–Stokes equation 🌐 Purcell, Life at Low Reynolds Number (a reprint) 📘 Steven Vogel, Life in Moving Fluids Step 15. Heat Transfer: Three Roads for Heat -------------------------------------------- Why: Conduction, convection, radiation: how heat escapes, and how to make it. Conduction, convection and radiation: why a metal rail feels colder than a wooden step, why frost forms above freezing, and why a laptop has fins. You step out on a still, clear night in early spring. The air is a few degrees above freezing, yet the car roof is furred with frost. The next night is cloudy and the same car is dry. Your laptop, meanwhile, has a grille on the side and a fan that spins up when you open too many tabs. And the metal rail by your front door feels colder than the wooden step beside it, though both have sat out all night at the same temperature. Those three puzzles have one answer: heat moves in exactly three ways, each with its own rule, and an engineer who has taken a heat transfer course sees which one is in charge. Thermodynamics tells you heat will flow from hot to cold. Heat transfer tells you how fast. The course in one box Who takes itMechanical, aerospace, chemical and nuclear engineers, usually in year 3. What it is forPredicting how quickly heat moves through, off and between things, so you can stop a house, an engine or a chip from getting too hot or too cold. Comes afterThermodynamics (34/14), Fluid Mechanics (34/15) and Differential Equations (34/03). Leads toNo later subject in this curriculum lists it as a prerequisite, but it is behind turbine-blade cooling in Propulsion (34/21), the thermal design in Machine Design (34/19), and almost every capstone project (34/22) with a motor or a battery. What this piece gives youYou will not be able to design a heat exchanger. You will see why a thick coat works, why a fin helps, and why a hot steel ball and a hot potato cool differently. Easier first read: The Physics of the Greenhouse Effect (13/15), which is radiation at planet scale. Conduction: heat creeping through stuff Conduction is heat passing through a solid (or still fluid) by jostling neighbours. Hot atoms vibrate harder and pass the shaking along. No bulk motion, just a relay. The rule is Fourier's law, set out by Joseph Fourier in his 1822 Théorie analytique de la chaleur: the heat flowing is proportional to how steep the temperature gradient is, and to a property of the material called its thermal conductivity, k. Q̇ = k A ΔT ÷ Lheat rate (watts) = conductivity × area × temperature difference ÷ thickness Thicker walls slow heat; bigger temperature differences speed it up; the material sets the scale. And the scale is enormous. Copper has k of about 400 W/m·K, aluminium about 240, steel about 45, glass about 1, brick about 0.7, wood about 0.15, fibreglass insulation about 0.04 and still air about 0.026. Copper conducts ten thousand times better than fibreglass. That also solves the cold rail. Metal and wood are the same temperature, but your hand is warmer than both. Metal drains heat out of your skin much faster, so it feels colder. Your nerves sense the rate of heat loss, not the temperature. The rate is what conduction sets. Interactive · Heat through a slab Copper Aluminium Steel Glass Brick Wood Fibreglass Still air Material Thickness Temperature difference across it Typical rounded conductivities near room temperature. Real values vary with alloy, density, moisture and temperature. Try copper at 5 cm and 50 K: 401 kW through every square metre. Switch to fibreglass: 40 W. The ratio is the whole reason insulation exists. And notice still air: it is a superb insulator, which is why most insulation is just a way of trapping air so it cannot move. The moment air moves, we are in the next mode. Thermal resistance: a wall is a circuit Rearrange Fourier's law and it looks like Ohm's law. Temperature difference plays voltage, heat rate plays current, and the rest is a thermal resistance: Q̇ = ΔT ÷ R, with R = L ÷ (k A)thicker or less conductive means more resistance. Layers in series simply add. So a wall of brick, insulation and plasterboard is three resistors in a row, plus a thin film of nearly-still air on each face. The temperature drops across the layers like voltage across resistors, and most of the drop lands where the resistance is biggest. Engineers usually work per square metre and quote the U-value, which is 1 divided by the total resistance. Lower is better. Interactive · Build a wall Brick Fibreglass insulation Plasterboard Inside minus outside temperature k: brick 0.72, fibreglass 0.04, plasterboard 0.17 W/m·K. Air-film resistances 0.13 inside and 0.04 outside m²·K/W, the standard values for a vertical wall. Ignores studs, which leak extra heat. The default wall loses about 870 W through 100 m² when it is 25 K warmer inside. Slide the insulation to zero and the loss jumps to about 6,490 W, seven and a half times more. Ten centimetres of an insulator 18 times less conductive than brick beats the brick completely. The hidden lesson: resistance adds, so the biggest resistor dominates, and nothing you do to a small one matters much. Convection: a moving fluid carries the heat away Where air or water touches a hot surface, it warms, then moves away and is replaced by cooler fluid. That is convection: conduction into the fluid, plus the carrying. If the fluid moves by its own buoyancy (hot air rises) it is natural convection. If a fan, pump or wind drives it, it is forced. Newton's law of cooling bundles all that messy fluid behaviour into one number: Q̇ = h A (Tsurface − Tfluid)h is the heat transfer coefficient in W/m²·K The catch: h is not a property of the material. It depends on the flow, the shape and the fluid, which is why an entire course exists to estimate it. Typical textbook ranges are roughly 2 to 25 for natural convection in air, 25 to 250 for forced air, and 100 to 20,000 for forced liquids. A thin, nearly stuck layer of fluid clings to the surface and acts like a blanket. Anything that thins it (wind, a fan, stirring) raises h. Interactive · Same patch, different fluid Still air Fan or wind Moving water Heat transfer coefficient h Surface warmer than the fluid by h values are illustrative round numbers inside the textbook ranges, not a prediction for a particular flow. Set the difference to 15 K. Still air takes 1.5 W off your 10 cm square of skin; moving water takes 150 W. That is why 15 °C water saps you far faster than 15 °C air, and why wind chill is just convection with a higher h. It is also why a fan cools a chip without making the air any colder: it raises h. Radiation: the sky is cold The third mode needs no fluid and no solid. Everything above absolute zero emits electromagnetic radiation, mostly infrared at everyday temperatures, and the power climbs with the fourth power of the absolute temperature. Josef Stefan found the T⁴ rule from measurements in 1879, and Ludwig Boltzmann derived it from theory in 1884. Two surfaces exchange heat: q = ε σ (Ts⁴ − Tsurroundings⁴)σ = 5.67 × 10⁻⁸ W/m²·K⁴, temperatures in kelvin, ε = emissivity between 0 and 1 Emissivity ε says how well a surface radiates (and absorbs). Matte paint, skin, wood and water are around 0.9. Polished metal can be a few percent, which is why a space blanket works and why a thermos is silvered inside. The Sun warms the Earth across empty space this way, and a spacecraft's radiator is the only way to get rid of its waste heat. Now the car roof. At night the surface looks up at the sky, which is far colder than the air: much of the sky's radiation comes from very cold upper layers, and dry air passes the roof's infrared straight through to space. So the roof radiates away more than it gets back. It cools until warmth from the surrounding air, by convection, balances the loss. A cloudy night is a blanket of warm, wet air that radiates back down, so the roof stays near air temperature. Interactive · Why frost forms above freezing Clear, dry, calm Clear, light breeze Hazy and humid Overcast Air temperature Emissivity of the surface Sky colder than the air by (assumed) Air contact, h A thin, insulated surface that gets no heat from below, like a car roof's outer skin or a blade of grass. The effective sky temperature is not one fixed number: it depends on humidity, clouds and altitude, so the 20 K, 10 K and 3 K drops are illustrative assumptions. Real surfaces usually dip less because heat also leaks up from below. With 3 °C air, a clear calm sky and paint-like emissivity, the model's skin settles near −2.4 °C: frost, with the air above freezing. Add a light breeze or a bit of cloud and it stays near 0 °C or above. That matches what you see: frost needs clear, calm nights. Emissivity matters too. Cover the roof with shiny foil (ε near 0.05) and it radiates little, so it barely cools. Fins and heat sinks: buy surface area Look at Newton's law again: Q̇ = h A ΔT. If h is stuck (air gives you what it gives you) and ΔT is capped (a chip must stay under about 100 °C), the only dial left is A. A fin is a thin strip of metal that sticks out from a hot surface to add area. A heat sink is a bunch of fins. The laptop grille hides one, and a car radiator is a heat sink with water inside. But heat must travel along the fin by conduction, losing heat to the air as it goes, so the tip is cooler than the base. A long skinny fin of poor metal is mostly wasted: the far end barely helps. The standard result for a fin with an insulated tip is: Q̇fin = √(h P k Ac) · θbase · tanh(mL), m = √(hP ÷ kAc)P = perimeter, Ac = cross-section area, θbase = base temperature minus air, L = length Interactive · One fin Copper Aluminium Steel Fin length Air flow, h A straight rectangular fin, 2 mm thick and 50 mm wide, base 50 K above the air, insulated tip. Metal k: copper 401, aluminium 237, steel 45 W/m·K. Drag the length and watch the curve flatten: the first centimetres carry most of the heat. Switch to steel and the curve saturates sooner and lower, because heat cannot reach the far end. Raise h (blow on it) and the fin gets less efficient, because it dumps heat before it travels far, but the fin moves more heat in total. That trade-off, between area, conductivity and airflow, is the whole design of a heat sink. Transient cooling: the time constant and the Biot number Everything so far held still. But a hot cup, a quenched blade or a switched-off chip cools over time. The simplest model treats the object as one lump at a single temperature, shedding heat through its surface. The lump has a thermal "capacitance" (how much energy it stores per kelvin: density × specific heat × volume) and a thermal "resistance" to the surroundings (1 ÷ hA). It is an RC circuit, and so it cools like one: T(t) − T∞ = (Ti − T∞) e−t/τ, τ = ρ c V ÷ (h A)after one time constant τ, 37% of the original temperature gap is left; after 3τ, 5% The model has a catch. It assumes the inside is the same temperature as the skin, so heat flows out easily compared with how fast the outside sheds it. The Biot number, Bi = h Lc ÷ k (with Lc = volume ÷ surface area), compares the two resistances: heat having to get out of the object versus heat leaving its surface. If Bi is below about 0.1, the lump model is fine. Above that, the inside lags behind the skin. Interactive · Cooling ball Copper Steel Water-like (a potato or a plum) Ball radius Air flow, h Starts at 100 °C in 20 °C air. ρ, c, k: copper 8960, 385, 401; steel 7850, 434, 60; water-like 1000, 4180, 0.6 (SI units). Ignores radiation. A 10 mm steel ball in still air takes about 7.6 minutes per time constant, so about 23 minutes to come within 5% of room temperature. Biot number: 0.0014, so the lump model is excellent. Now choose "water-like" at 25 mm: τ is about 23 minutes but the Biot number is 0.35. Real potatoes stay hot in the middle long after the skin is cool, so the lump model underestimates how long they stay hot inside. Copper's huge k means any copper ball is almost always a lump, even in water. Metals pass the test; food does not. What a graduate just sees Heat has three roads: through solids (conduction), by carried fluid (convection) and across space (radiation). A thing feels cold because it takes heat away quickly, not because it is colder. Resistances in series add, and the biggest one decides everything. Fix the big one. Insulation is mostly trapped still air. The moment the air can move, it stops working. Water cools you far faster than air at the same temperature, because h is a hundred times bigger. Radiation scales with temperature to the fourth power, so it dominates when things get hot or when nothing else can carry heat, as in space. When area is the only dial left, add fins. They have diminishing returns past a certain length. Everything cools exponentially, with a time constant that is the product of a thermal resistance and a thermal capacity, as long as Bi is small. Where it shows up next Propulsion (34/21): a modern turbine runs hotter than its metal would survive, so blades are riddled with cooling passages and wrapped in a film of cool air, all heat-transfer design. Machine Design (34/19) and Materials Science & Manufacturing (34/13): a bearing, a brake disc or a motor winding has a thermal limit as real as its strength limit, and k is a material property. Control Systems (34/20): a thermostat loop is a time constant being chased, using the same exponential as the cooling ball. In the world: house insulation and windows, CPUs and phones, batteries and EV cooling, cooking, spacecraft radiators and the Earth's own energy balance. If you want to go deeper 📘 Textbook. Theodore Bergman, Adrienne Lavine, Frank Incropera and David DeWitt, Fundamentals of Heat and Mass Transfer (Wiley, 8th edition). The standard text, thorough and full of tables. 📘 Textbook. Yunus Çengel and Afshin Ghajar, Heat and Mass Transfer: Fundamentals and Applications (McGraw Hill, 6th edition in 2020). Chattier, with many worked examples. 📘 Free to read alone. John Lienhard V and John Lienhard IV, A Heat Transfer Textbook (6th edition, free to download for personal use). 🌐 Free course. MIT OpenCourseWare 2.051 Introduction to Heat Transfer (Fall 2015, undergraduate). A shorter route is the heat-transfer part of 16.050 Thermal Energy. The takeaway Heat moves by conduction (Fourier: k A ΔT ÷ L), convection (Newton: h A ΔT) and radiation (Stefan–Boltzmann: ε σ T⁴). Treat a wall like a circuit, where resistances in series add and the biggest wins. Add area with fins when the other dials are stuck. And check the Biot number before treating an object as a single lump with one temperature and one time constant. Try this: on a cold, clear night, feel the roof of a parked car and the roof of a car parked under a tree or a carport. Then check which one has frost. The difference is a clear sky versus a ceiling that radiates back. Sources & Further Reading Sources. The three modes, thermal resistance, fins and the lumped-capacitance model are standard textbook heat transfer (Incropera; Çengel & Ghajar; Lienhard). Fourier's 1822 Théorie analytique de la chaleur; Stefan 1879 and Boltzmann 1884 for the T⁴ law. σ = 2π⁵k⁴ ÷ 15h³c² = 5.670374 × 10⁻⁸ W/m²·K⁴ from the exact SI values of k, h, c. Thermal conductivities: rounded typical values (Incropera appendix; Engineering Toolbox: window glass 0.96, plasterboard 0.17, fibreglass 0.04, still air 0.026). Convection ranges: Incropera Table 1.1 (free convection in gases 2 to 25, forced in gases 25 to 250, forced in liquids 100 to 20,000 W/m²·K). Air-film resistances 0.13 and 0.04 m²·K/W: ISO 6946 values for horizontal heat flow. Frost above freezing on clear nights: radiative cooling to a sky colder than the air; the effective sky temperature is a convenient and approximate concept and is set by assumption in the interactive. Every number was computed in node: the 100 m² wall loses 866 W with 10 cm of fibreglass (U = 0.347 W/m²·K, R = 2.885) and 6,487 W without (R = 0.385); copper 5 cm slab at 50 K carries 401 kW/m²; the roof skin settles at −2.4 °C (clear, calm, h = 10), 0.4 °C (h = 25), 0.1 °C (10 K sky), 2.1 °C (3 K sky); aluminium fin, 50 mm, h = 25, carries 5.96 W (efficiency 92%); steel ball r = 10 mm, h = 25: τ = 454 s, Bi = 0.0014; water-like ball r = 25 mm: τ = 1,393 s, Bi = 0.347. 🌐 Lienhard, A Heat Transfer Textbook (free download) 🌐 MIT OCW 2.051 Introduction to Heat Transfer 📘 Bergman, Lavine, Incropera & DeWitt, Fundamentals of Heat and Mass Transfer Step 16. Vibrations: Everything Is a Spring and a Mass ------------------------------------------------------ Why: Everything vibrates; resonance is how small pushes break big things. Why a guitar string has a pitch, a washing machine walks, and a bridge can wobble itself apart: one equation, and the number that decides how badly things shake. Pluck a guitar string and it rings at one pitch, no matter how hard you pluck it. Push a child on a swing in time with the swing and a gentle shove builds a big arc. Drive over a washboard road at the wrong speed and the whole car shudders. Different things, one idea: almost every object has a frequency it prefers to move at, and it responds wildly to being pushed at that rhythm. A vibrations graduate sees this everywhere. A phone buzzing on a table, a washing machine walking across the floor, a skyscraper swaying in a typhoon, a tall bridge that wobbles under a crowd: each is a mass, a spring and some friction, and each has a number that says how badly it will misbehave. The course in one box Who takes it: mechanical, aerospace and civil engineers, usually in year 3. What it's for: predicting how a structure or machine moves when it is pushed, and how to stop that motion from becoming a failure. Comes after: Dynamics (34/12) and Differential Equations (34/03). Leads to: Control Systems (34/20), which uses the very same equation, plus Machine Design (34/19) for shafts, gears and bearings, and Propulsion (34/21) for blades that must not flutter. Fluid Mechanics (34/15) joins it whenever wind or water drives the motion. Honest line: you won't be able to design a damper for a tower after this. You will see why a bridge can wobble, why soft mounts protect a machine, and what Taipei 101's giant golden ball is for. Everything is a spring and a mass Take anything that springs back when displaced: a diving board, a building, a guitar string, a car on its tyres. Close up, it behaves like a mass on a spring. Newton's law says that the spring pushes back in proportion to how far you stretch it, so the acceleration is proportional to the displacement and points the other way. That gives a motion that repeats, at a frequency that depends on only two things. ωn = √(k / m)natural frequency (rad/s) = square root of stiffness ÷ mass Stiffer means faster; heavier means slower. Divide by 2π to get hertz. Worked example: a 1 kg mass on a 100 N/m spring has ω = 10 rad/s, so f = 1.59 Hz, one swing every 0.63 s. A car body of about 400 kg per wheel on a 20,000 N/m spring comes to 1.1 Hz, roughly a walking pace, which is why a car feels like it is floating. Four times the stiffness, or a quarter of the mass, doubles the frequency, because of the square root. The size of the pluck doesn't appear in the formula. The frequency is a property of the object itself, which is what makes it so useful. Interactive · Mass and spring 1 kg, 100 N/m4 kg, 400 N/m16 kg, 400 N/m Mass Spring stiffness Pause ω = √(k/m) = 10 rad/s → 1.59 Hz, one swing every 0.63 s. Ideal, undamped, computed live. Note that 4 kg on 400 N/m has the same frequency as 1 kg on 100 N/m: only the ratio k/m counts. Quadruple the mass and the frequency halves. Damping: where the energy goes Real things stop ringing because friction, air and bending metal turn motion into heat. That is damping. Engineers measure it with a single dimensionless number, the damping ratio ζ (zeta): the actual damping divided by the critical damping, the exact amount that just prevents any overshoot. ζ = c / (2√(k·m))damping ratio = damper strength ÷ the critical value ζ below 1 (underdamped): it overshoots and rings, each swing smaller. A guitar string has tiny ζ and rings for seconds; a car's shock absorbers are set well below 1 so the wheels still follow the road. ζ = 1 (critical): the quickest return to rest with no overshoot. A good door closer aims here. ζ above 1 (overdamped): no overshoot, but a slow, syrupy creep. For large ζ it behaves like a first-order system, a single exponential with a time constant, which is exactly what the next subject uses to describe cruise control. For ζ below 1 each swing is smaller than the last by a fixed factor. At ζ = 0.05 each swing is 73% of the one before, so it rings for dozens of cycles. At ζ = 0.2 it is 28%: three swings and it's gone. Interactive · Let go and watch 0.020.20.71 (critical)2 (overdamped) Damping ratio Underdamped: each swing is 28% of the last. Exact solution of m·x″ + c·x′ + k·x = 0, released from rest, checked against a step-by-step numerical solution. Forced vibration and resonance Now push the mass back and forth at a steady frequency. At first the mass just follows the push. If you push very fast, it barely moves, because there isn't time. But push at the natural frequency and each push arrives just as the mass is moving the right way, so the energy adds up cycle after cycle until damping drains energy as fast as you put it in. That is resonance. The amplitude, as a multiple of what a steady push of the same size would do, is the famous curve: amplitude ÷ static = 1 / √[(1 − r²)² + (2ζr)²]r = driving frequency ÷ natural frequency At r = 1 the first bracket vanishes and the amplification is 1/(2ζ). A lightly damped structure with ζ = 0.02 moves 25 times further than a static push; with ζ = 0.5 barely more than a static push. So damping, not the size of the push, decides how bad resonance is. The motion also shifts from moving with the push (below resonance) to moving against it (above), with a 90° lag exactly at resonance. Interactive · The resonance curve Slow pushAt resonanceFast push Damping ratio ζ Driving frequency At resonance with ζ = 0.1 the motion is 5× a static push. Steady-state response of a damped mass-spring to a sinusoidal force. For ζ = 0.05 the peak is 10×; for ζ = 0.5 it shrinks to 1.15× and moves to lower frequency. Isolation: soft mounts and the √2 rule Sometimes the push is something you want to keep away from a structure: an engine shaking a car body, a washing machine's drum shaking a floor, a seismometer ignoring the ground. The trick is to put a spring between them. The fraction of shaking that gets through is called transmissibility, and it has a surprising shape: below r = √2 (about 1.41) the mount makes things worse, and only above it does isolation begin. Above √2, more damping lets more through at high frequency. So the rule: make the mount's natural frequency well below the frequency you want to block. To cut a 30 Hz engine's force to 10%, with light damping you need r ≈ 3.3, so a mount with a natural frequency near 9 Hz. In practice that means a soft spring that sags a few millimetres under the load. The price is that every time the machine starts or stops it sweeps through the mount's natural frequency, so you want some damping to survive the crossing. Interactive · An engine on mounts Mount natural frequency Mount damping ratio A 9 Hz mount under a 30 Hz engine lets about 13% of the force through. The curve is the standard transmissibility of a damped mass-spring. Try moving the mount to 25 Hz and watch it amplify. Modes: a structure is a choir of springs A single mass has one natural frequency. A string, beam, plate or aircraft wing is a continuous chain of masses and springs, so it has many: a mode is one pattern of motion in which every part moves in step at one shared frequency. The lowest mode is the whole string bowing out. The second has a still point in the middle (a node), the third has two, and so on. Pluck the string and you excite several at once; that blend is the timbre of the instrument. For an ideal string the modes are whole-number multiples of the lowest, a harmonic series. A guitar's low E at 82.41 Hz also rings at 164.8, 247.2 and so on. A beam fixed at one end (a diving board, a wing, a skyscraper) is different: its second mode is 6.27 times the first and its third 17.5 times, so the overtones are not neat multiples. Engineers find these modes first, then keep forces away from them. Interactive · Mode shapes Guitar stringCantilever beam Mode 1Mode 2Mode 3Mode 4Mode 5Mode 6 Pause Mode 2 of a string tuned to 82.41 Hz: 164.8 Hz, with 1 still point (node). Exact mode shapes (sine for the string; the standard cosh/cos formula for the beam, using βL = 1.875, 4.694, 7.855, 10.996, 14.137, 17.279). Shown in slow motion, with the amplitude exaggerated. Orange dots are nodes. When it goes wrong: Tacoma Narrows and the wobbly bridge Tacoma Narrows (1940). The usual textbook story says the wind shook the bridge at its natural frequency and resonance tore it down. That version is wrong. The bridge opened on 1 July 1940 and fell on 7 November, in a wind of roughly 65 km/h (reports range from 35 to 42 mph). The deck was slender and flexible in twisting, and on the day it went from bouncing to twisting about once every four seconds. In 1991 K. Y. Billah and R. H. Scanlan, in the American Journal of Physics, argued that the cause was aeroelastic flutter: the twisting itself changed the wind's forces on the deck so that they fed energy into the twist. This is a self-excited motion, with no steady external beat to resonate with. Once the wind passed a threshold speed, the aerodynamic effect acted like negative damping. Some details are still argued over, but "simple resonance" is not the accepted account. London Millennium Bridge (2000). The footbridge opened on 10 June 2000 and was closed two days later after it swayed sideways under crowds. Walking pushes sideways on the deck about once per two steps. If the deck sways, people adjust their stride, and on this bridge their combined pushes ended up in phase with the sway, adding energy instead of removing it. The effect only began once the crowd passed a critical size. The Arup engineers' study showed that any bridge with a lateral frequency below about 1.3 Hz could suffer the same if enough people were on it. The fix was not to stiffen it but to add dampers (viscous and tuned mass dampers) and it reopened in early 2002. Engineers still debate how much of the effect is people falling into step with each other and how much is each person balancing against the sway. Both stories share one mechanism: something that pushes in a way that reduces the effective damping. Once the damping reaches zero, any disturbance grows. Try it. Interactive · Crowd, damping and the tipping point As openedWith added dampers People on the deck With 100 people the sway still dies out. The tipping point is about 157 people. An illustrative model, not the real bridge's data: 100 tonnes of moving mass, 0.5 Hz, 0.5% natural damping, each person adding 20 N·s/m of negative damping, and +5% damping for the "added dampers". It shows the mechanism: the threshold is where the crowd's push equals the structure's own damping. Tuned mass dampers: a second spring that eats the vibration You can fight resonance by adding damping, or by adding a second mass-spring tuned to the first. A tuned mass damper is a small mass on a spring and damper, attached to the structure and tuned to its natural frequency. When the structure resonates, the small mass swings against it and absorbs energy. The best tuning, worked out by J. P. Den Hartog and still the starting point, is simple: for a mass ratio μ, tune the damper to 1/(1 + μ) of the building's frequency, with damping ratio √(3μ/(8(1+μ)³)). Taipei 101 is the famous case: a 660-tonne steel sphere about 5.5 m across (728 short tons) hangs like a pendulum in the upper floors, and in Typhoon Soudelor in 2015 it swung about a metre. The makers claim it can reduce sway by up to 40% (a company figure). In the simulator below, a damper that weighs only 2% of the structure cuts the peak response from 50× to about 9×. Interactive · Tune the damper Damper mass Damper tuning (× building frequency) Damper damping ratio Optimal settingsMistuned (15% off) Peak response without the damper: 50×. With it: about 8.7×. Two-mass model computed with complex arithmetic; the structure has 1% damping. The damper's mass is the slider's value in tenths of a percent, so 20 means 2%. At 100 (10%) you can see the response split into two lower humps. What a graduate just sees Everything with mass and springiness has a natural frequency, √(k/m), set by the object and not by how hard it is hit. Stiffer goes up, heavier goes down, and by the square root: four times the stiffness doubles the frequency. Damping ratio, not the force, decides how bad resonance is: the peak is about 1/(2ζ). Below √2 times its natural frequency a soft mount amplifies; above it, it protects. Any structure has a whole family of modes; engineers keep forces away from the low ones. Tacoma Narrows was flutter, a self-excited instability, not simple resonance. When something fails by growing vibration, look for what has made the damping effectively negative. If you can't remove the push, add damping (or a tuned damper) rather than making it stiffer. Where it shows up next Control Systems (34/20): the equation of a mass with damping and a spring is the standard second-order system, with the same ζ and ωn. Feedback is how you add damping by software: a flight controller does for a drone what a viscous damper does for a bridge. Machine Design (34/19): shafts and gearboxes have critical speeds at which they whip; turbines and rotors are balanced so their push stays small. Propulsion (34/21) and Fluid Mechanics (34/15): compressor and turbine blades and wings can flutter, the Tacoma mechanism at jet speeds. The real world: earthquake-resistant towers, car and aircraft ride, hard-drive heads, noise and comfort engineering, musical instrument design. Computer Science: analysing vibrations is Fourier analysis (see Fourier Transform, 12/12), and software simulating springs underlies physics engines, animation and robotics. If you want to go deeper 📘 Textbook: Singiresu S. Rao, Mechanical Vibrations (Pearson): the most widely assigned undergraduate text, with many worked examples. 📘 Textbook: Daniel J. Inman, Engineering Vibration (Pearson): clearer on modelling and on MATLAB use. J. P. Den Hartog's Mechanical Vibrations (Dover) is the short classic that gave us the tuned damper. 📘 Gentler: The Feynman Lectures on Physics, Vol. I, ch. 21 "The Harmonic Oscillator" and ch. 23 "Resonance", free online. ▶️ Free course: MIT OpenCourseWare 2.003SC Engineering Dynamics (Fall 2011), an introduction to vibrations and dynamics of lumped mechanical models, with recitations on frequency response. 🌐 Easier first read in this library: Waves and Sound (13/13) and Fourier Transform (12/12). The takeaway Everything springy is a mass and a spring with natural frequency √(k/m). Damping (ζ) decides whether it rings, creeps or returns cleanly, and sets the peak of resonance at about 1/(2ζ). A soft mount protects only above √2 times its natural frequency. Structures have many modes, and the dramatic failures, such as Tacoma Narrows and the Millennium Bridge, came from forces that made the effective damping negative, not from a simple beat. A tuned mass damper is a second spring placed where it can steal the energy. Try this: hold a ruler overhanging a desk and twang it. Shorten the overhang and hear the pitch rise. Then stick a lump of putty on the tip and hear it drop: stiffness up, frequency up; mass up, frequency down. Sources & Further Reading Sources. Single-degree-of-freedom theory, modes, isolation and the Den Hartog absorber are standard textbook material (Rao; Inman; Den Hartog). Tacoma Narrows: K. Y. Billah and R. H. Scanlan, "Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks", American Journal of Physics 59 (1991) 118–124; MIT 18.03 notes (Appendix B); dates and wind reports vary by source (35–42 mph). Millennium Bridge: P. Dallard et al., The Structural Engineer 79(22), 2001; Ingenia, "Stabilising the London Millennium Bridge"; Wikipedia, "Synchronous lateral excitation". Taipei 101: Britannica and the Guinness record page (660 metric tons; floors given as 87th to 92nd by different sources; the 40% is a manufacturer's claim). Computed in node: all frequencies, ratios, peaks and the damper optimum (checked against a numerical scan of the two-mass model); the release response was checked against a step-by-step numerical solution. 🌐 MIT OpenCourseWare 2.003SC Engineering Dynamics 🌐 Billah and Scanlan (1991), scanned paper and Ingenia: Stabilising the London Millennium Bridge 📘 Rao, Mechanical Vibrations · Inman, Engineering Vibration · Den Hartog, Mechanical Vibrations Step 17. Probability & Statistics: Reasoning with Uncertainty ------------------------------------------------------------- Why: Tolerances, test data and failure rates are statistics. Nothing you measure is exact, and no part comes out the same twice. Statistics is the discipline of knowing how much to trust a number. A machinist turns a hundred steel shafts to "10 millimetres". Measure them with a good micrometer and no two are the same: 9.981, 10.012, 10.004. Nothing is wrong. That scatter is a fact of every process, every material and every sensor. Now stack five of those parts end to end in a gearbox. Will the stack still fit the housing? A pollster calls a thousand people and says 52% will vote yes, give or take three. A drug trial says "significant". A forecast says 30% chance of rain. Every one of these is the same question: given scatter, how much should I trust this number? A graduate of probability and statistics doesn't see a single number any more. They see the shape of the scatter behind it. This piece builds that sight with eight things you can run: dice, tolerances, polls, a drug-trial trap, a belief that updates, and a gearbox stack. The course in one box Who takes it: almost every engineering and science student, usually in the second year; computer scientists meet it again before machine learning. What it's for: deciding what to believe, and how far to trust a measurement, when outcomes are uncertain. Comes after: Calculus I (34/01), which helps with continuous distributions, though the ideas here need little more than arithmetic. Leads to: Machine Learning (34/32) needs it directly. It also shows up in Materials Science & Manufacturing (34/13: process control and defects), Systems Engineering (34/22: reliability) and Computer Security (34/33: randomness and guessing). Honest line: you won't be able to design a clinical trial after this. You will see why "significant" doesn't mean "important", why a bigger sample beats a cleverer guess, and why most real quantities look like a bell curve. For easier first reads, see the library's Probability: How to Not Fool Yourself (12/03), Statistics for Reading the World (12/05) and Bayes' Theorem (12/17); this piece goes further into the engineering side. Random variables: a shape with a centre and a spread A random variable is a number that comes out of a process you can't predict one run at a time. The key move is to stop asking "what will it be?" and ask "what is the shape of what it could be?" Roll two dice and add them: 7 is six times as likely as 2 or 12, so the shape is a tent. Roll many times and the tent appears. Interactive · Roll dice, grow a shape Dice added together Roll 10 timesRoll 1,000 timesClear Without JavaScript: two dice add to 7 with probability 6/36 and to 2 or 12 with probability 1/36 each. The mean total is 7 and the standard deviation is about 2.42. Two numbers summarise a shape. The mean is its balance point: 7 for two dice. The standard deviation (the "spread") says how far values typically stray from the mean: about 2.4 for two dice, which means totals usually land between roughly 5 and 9. The variance is the standard deviation squared, and it is the one that adds up neatly when you combine independent things. standard deviation = √( average of (value − mean)² ) Now the shape that matters most. Measured quantities (shaft diameters, heights, sensor noise) tend to form the normal curve, the bell. Its great convenience is a fixed rule: about 68.3% of values fall within one standard deviation of the mean, 95.4% within two, and 99.7% within three. Engineers use it every day. Say the shaft's spec is 10.00 mm ± 0.05 mm. Slide the process around and see what share of parts get thrown away. Interactive · How many parts fail the spec? Average diameter Spread (standard deviation) Without JavaScript: a centred process with standard deviation 0.025 mm (the spec limits are exactly ±2 standard deviations) scraps about 4.55% of parts. Cut the spread to 0.01 mm and about 0.00006% are scrapped. Shift the average 0.02 mm off-centre with spread 0.02 mm and about 6.7% are scrapped. Two lessons graduates keep. First, spread, not average, decides quality: a process centred perfectly but wide still scraps parts. Second, being off-centre costs a lot even when the spread is fine. That is why factories track both. The central limit theorem: why the bell keeps appearing Why should so many things be bell-shaped? The answer is one of the most useful theorems in mathematics. Take almost any random process, however lopsided, and average a few dozen independent draws. The averages pile up in a bell, centred on the true mean, with spread shrinking like 1/√n. Early forms are due to de Moivre in 1733 (for coin flips) and Laplace, whose general version dates from about 1810. Pick a lopsided source and drag the sample size. Interactive · Average anything, get a bell Flat (0 to 1) Very lopsided Coin (0 or 1) Draws averaged together: n Without JavaScript: for the lopsided source (mean 1, standard deviation 1) the average of n = 30 draws has standard deviation 1/√30 = 0.183, and the 3,000 averages form a close-to-normal bell around 1. At n = 1 you see the raw source. By n = 30 even the lopsided one is a near-bell. The prediction (the curve) uses only two numbers from the source, its mean and its standard deviation. The shape of the source is forgotten. That is why a measurement made of many small independent errors is normal, and why so much of statistics can pretend things are bell-shaped. standard error of an average = σ / √n In words: the average of n draws wobbles less than one draw, by the square root of n. To halve the wobble you need four times the data, not twice. It is the quietest expensive fact in all of experimental science. Sampling and confidence intervals: how sure is a poll? You can't ask everyone, so you ask n people and treat the share saying yes as an estimate. How far off might it be? The standard error from the last section gives the answer. For a share p the wobble is √(p(1−p)/n), and a 95% confidence interval is the estimate plus and minus about two of those. For n = 1,000 and a share near half, that is ±3.1 percentage points: the familiar "margin of error". The wording is subtle, and the interactive makes it concrete. "95% confidence" is a statement about the method, not about one interval: if you ran the procedure again and again, about 95 intervals in 100 would capture the true value. Here the truth is secretly 52%. Run 100 polls. Interactive · 100 polls, one truth People asked in each poll Run 100 polls Without JavaScript: with 1,000 people per poll the margin of error is about 3.1 points; run 100 polls and about 95 of the intervals cover the true value. With 4,000 people the margin halves to about 1.5 points. Watch two things. Roughly five in a hundred intervals miss the truth even when everything is done correctly. And quadrupling the sample from 1,000 to 4,000 only halves the interval. A poll of a country and a poll of a town need the same sample size for the same precision, as long as the sample is random. The real danger is not the margin of error; it is a sample that isn't random at all, which no formula fixes. P-values and Bayes: two ways to weigh evidence A hypothesis test asks whether data are surprising if nothing is going on. The p-value is the probability, assuming nothing is going on, of seeing a result at least as extreme as yours. Flip a coin 100 times and get 60 heads: for a fair coin the chance of a split at least that lopsided in either direction is about 5.7%. It is not the probability that the coin is fair, and it is not the probability that the result was "due to chance". The American Statistical Association issued a rare formal statement on this in 2016, with six principles. Two of them: p-values do not measure the probability that a hypothesis is true, and a p-value does not measure the size or importance of an effect. The classic trap is testing many things. At the usual cut-off of 0.05, a test of a true "nothing" comes out "significant" one time in 20 by luck. Test twenty unrelated things and the chance that at least one lights up is 1 − 0.9520, about 64%. Try it: each press tests twenty effects that are, by construction, nothing. Interactive · Test twenty things that do nothing Run 20 testsReset the tally Without JavaScript: twenty tests of pure noise at the 0.05 level give at least one false "discovery" about 64% of the time, and one in 20 tests on average. If you report only the one that lit up, you have a headline and no finding. This is part of why many published results fail to repeat: in the Open Science Collaboration's 2015 attempt to replicate 100 psychology studies, 97% of the originals had statistically significant results but only 36% of the replications did. Bayes' rule answers a different question: given the data, how should my belief move? Start with a prior belief, multiply by how well each possibility explains the data, and renormalise. Say you test parts and want the defect rate. Know nothing at first (any rate from 0% to 100% equally likely) and add parts one at a time. Interactive · Learn a defect rate from parts Add a good partAdd 10 good partsAdd a defective partReset Without JavaScript: after 20 good parts and no defects, a flat starting belief gives a most-plausible defect rate near 0%, an average of 4.5%, and a 90% credible range of roughly a quarter of a percent to 13%. Twenty clean parts cannot rule out a defect rate near one in eight. Frequentists and Bayesians disagree about the philosophy: one treats probability as long-run frequency and never assigns it to a hypothesis, the other as degrees of belief that data update. In practice both are used, often on the same project, and on big data sets they usually agree. The Bayesian version has one practical advantage here: after twenty clean parts it honestly says "we still can't rule out one in eight". Correlation, regression and the trap of causation Correlation (called r) measures how tightly two quantities move together along a straight line, from −1 to +1. Regression fits the best line through the cloud (by least squares, as in Linear Algebra, 34/04). Both are summaries, and any summary can hide the story. In 1973 the statistician Francis Anscombe made four data sets with, to two decimals, the same mean, the same spread, the same r (0.82) and the same line (y = 3 + 0.5x). Look at each. Interactive · Four data sets, one set of numbers Set 1Set 2Set 3Set 4 Without JavaScript: all four Anscombe sets have mean x = 9, mean y = 7.50, correlation 0.816 and line y = 3.00 + 0.50x. Set 1 is a noisy line, set 2 a curve, set 3 a line with one outlier, and set 4 a single far point that creates the whole slope. The numbers agree and the stories do not: one noisy line, one curve, one line with a rogue point, one vertical stack held up by a single point. Always plot first. The other trap is the famous one. Ice-cream sales and drownings rise together, because summer heat drives both. A confounder (a hidden third factor) makes two things move together without either causing the other. The remedy graduates know is randomised assignment: if a coin decides who gets the treatment, nothing else can be systematically linked to it. Where you can't randomise, causal claims need much more than a line through a cloud. Engineering: tolerance stacks, reliability and Monte Carlo Five parts, each 10 mm ± 0.1 mm, line up in an assembly. What is the worst the total can be out? Worst-case stacking adds the tolerances: ±0.5 mm. It is safe and expensive, because it assumes every part is at its extreme in the same direction at once. If the parts vary independently, variances add, so the standard deviations add in quadrature: root-sum-square, √5 × 0.1 = ±0.224 mm. For five parts that is under half the worst-case figure. When the formulas get hard, engineers use Monte Carlo: let the computer build the assembly ten thousand times with random parts and count how often it fails. Here each part's tolerance is treated as three standard deviations. Choose how many parts are stacked. Interactive · Monte Carlo tolerance stack Parts in the stack Build 10,000 assemblies again Without JavaScript: five parts at ±0.1 mm (three standard deviations) give a worst-case limit of ±0.5 mm, a root-sum-square limit of ±0.224 mm, and in simulation only about 0.3% of assemblies exceed the root-sum-square limit; none come close to 0.5 mm. The honest caveat is that root-sum-square assumes the parts vary independently and are centred. A tool wearing in the same direction on every part breaks that assumption, and so does one supplier's batch. Engineers disagree about when to trust statistical stacking over worst-case; safety-critical designs often insist on worst-case. Reliability follows the same arithmetic. A system that needs all of its parts is a series system, and probabilities multiply: ten components that each work 99% of the time give 0.9910 ≈ 90.4% for the whole. Add a backup in parallel and failure needs both to fail: two units that each work 90% of the time give 1 − 0.1² = 99%. The two rules drive why aircraft duplicate systems and why complicated machines need very reliable components. What a graduate just sees Every number has a spread behind it; the first question is "how scattered?", not "what is it?". Spread, not the average, decides how many parts are scrapped, and centring matters too. Averages of many independent things are bell-shaped, whatever the source looked like. Halving the uncertainty costs four times the data. A 95% interval is a claim about a method: it misses one time in twenty by design. A p-value is "how surprising is this if nothing is going on", not "how likely is it true". Test twenty things and one will look significant; ask how many were tried. Plot the data; correlation is not causation, and only randomising breaks a confounder. Independent errors add in quadrature; series reliabilities multiply. Where it shows up next Machine Learning (34/32): training is fitting a model to noisy data; ideas of overfitting, error bars and Bayes are this course made bigger. Materials Science & Manufacturing (34/13): process capability and scrap rates are the normal curve against a spec limit. Mechanics of Materials (34/11) and Machine Design (34/19): safety factors exist because loads and strengths are distributions, not numbers. Systems Engineering (34/22): reliability budgets multiply series probabilities and add redundancy to push them back up. Computer Security (34/33) and Algorithms (34/24): good random numbers, randomised algorithms, and the odds of guessing a secret. If you want to go deeper 📘 Douglas Montgomery and George Runger, Applied Statistics and Probability for Engineers (Wiley; 7th edition). The standard engineering course text, with plenty of tolerance, quality and reliability problems. 📘 Joseph Blitzstein and Jessica Hwang, Introduction to Probability (CRC Press; 2nd edition, 2019). Best for deep intuition about probability itself, from Harvard's Stat 110. 📖 David Spiegelhalter, The Art of Statistics (2019): the most readable route for learning alone, built around real cases and honest about uncertainty. 🌐 MIT OpenCourseWare 18.05, Introduction to Probability and Statistics (Spring 2022): free notes and problem sets, with Bayesian and classical statistics side by side. The takeaway Nothing you measure is exact, so every number is a shape. Summarise a shape by its centre and its spread, trust averages more as the square root of the sample, read a confidence interval as a promise about a method, treat a p-value as a measure of surprise, and plot before you believe a correlation. Try this: next time you see a poll or a "significant" headline, find the sample size and ask how many things were tested. Then, in the dice widget above, add four dice and watch the tent turn into a bell. Sources & Further Reading Sources. The normal-curve rule (68.3%, 95.4%, 99.7%), the central limit theorem, confidence intervals, the binomial test, the Beta posterior, root-sum-square stacking and series/parallel reliability are standard textbook results (Montgomery and Runger; MIT 18.05); every number shown was recomputed in code. Wasserstein and Lazar, "The ASA's Statement on p-Values: Context, Process, and Purpose", The American Statistician (2016). Open Science Collaboration, "Estimating the reproducibility of psychological science", Science (2015): 97% of originals significant against 36% of replications. Anscombe, "Graphs in Statistical Analysis", The American Statistician (1973). Early forms of the central limit theorem: de Moivre (1733) and Laplace (c. 1810), as surveyed in recent histories such as Ulyanov, "From classical to modern central limit theorems" (arXiv 2405.19828). Go deeper. The books and course above; Wasserstein and Lazar's statement is free and short (doi.org/10.1080/00031305.2016.1154108). Step 18. Machine Design: The Parts Every Machine Is Built From -------------------------------------------------------------- Why: Shafts, gears, bearings, bolts: the parts every machine is built from. A bicycle has a chain, bearings, bolts, springs, an axle and a slowly growing crack. Machine design is knowing which part does what, and how each one breaks. Look at a bicycle and you are looking at almost everything in this course. A chain and two sprockets turn pedalling into driving force. Bearings in the wheels and the pedals let things spin with almost no friction. Bolts clamp the handlebar stem. Springs sit in the saddle and the brakes. A shaft (the axle) carries the wheel's load. And somewhere in the frame, at a weld or a drilled hole, a crack may be growing very slowly with every pedal stroke. Almost every machine, from a drill to a jet engine to a washing machine, is built from the same short list of standard parts, and from the same short list of ways of failing. Machine design is the craft of choosing, sizing and combining those parts so the machine does its job, for its intended life, without surprises. The course in one box Who takes it: mechanical engineers, usually in year 4, and many aerospace and manufacturing engineers. What it's for: designing real machines out of standard parts, and making sure they don't break. Comes after: Mechanics of Materials (34/11), Materials Science & Manufacturing (34/13), Engineering Drawing & CAD (34/18). Leads to: Systems Engineering & the Capstone (34/22), the project where a team designs a whole machine. It leans every day on Dynamics (34/12), Vibrations (34/17) and Control Systems (34/20). Honest line: you won't be able to certify a gearbox after this; you will see why the shaft has a step in it, why the bolt is tightened so hard, and why the crack starts at the corner. Power is torque times speed Every machine moves energy from a source (a motor, an engine, your legs) to a job. The rate at which it does so is power, and for anything that rotates, power is the product of two things: how hard it twists and how fast it turns. power = torque × angular speed    (P = T ω, with ω in radians per second) That has a consequence that drives the whole subject: for a fixed amount of power, speed and torque trade against each other. A gear pair is a trade machine. If the driven gear has twice as many teeth, it turns at half the speed and delivers twice the torque, less the small losses in the mesh. A bicycle's low gear does exactly this: slower wheel, more force at the road, so you can climb. Worked number: a 1 kW motor at 3,000 rpm delivers 3.18 N·m. Through a two-stage reducer of 15:60 teeth and 15:60 teeth, the overall ratio is 4 × 4 = 16, so the output turns at 187.5 rpm and, with about 98% efficiency per mesh, delivers about 48.9 N·m. Power out is about 0.96 kW; the missing 40 W is heat in the oil. Interactive · A two-stage gearbox Reducer, 16:1 Reducer, 6.25:1 Overdrive, 1:4 Gear 1 (on the input shaft) Gear 2 (meshes with gear 1) Gear 3 (on the same shaft as gear 2) Gear 4 (output) Motor power Motor speed Gear sizes are drawn in proportion to tooth count. Efficiency is taken as 98% per mesh, typical of a well-made gear pair. Gears 2 and 3 share one shaft, so they turn at the same speed (gear 4 is drawn in front of gear 2). The same idea has other forms. A belt or a chain moves power between shafts that are far apart: a chain or a toothed belt doesn't slip, so the ratio is fixed by the tooth counts; a smooth or V-belt can slip, which can protect the machine and also loses a little. Rough rule of thumb: gears for compact, exact, high-power drives with oil; chains for long, strong, rougher drives; belts for quiet, light, lubrication-free ones. Shafts and bearings: the parts that turn A shaft carries torque from one part to another: a motor to a gear, a gear to a wheel. When a shaft twists, the outer fibres are stretched in shear the most, and the stress there is: shear stress = 16 × torque ÷ ( π × diameter³ ) The cube is the important thing. Double the diameter and the same torque produces one-eighth of the stress, so a shaft that is 26% fatter (the cube root of 2) carries twice the torque. That is why shafts are thick in the middle where torque and bending are highest, and step down where the loads are lower: material is expensive and heavy, and diameter is a cheap lever. To size a shaft, find the torque from P = T ω, decide how much stress the material may carry (a fraction of its yield, divided by a factor of safety we'll meet below), and solve for the diameter. Try it. This is the twisting load only; real shafts also bend and have keyways, which is what the later sections are about. Interactive · How thick must the shaft be? Mild steel Aluminium 6061-T6 Alloy steel, heat treated Power Speed Factor of safety Allowed shear stress = 0.577 × yield strength ÷ factor of safety (the standard distortion-energy estimate). Yields are typical values; the alloy steel is illustrative. Pure twist only. A bearing lets a shaft turn while something else holds it in place. Bearings don't fail all at once: the steel balls and tracks wear a fatigue pit after so many revolutions. Makers rate each bearing with a load C at which 90% survive a million revolutions, and life scales with a cube for ball bearings: life (millions of revolutions) = ( C ÷ P )³ Halve the load and life goes up eightfold; double it and it drops to an eighth. Interactive · Bearing life Load on the bearing Speed Ball bearing, the 90%-survival rating life (L10), no allowance for contamination or lubrication quality. Roller bearings use a different exponent. A bolt is a spring you stretch on purpose A spring is simply a part designed to deflect: force equals stiffness times stretch, F = k x, and it stores energy. Two springs side by side add their stiffnesses; two in a chain are softer than either. Now look at a bolt differently: a tightened bolt is a long, thin spring, stretched by the nut, and what it squeezes (the plates) is a short, fat spring, compressed. The two forces are equal and opposite, which is the clamping force, called the preload. Here is the surprise. When an outside load tries to pull the joint apart, the bolt doesn't carry most of it. The plates are far stiffer than the bolt, so they simply relax a little as the bolt stretches. The bolt's load rises only by its stiffness share C of the outside pull (often around 0.2), while the clamp force falls by the rest. As long as the joint stays clamped, the bolt sees a small fraction of any cycling load, which is why a well-preloaded bolt survives vibration that a loose one would not. Interactive · What a preloaded bolt feels M8 M10 M12 M16 Tightening (preload) Bolt's share of joint stiffness Outside pull on the joint Class 8.8 metric bolts: proof stress 580 MPa up to 16 mm, with standard tensile-stress areas. Torque from the common rule T = 0.2 × F × d; the factor 0.2 varies from about 0.15 to 0.3 with finish and lubrication, so torque is only a rough guide to preload. A common rule is a preload of about 75% of proof load for reusable joints. Welded joints are the permanent alternative: no preload, no loosening, but the weld and the metal beside it (the heat-affected zone) are often the weakest and most sharply shaped part of the structure, which matters a great deal in the next section. Factor of safety: a number you choose Designing against failure starts with the simplest comparison: factor of safety = strength ÷ stress. A part with a yield strength of 300 MPa carrying 100 MPa has a factor of safety of 3. For a steady, static load, that's almost the whole story: stay below yield, with margin. The margin isn't a law of nature; it is a judgement. It covers loads you can't predict exactly, material scatter, manufacturing error, and what happens if the part fails. A garden chair can take a large one; an aircraft cannot afford to, so it takes a small one with very careful analysis and testing. US rules for large transport aircraft (14 CFR 25.303) apply a factor of 1.5 to the maximum loads expected in service. The same engineers who are generous on a hoist hook are tight on an aircraft wing, and both are rational. Fatigue and stress concentration: why parts break below their strength Static design has a trap. A part loaded and unloaded many times can break at a stress far below its strength. This is fatigue. It was first noticed on the railways: in 1842, a locomotive axle broke on the line from Versailles to Paris, near Meudon, and the carriages behind piled into it and caught fire. At least 55 people died (some accounts give far more). The engineer William Rankine, studying axle failures, recognised the importance of stress concentrations, and from about 1860 August Wöhler tested railway axles under repeated loading and showed that the swing in stress matters more than the peak, and that for steel there is a stress swing below which life seems unlimited: the endurance limit. For many steels it is roughly half the tensile strength. The second half of the trap is stress concentration. Force flows through a part like water through a pipe, and a hole, a sharp corner or a notch makes it bunch up. For a hole in a wide plate in tension, the elastic theory (Inglis, 1913) gives the peak stress at the hole as the average stress times (1 + 2a/b), where a is half the hole's width across the load, b its depth along it. For a round hole (a = b) that is exactly 3. For a thin sharp slit it is enormous. Fatigue cracks always start at such places. Worked number for steel with a tensile strength of 500 MPa, with its endurance limit set at 250 MPa. A smooth bar swinging ±100 MPa is far below the limit and should last indefinitely. Drill a round hole in it: the peak at the hole is 300 MPa, above the limit, and the bar lasts about 117,000 cycles, which is about 1.3 hours at 25 cycles per second. Same load, same metal, one hole. Interactive · Make a notch, count the cycles Smooth bar Round hole Slot, 3:1 Crack-like, 10:1 Nominal stress swing (±) Notch sharpness a/b A model steel: tensile strength 500 MPa, endurance limit 250 MPa (half), life curve fitted through 450 MPa at 1,000 cycles and 250 MPa at a million. Real parts also lose strength from surface finish, size and temperature, and a real notch is partly smoothed by yielding. The slider is a teaching model, not a design tool. The de Havilland Comet, the first jet airliner, showed it again. Two Comet 1s broke up in the air in January and April 1954 (35 and 21 people died). Investigators at the Royal Aircraft Establishment pressurised a whole airframe in a water tank, over and over, and it burst after about 3,000 pressurisation cycles (real and simulated flights). The failure began at a fatigue crack growing from a rivet hole at the corner of a window-like cut-out for the direction-finder aerial in the roof, where the stress was concentrated. A popular version of this story blames square passenger windows, but the investigators' report was about those aerial cut-outs, and the passenger windows were not the cause. The real lessons were to design for the stress concentration, to test for fatigue, and to build in ways to stop a crack from running. The design process: from requirements to prototype Machine design is not one calculation; it is a loop. Requirements come first, in numbers: how much load, how fast, for how many years, at what size and cost. Then concepts: several ways to do it (gear, chain or belt? bolted or welded?), compared honestly against the requirements. Then analysis: the sizing and fatigue checks above, plus simulation, on the best concept. Then a prototype and test, which will almost always show something the analysis missed. Then round again. Engineers genuinely differ on the balance. Some favour analysing carefully before building anything; others argue for building and testing cheaply and often, which 3D printing has made easier. Most agree that where a failure kills people, the analysis comes first. The best designs usually come from doing both. What a graduate just sees Power is torque times speed, so gears and belts trade one for the other. A shaft's strength goes with diameter cubed; bearing life goes with load to the minus third. A bolt is a stretched spring; preload makes it feel a small fraction of the outside load. A factor of safety is a decision about what is unknown and what failure costs. Repeated stress breaks parts far below their strength; steel has an endurance limit of roughly half its tensile strength. A hole concentrates stress by 3, a sharp notch by far more, and fatigue cracks start there. Welds, corners and keyways are where machines break. Design is a loop of requirements, concepts, analysis and prototype. Where it shows up next Systems Engineering & the Capstone (34/22): choosing a drive train and fastening scheme is the first big decision a capstone team makes. Vibrations (34/17) and Dynamics (34/12): rotating shafts have critical speeds, and gear teeth meshing is a vibration source. Control Systems (34/20): a motor and gearbox behind a feedback loop; gear backlash and inertia change how the loop behaves. Materials Science & Manufacturing (34/13) and Engineering Drawing & CAD (34/18): the steel grade, heat treatment, fits and surface finish you choose appear on the drawing. In the real world: why a wind turbine's gearbox is among its most expensive repairs, why cranes are inspected for cracks, why a loose wheel nut is dangerous. If you want to go deeper 📘 Standard texts: Richard Budynas and Keith Nisbett, Shigley's Mechanical Engineering Design (McGraw Hill, 11th edition 2020; a 12th edition is out), the usual course text for everything above. Robert Norton, Machine Design: An Integrated Approach (Pearson), more worked examples and design-case flavour. 📘 Gentler: Henry Petroski, To Engineer Is Human: The Role of Failure in Successful Design (1985), how failures teach engineers, in plain language. J. E. Gordon, Structures: Or Why Things Don't Fall Down. 🌐 Free course: MIT OpenCourseWare 2.72, Elements of Mechanical Design (bearings, springs, gears, cams, mechanisms, with a lab project). The takeaway Machines are built from a short list of parts, each with one governing equation: power is torque times speed, a shaft's strength goes with diameter cubed, a bearing's life with load to the minus third, and a preloaded bolt is a spring that shields itself from the load. Designing against failure means a deliberate factor of safety for steady loads and, for repeated ones, respect for fatigue and above all for notches. Break the part in your head before it breaks in the field. Try this: look at a pair of scissors, a bicycle crank or a door hinge and find the standard parts in it (shaft, bearing, bolt, spring, gear). Then find where it would crack first: look for a sharp internal corner, a hole, or a weld. Sources & Further Reading Sources. Versailles rail accident, 8 May 1842 (at least 55 dead; accounts range up to about 200), Rankine and stress concentration, Wöhler's axle tests from about 1860 and 1870 summary: Wikipedia, "Fatigue (material)" and "Versailles rail accident", and the Open University's Tay Bridge materials. Comet: Wikipedia, "de Havilland Comet" (G-ALYP, 10 January 1954, 35 dead, 1,290 pressurised flights; G-ALYY, 8 April 1954, 21 dead, 900 flights; test airframe G-ALYU burst after 3,057 cycles; cause at the ADF window cut-out; square-window account described there as a myth). Inglis stress concentration (1 + 2a/b, 3 for a circle): Wikipedia, "Stress concentration". Endurance limit of steel about 0.5 × tensile strength up to 1400 MPa: Shigley-based lecture notes. Factor 1.5: 14 CFR 25.303. Class 8.8 bolts (proof stress 580 MPa up to M16, M12 stress area 84.3 mm²): ISO 898-1 summaries. T = K F d with K about 0.2 and preload 75% of proof for reusable joints: Shigley as summarised in university case studies and lecture notes. Computed in node: gearbox numbers, shaft diameters (5 kW at 1,500 rpm: 31.8 N·m, 15.0 mm in mild steel at a factor of 3), bearing lives, bolt loads and torques, and the fatigue lives (117,000 and 2,250 cycles). 🌐 MIT OpenCourseWare: 2.72 Elements of Mechanical Design 📘 Budynas and Nisbett, Shigley's Mechanical Engineering Design 📘 Petroski, To Engineer Is Human 📘 Norton, Machine Design: An Integrated Approach Step 19. Control Systems: How Machines Hold a Setpoint ------------------------------------------------------ Why: Feedback is how a machine holds a speed, a heading or a temperature on its own. A thermostat, a cruise control and a hovering drone do the same thing: measure, compare, correct. Why that works, and why too much of it makes things oscillate. You set the cruise control to 100 km/h and drive up a long hill. The engine works harder and the speed barely moves. Nobody told the car about the hill. A thermostat holds a room at 21 °C through a heatwave and a cold snap. A drone hovers in a gusty wind as if hung from a hook. A rocket booster lands upright. All of them do the same thing: they measure what is happening, compare it with what should be happening, and push on the difference. That loop is feedback, and control systems is the course on how to do it well, and what happens when you do it badly (it oscillates, sometimes violently). The course in one box Who takes it: mechanical, aerospace, electrical and chemical engineers, usually in year 4 (sometimes late year 3). What it's for: making a machine hold a speed, a heading, a temperature or a position on its own, even when the world pushes back. Comes after: Differential Equations (34/03), Vibrations (34/17) and Linear Algebra (34/04). Leads to: Systems Engineering & the Capstone (34/22), where the controller has to agree with every other subsystem. In Computer Science it meets Software Engineering (34/31) in embedded code and Machine Learning (34/32) in reinforcement learning, which is control with a learned rule. Honest line: you won't be able to design a flight controller after this. You will see why a loop oscillates when its gain is too high, and what the three letters PID are doing in the car you drive. Open loop versus closed loop A toaster is open loop: you pick a time, it heats for that long and stops, whether the bread is pale or black. It never looks at the result. A thermostat is closed loop: it measures the room and decides, again and again, whether to heat. Open loop works when you know the system exactly and nothing disturbs it. The real world is not like that, and closed loop is how machines cope. Here is the cruise control, as a toy car with a first-order lag (a time constant of 25 s). In open-loop mode the throttle is set once to hold 25 m/s on flat ground. Add a hill and watch. Interactive · Open loop, then closed Open loopClosed loop (P) Hill steepness Feedback gain Open loop on a 2% hill: speed drops about 4.7 m/s (17 km/h) and stays down. A toy car: 1,200 kg, drag 48 N per m/s, engine force 0 to 3,000 N, starts at 20 m/s. Closed loop adds force in proportion to the speed error. Gain 600 cuts the loss on a 2% hill to about 0.36 m/s. Feedback and the block diagram Control engineers draw every loop as a block diagram: boxes for things that transform a signal, arrows for the signals. The set speed goes in; a comparator subtracts the measured speed to give the error; the controller turns the error into a command; the actuator (here the engine) does the pushing; the plant (the car) responds; the sensor (a speed sensor) reports back. A disturbance, the hill, enters from the side. ControllerActuatorPlantSensor set speederrorPIDenginecarhillspeed−+ Why is this so powerful? Suppose the plant's gain G is uncertain by ±30% (a heavy load, a worn part). Open loop passes that uncertainty straight to the output. With feedback and a high controller gain K, the output is KG/(1 + KG), which is close to 1 for any G, so the loop all but ignores the uncertainty. That is also why feedback rejects disturbances: the error shows up and the controller pushes back. Interactive · Feedback shrinks uncertainty Controller gain K With K = 10, plant gains of 0.7, 1.0 and 1.3 give 0.875, 0.909 and 0.929 instead of 0.7, 1.0 and 1.3. Command = 1. Open loop output = G; closed loop output = KG ÷ (1 + KG). The leftover shortfall is what integral control, below, removes. First- and second-order response: time constant, overshoot, settling To judge a controller you look at its step response: how the output follows when the command suddenly jumps. Two shapes cover most of what you meet. First order is a smooth exponential approach, like the car's speed or a cooling cup of tea. It has one number, the time constant τ: the output covers 63% of the gap in one τ, 95% in 3τ and 98% in 4τ, which engineers call "settled". It can never overshoot. Second order has inertia as well as lag, like the mass-spring of Vibrations (34/17), and it is described by the very same two numbers: the natural frequency ωn and the damping ratio ζ. Low ζ gives fast but overshooting ringing; ζ near 0.7 gives about 5% overshoot and is a common design target. The overshoot depends only on ζ: 53% at ζ = 0.2, 16% at 0.5, 4.6% at 0.7. Settling time is roughly 4 ÷ (ζωn). Interactive · Step response First orderSecond order Time constant τ Natural frequency ωₙ Damping ratio ζ First order, τ = 2 s: 63% at 2 s, 98% at 8 s. Never overshoots. Exact formulas, checked against a step-by-step numerical solution. In second order: ζ = 0.2 gives 52.7% overshoot, 0.5 gives 16.3%, 0.7 gives 4.6%. Transfer functions and Laplace, in plain words Here is the trick that makes control theory work. Differential equations are awkward to combine: put two systems in a chain and you must solve their equations together. The Laplace transform turns calculus into algebra. It replaces "take the derivative" with "multiply by s", so that a differential equation becomes a fraction, the transfer function: output divided by input, written in terms of s. Chains of systems become products, and feedback loops become the formula you already met, G ÷ (1 + GC). The part a graduate carries for life is what the fraction's roots, the poles, mean. Each pole is an exponent the system can "ring" with: a pole at s = −σ gives e−σt, a decay with time constant 1/σ. A pair at s = −σ ± jω gives a decaying oscillation at frequency ω. The first-order car has one pole at −1/τ. The mass-spring has the pair −ζωn ± jωn√(1 − ζ²). If every pole sits on the left side of the s-plane, every response dies out: the system is stable. One pole on the right and something grows without limit. Interactive · Place a pole Real part (decay or growth) Imaginary part (ringing) Pole at s = −0.8 ± 3.0j: stable, ringing about 0.5 times a second. The response plotted is eσt·cos(ωt). Left of the vertical axis everything decays; right of it, everything grows. PID control, explained with a cruise control The workhorse controller turns the error into a command with three terms: P (proportional): the present. Push in proportion to the error now. Simple, but it leaves a permanent leftover error when something (the hill) needs a steady push, because a push needs an error to exist. I (integral): the past. Add up the error over time and push on the total. A leftover error keeps growing the sum until the leftover is gone. This is what removes the steady-state error, but too much of it overshoots. D (derivative): the future. Push against how fast the error is changing, a guess at where it is heading. It adds damping, but it amplifies sensor noise. u = Kp·e + Ki·∫e dt + Kd·de/dtcommand = P × error + I × accumulated error + D × rate of change of error This three-term rule runs most of industry. Åström and Hägglund, authors of the standard text on PID, report that more than 95% of the control loops in process industries are of the PID type, and most of those are really PI. Ziegler and Nichols published the classic tuning rules in 1942 and they are still taught. Try tuning the cruise control yourself. Start with P only: see the leftover. Switch to PI: the leftover vanishes. Then switch on sensor noise and add D. Interactive · Tune the cruise control P onlyPIToo much IPID Clean sensorNoisy sensor Hill steepness P gain I gain D gain P only on a 2% hill: about 0.36 m/s below the set speed, for good. With PI: back to the set speed. The same toy car, with a feed-forward trim of 1,200 N plus PID on the measured speed (anti-windup included). Noise: 0.15 m/s of random error on the speed sensor, fixed seed so runs repeat. Stability: why too much gain oscillates If a little feedback is good, is more better? Not forever. Imagine steering a car by looking only at your distance from the centre line. If you correct with too much force, you overshoot, correct back too hard, and weave. A real loop has delays: sensors take time, computers take time, motors take time. A correction that arrives late can push at the wrong moment, adding energy instead of removing it. Past a certain gain a loop goes from settling to ringing to growing: it has gone unstable. It is the same negative-damping story as the bridges in Vibrations. The drone below holds an altitude. Its plant is almost pure inertia (thrust makes acceleration), so P alone swings it up and down forever: there is no damping in the plant. The D term is the missing damper (ζ = Kd ÷ 2√Kp). Now add what real hardware does: delay the commands, or update them only every so often. Interactive · Hold the altitude P onlyWell damped250 ms delayUpdates every 0.3 s P gain D gain Delay in the loop Controller update interval Well damped: settles on the target with no overshoot. A 1 kg drone, thrust limited to ±1 g about hover, 0.3 m step. P only never settles (it swings at √Kp rad/s). With the D term, a 250 ms delay is enough to make it grow, and a 0.3 s update interval makes the higher-gain setting ring. Sensors, actuators and digital control A loop is only as good as what it can see and what it can do. Sensors (encoders, gyros, thermocouples, pressure and GPS) set the noise floor, and you can't control what you can't measure. Actuators (motors, valves, heaters, control surfaces) have limits: the engine in the sim can't produce negative force or more than 3,000 N. When an actuator saturates while the integral keeps adding up, the controller "winds up" and overshoots badly, so real controllers freeze the integral at the limits. The sim does exactly that. Today nearly every controller is digital: a microcontroller samples the sensor, computes, writes an output, and holds it until the next sample. The delay and the holding are real and are what the "update interval" slider is imitating. A textbook rule (Franklin, Powell and Emami-Naeini) is to sample about 20 times faster than the closed-loop bandwidth you want, and other texts quote anywhere from 10 to 30. So a flight controller usually runs its loop tens to hundreds of times a second. This is where this course meets Computer Science: a control loop is a real-time program with a deadline, and missing the deadline is a stability problem, not a slowdown. What a graduate just sees Every automatic machine is a loop: measure, compare, correct. Ask what the sensor sees and what the actuator can do. Open loop is a bet that nothing changes; feedback is how you stay right when it does. A first-order system has a time constant (98% settled in four of them); a second-order one has ζ and ωn, and ζ sets the overshoot. Poles tell the story: left of the axis it decays, right of it it grows, and the imaginary part is the ringing. P alone leaves an error, I removes it, D adds damping, and each has a cost (overshoot, windup, noise). More gain is faster until delay turns correction into a push in the wrong direction. Real actuators saturate, real sensors are noisy, and real computers sample. A controller is a damper you can reprogram. Where it shows up next Systems Engineering & the Capstone (34/22): a controller is useless without sensors, actuators, power and software that agree with each other; the capstone is where they are tested together. Vibrations (34/17): feedback is active damping; a flight controller does for a drone what a viscous damper does for a bridge. Propulsion (34/21): engines run on control loops for fuel flow and turbine speed. The real world: thermostats, cruise control, 3-D printers, robot arms, drones, aircraft autopilots, rockets, chemical plants, insulin pumps. Computer Science and Electrical Engineering: embedded and real-time software (Embedded Systems, 34/41; Operating Systems, 34/26) run the loops; reinforcement learning (34/32) learns a control rule from experience instead of deriving it. If you want to go deeper 📘 Textbook: Franklin, Powell and Emami-Naeini, Feedback Control of Dynamic Systems (Pearson; 8th edition): the standard undergraduate text, with a full digital control chapter. 📘 Textbook: Katsuhiko Ogata, Modern Control Engineering (Pearson): a long-time classroom favourite, strong on transfer functions and root locus. 📘 Gentler (free): Karl Johan Åström and Richard Murray, Feedback Systems: An Introduction for Scientists and Engineers (Princeton University Press; 2nd edition, 2021), which its authors keep available free online. Also Brian Douglas's Control Systems Lectures on YouTube and Steve Brunton's Control Bootcamp, both free video series. ▶️ Free course: MIT OpenCourseWare 2.04A Systems and Controls (Spring 2013): from differential equations to Laplace, transfer functions and root locus, with a lab on a rotating flywheel. 🌐 Easier first read in this library: Fourier Transform (12/12). The takeaway Feedback means measuring, comparing and correcting, and it lets a machine hold a setpoint despite hills, loads and uncertainty. Systems respond with a time constant (first order) or with overshoot set by the damping ratio (second order), and their poles show whether they will decay, ring or blow up. PID combines the present, the past and the future of the error, and every term costs something. Too much gain, or too much delay, turns correction into oscillation, which is why stability comes before performance. Try this: take a hot shower and adjust the tap. Notice how long the water takes to answer. If you turn it again before the last change arrives, you swing between scalding and freezing. That is a loop with delay and too much gain, and you are the controller. Sources & Further Reading Sources. Block diagrams, step response, Laplace and stability are standard textbook material (Franklin et al.; Ogata; Åström and Murray). PID prevalence (over 95% of process-control loops, mostly PI): K. J. Åström and T. Hägglund, PID Controllers: Theory, Design and Tuning and Advanced PID Control. Tuning rules: J. G. Ziegler and N. B. Nichols, "Optimum settings for automatic controllers", Transactions of the ASME 64 (1942) 759–768. Sampling rule of thumb: Franklin, Powell and Workman / Emami-Naeini, discussion of digital control (about 20 times the bandwidth; other texts say 10 to 30). Computed in node: the exact step responses (checked against numerical integration), overshoot and settling figures, the cruise-control and drone simulations and the stability thresholds quoted in the notes. 🌐 MIT OpenCourseWare 2.04A Systems and Controls 🌐 Åström and Murray, Feedback Systems (2nd edition site) ▶️ Steve Brunton, Control Bootcamp 📘 Franklin, Powell and Emami-Naeini, Feedback Control of Dynamic Systems Step 20. Propulsion: Throwing Mass Backwards -------------------------------------------- Why: Jets and rockets: pushing mass backwards to go forwards. A balloon, a jet, a rocket and an ion thruster are the same machine: push mass backwards and go forwards. The rest is a fight to do it with the least fuel. Blow up a balloon, pinch the neck, let go. It shoots around the room, and nothing is pushing against anything: there is no ground in the air for it to shove off. A fire hose is the same idea with muscle. Firefighters brace themselves because the water leaving the nozzle shoves the hose backwards. That is the whole of propulsion. A jet engine, a rocket and an ion thruster are the same machine at three scales: throw mass backwards, and the machine goes forwards. The rest of the subject is about how to do that cheaply, with the least fuel for the push you need. The answers are surprising: an airliner wants huge, gentle exhaust, a rocket wants a hot, fast one, and a satellite wants a nearly invisible one. The course in one box Who takes itAerospace engineers, in year 4, and mechanical engineers who choose the energy or aerospace track. What it is forWorking out how much push a given engine can make, and how much propellant a given journey costs. Comes afterThermodynamics (34/14) and Fluid Mechanics (34/15). Leads toSystems Engineering & the Capstone (34/22), where an engine is one subsystem of a whole vehicle. Heat management from Heat Transfer (34/16) matters in every engine. What this piece gives youYou will not be able to design a turbine. You will see why a rocket is mostly fuel, why an airliner's engine is mostly a fan, and why a spacecraft can use a thruster that pushes with the force of a sheet of paper. Easier first reads in the library: Flying Machines (29/05) and Rockets and Spaceflight (29/06) for the history; this piece is the physics. Thrust is momentum flux: push mass backwards Newton's second law says force equals the rate at which momentum changes. An engine takes in some mass each second, speeds it up, and shoves it out the back. The momentum it adds per second is the force on the engine. For a rocket standing still, it is as simple as it gets: F = ṁ × vethrust = mass thrown per second × speed it is thrown at. A jet moving at speed v0 uses (ve − v0), since the air arrived already moving. Notice you can get the same thrust from many different recipes. A lot of mass thrown slowly, or a little mass thrown very fast. They are not equal in cost, though. Making the exhaust move costs kinetic energy, ½ ṁ ve², and that grows with the square of the speed. Slide the exhaust speed below for a fixed 100 kN push. Interactive · The same push, different bills Exhaust speed A stationary engine, thrust fixed at 100 kN. Mass flow = F ÷ ve; power = ½ F ve. At 1,000 m/s you throw 100 kg a second and pay 50 MW. At 10,000 m/s you throw only 10 kg a second, but the exhaust carries away 500 MW. Fast exhaust saves propellant and costs power. Which one matters more depends on whether you are carrying the propellant (a rocket) or scooping it from the air for free (a jet). Propulsive efficiency: why airliners use big, slow jets A moving vehicle has a second problem: the exhaust is not wasted only if it ends up still. If a jet flies at 250 m/s and throws gas back at 600 m/s, the gas leaves the scene still moving at 350 m/s relative to the ground. That leftover kinetic energy is lost. The fraction of the engine's energy that does useful pushing is the propulsive efficiency: ηp = 2 ÷ (1 + ve ÷ v0)best when the exhaust is only a little faster than the vehicle. At ve = v0 it reaches 100% (and thrust vanishes). So at the same thrust, the best engine for a slow vehicle moves a large amount of air a small amount faster. That is the logic of the propeller: a huge disc that grabs a lot of air and speeds it up gently. It is also the logic of the turbofan. A jet engine's core makes hot gas. Instead of sending it all out the back, most of its power drives a big fan at the front, which pushes a stream of air around the core. The bypass ratio is the mass of air going around the core divided by the mass going through it. Interactive · Add a fan, keep the core Bypass ratio Flight speed Idealised: the same core (50 kg/s of air) adds the same jet power at every setting, shared evenly over core and bypass air. Ignores fan and nozzle losses. It shows the trend, not a real engine. At 250 m/s (about 900 km/h, an airliner's cruise), a bare core gives 17.5 kN at 59% efficiency. Bypass ratio 8, with the same core, gives about 26.6 kN at 89%: more push for the same fuel. Now raise the flight speed towards 400 m/s: the advantage shrinks, and the fan, which is fat and drag-heavy, stops paying for itself. That is why airliners cruise at high subsonic speed with high-bypass fans and fighters use thin, low-bypass engines. Published ratios run roughly 0.3 to 0.7 for combat engines and about 5 to 12 for airliners: the LEAP-1A is around 11:1, the GE9X around 10:1 and the PW1000G about 12:1. Inside a jet: the Brayton cycle and the engine family The core of every gas-turbine engine is the Brayton cycle you met in Thermodynamics (34/14): a compressor squeezes the air, a combustor burns fuel in it continuously, and a turbine takes some of the hot gas's energy to turn the compressor. What is left either shoots out as a jet or turns a fan or a propeller. The ideal efficiency 1 − (pressure ratio)−0.286 rises with pressure ratio, which is why engines push the compressor harder and the combustor hotter. The turbine runs in gas hotter than its blade metal would survive uncooled, which is where Heat Transfer (34/16) comes in. EngineWhat gives the pushSweet spot TurbopropCore drives a propeller through a gearboxSlower, lower-flying aircraft High-bypass turbofanMostly the big fan, a bit of core jetAirliners, just under the speed of sound Low-bypass turbofan, turbojetMostly the fast core jet (often with afterburner)Fighters, supersonic flight RamjetNo compressor: speed squeezes the airRoughly Mach 3 to 6, needs to be moving already ScramjetBurns fuel in air that stays supersonicFrom about Mach 5 or 6 upward Take away the compressor and the turbine, and let speed do the squeezing. That is a ramjet. Slow the incoming air to a stop and you recover its kinetic energy as pressure and heat. Interactive · Compression for free Flight speed Isentropic stagnation relations for air (γ = 1.4) at 216.65 K, the temperature at 11 km in the standard atmosphere: T₀ = T(1 + 0.2M²), p₀/p = (1 + 0.2M²)3.5. At Mach 3 the ideal pressure ratio is about 37 to 1, as good as a big compressor, with no moving parts. At Mach 5 slowing the air fully would heat it to about 1,300 K (over 1,000 °C), and faster still would heat it until its molecules start to break apart. That is why a scramjet keeps the air supersonic while it burns fuel. Scramjets are hard: NASA's X-43A reached about Mach 9.6 on 16 November 2004, the record for an air-breathing aircraft, in a flight lasting seconds. Both ramjets and scramjets produce no thrust standing still, so they need a rocket or another engine to get them up to speed. The rocket equation: why rockets come in stages A rocket carries its own propellant, and that is the catch. To speed up, it has to accelerate its fuel too, and the fuel it has not yet burned is dead weight until it does. Konstantin Tsiolkovsky wrote the answer down in 1903: Δv = ve × ln(mstart ÷ mend)speed gained = exhaust speed × natural log of the mass ratio. Independent of thrust or time. The logarithm is the bad news. To add a fixed amount of speed you must multiply the starting mass by the same factor each time. With an exhaust speed of 3.4 km/s (Isp 350 s), every extra 3.4 km/s takes you from a mass ratio of e (2.7) to e² (7.4) to e³ (20). Reaching low orbit takes about 9.4 km/s once you include the energy lost to gravity and drag; published figures run about 9 to 10 km/s depending on the trajectory. A single tank-and-engine would need almost all of its mass to be propellant. The fix is to drop the empty tanks. When the first stage is empty, you throw its engines and tank away and continue with a lighter rocket, so the remaining fuel has less mass to shove. All orbital launchers in use today work this way. Try it. Interactive · Do you need stages? 1 stage 2 stages 3 stages 4 stages Specific impulse Empty weight of each stage, as a share of its tanks plus structure Payload as a share of lift-off mass Identical stages sharing the same structural fraction and payload ratio, Isp constant, no gravity or drag losses beyond the roughly 9.4 km/s target. A toy model: real stages differ in size, engines and structure. With the defaults, one stage reaches 7.3 km/s: short of orbit. Two stages reach 10.7 km/s. The same payload fraction and the same engines; the only change is throwing the empty tanks away. Now drag the structure slider up to 20%: a single stage cannot get anywhere near. Rocket designers fight for every percent of structure for exactly this reason. Specific impulse: miles per gallon for rockets A car has miles per gallon. A rocket has specific impulse: the thrust you get per unit weight of propellant burnt each second. Divide thrust by the propellant's weight flow and the answer comes out in seconds. Isp = F ÷ (ṁ g0) = ve ÷ g0g0 = 9.80665 m/s². Higher is better: more speed change from each kilogram of propellant. It is the exhaust speed in disguise: an Isp of 450 s means an exhaust speed of about 4.4 km/s. What sets it is the propellant and the engine. Hot, light exhaust is fastest, so hydrogen burnt with oxygen leads the chemical rockets. Solid rockets (propellant cast like a rubbery block): simple, storable, enormous thrust, roughly 250 to 270 s. Once lit they essentially cannot be throttled or shut down at will, so they work as boosters and missiles. The Space Shuttle's solid boosters were about 268 s in vacuum. Liquid kerosene and oxygen: roughly 300 to 350 s in vacuum. SpaceX's Merlin 1D is about 311 s in its sea-level form and about 348 s in the vacuum version, per published specs, which vary by source. Liquid hydrogen and oxygen: about 450 s. The Space Shuttle's RS-25 engine is rated 452 s in vacuum and 366 s at sea level. Hydrogen is very light, so the tanks are huge. Air-breathing jets do not carry their oxidiser, so counted the same way their Isp is several thousand seconds, which is why jets beat rockets inside the atmosphere. Sea level versus vacuum differ because the atmosphere pushes back on the nozzle; the same engine does better in space. Slide the Isp below and watch what a mission costs. Interactive · What a mission costs in propellant Solid Kerosene Hydrogen Ion thruster Specific impulse Speed change wanted For a 4 km/s speed change on a craft that ends at 1,000 kg: Isp 250 s needs about 4,110 kg of propellant, Isp 450 s needs 1,480 kg, and Isp 3,100 s needs 141 kg. That is why people care about every second of Isp. Electric propulsion: tiny thrust, huge Isp If exhaust speed is what matters, use electricity instead of chemistry. An ion thruster ionises a gas such as xenon and accelerates the ions through an electric field to tens of kilometres per second. NASA's NSTAR thruster, first flown on Deep Space 1 (launched 24 October 1998), reaches an Isp above 3,100 s with a maximum thrust of 92 mN at 2.3 kW of power. That is the weight of a couple of sheets of paper. The exhaust is 10 times faster than a chemical rocket's, and the propellant needed is a tenth as much. The price is time. With only a few kilowatts of solar power available, the push is tiny, so the thruster must run for months. Interactive · Chemical rocket or ion drive? Chemical Ion (NSTAR-like) Propellant carried A 1,000 kg craft in free space, ignoring gravity. The ion numbers follow NASA's NSTAR figures; the chemical thruster (320 s, 450 N) is an illustrative round number. With 100 kg of propellant the ion thruster gives a speed change of about 3.2 km/s, ten times the chemical thruster's 0.33 km/s, but it takes about 380 days of continuous thrusting, against under 12 minutes. The acceleration is about a hundred-thousandth of a metre per second squared. Electric propulsion is a long-haul tool: station-keeping, raising orbits and deep-space probes. It cannot lift anything off the ground. Check the arithmetic: thrust power ½ F ve is about 1.4 kW out of 2.3 kW in, a thruster efficiency of about 61%, in line with NASA's stated figure of above 61%. What a graduate just sees Every vehicle moves by throwing something backwards. Thrust is mass flow times velocity change. The same push can come from a lot of slow mass or a little fast mass, and the kinetic-energy bill climbs with the square of the speed. A vehicle wastes energy when its exhaust is much faster than itself. That is why airliners use huge fans and propellers use huge discs. A jet engine is a Brayton cycle in a tube. Pressure ratio and turbine temperature are the two numbers that matter. A rocket is mostly propellant, because the rocket equation is a logarithm: each extra bit of speed multiplies the mass. Staging works by throwing away dead weight at the right time. Isp is the exhaust speed in seconds. A higher Isp is a smaller tank for the same journey. Thrust and Isp are different goals. Chemical engines trade Isp for thrust; ion engines do the opposite. Where it shows up next Systems Engineering & the Capstone (34/22): engine mass, propellant and structure fight each other in a vehicle design, and the rocket equation is the referee. Heat Transfer (34/16), Machine Design (34/19): turbine blades, nozzles and bearings are heat transfer and machine design under extreme conditions. Control Systems (34/20): engines are throttled, gimballed and steered by feedback. In the world: the LEAP and GE9X engines on airliners, reusable first stages, GPS and communication satellites that keep station with electric thrusters, and any argument about what a fuel or an engine "should" cost per kilometre. If you want to go deeper 📘 Textbook. Jack Mattingly and Keith Boyer, Elements of Propulsion: Gas Turbines and Rockets (AIAA, 2nd edition, 2016). A standard undergraduate text for both jets and rockets. 📘 Textbook. George Sutton and Oscar Biblarz, Rocket Propulsion Elements (Wiley, 9th edition). The classic for liquid, solid and electric rockets. 📘 Gentler. John D. Clark, Ignition! An Informal History of Liquid Rocket Propellants (Rutgers University Press, 1972; reissued 2018): a chemist's funny, vivid account of how rocket fuels were found. Or the library's own Rockets and Spaceflight (29/06) for the history. 🌐 Free course. MIT OpenCourseWare 16.01 Unified Engineering: Thermodynamics and Propulsion (Fall 2005): thrust, efficiency, rocket performance and ideal cycle analysis of jet engines. For rockets in depth, 16.512 Rocket Propulsion (graduate). The takeaway Propulsion is throwing mass backwards: thrust is mass flow times speed. A fast exhaust saves propellant but costs power, and it wastes energy if it is much faster than the vehicle, so airliners use big slow fans. A rocket carries its own propellant, so the rocket equation, Δv = ve ln(mstart/mend), makes it mostly fuel, and staging is how it sheds dead weight. Isp measures exhaust speed, and electric engines buy a huge Isp with a tiny thrust. Try this: next time you are on a plane, notice the engine's front: that giant fan is most of the engine. Most of the push comes from air that never goes through the burner. Sources & Further Reading Sources. Thrust as momentum flux, propulsive efficiency, ideal Brayton and ram relations and the rocket equation are standard textbook propulsion (Mattingly; Sutton & Biblarz; MIT 16.01). Tsiolkovsky 1903: Wikipedia, "Tsiolkovsky rocket equation" (others, including William Moore in 1810, outlined it earlier; Goddard and Esnault-Pelterie rederived it). Delta-v to low orbit of about 9.4 km/s (published estimates 9 to 10 km/s): Wikipedia "Gravity loss" and course notes. Bypass ratios: LEAP-1A about 11:1, PW1000G about 12:1, GE9X about 10:1 (trade glossaries, Wikipedia); combat engines about 0.3 to 0.7, airliners about 5 to 12. RS-25: 452.3 s vacuum, 366 s sea level. Merlin 1D: about 311 s vacuum (sea-level engine) and about 348 s (vacuum variant), specs vary by source. Shuttle SRB about 268 s vacuum (The Space Review, citing NASA). NSTAR: NASA tables, maximum thrust 92 mN, Isp above 3,100 s at 2.3 kW, efficiency above 61%; Deep Space 1 launched 24 October 1998. X-43A: NASA and Guinness World Records, Mach 9.6 (Guinness lists 9.68) on 16 November 2004. Ramjet and scramjet Mach ranges vary by source (NATO RTO-EN-AVT-150). Everything an interactive shows was computed in node: 100 kN at 3,000 m/s needs 33.3 kg/s and 150 MW; bypass 0 → 17.5 kN, 58.8%, bypass 8 → 26.6 kN, 89.4% at 250 m/s; Mach 3 ideal ram ratio 36.7, T₀ = 607 K; Mach 5 T₀ = 1,300 K; staging with Isp 350 s, 10% structure, 2% payload: 7.35 km/s (1 stage), 10.66 km/s (2); propellant for 4 km/s on 1,000 kg: 4,112 kg (250 s), 1,475 kg (450 s), 141 kg (3,100 s), all with a 1,000 kg final mass; ion craft 3,203 m/s in 3.30 × 10⁷ s, chemical craft 331 m/s in 697 s. 🌐 MIT OCW 16.01: Thermodynamics and Propulsion 🌐 MIT OCW 16.512 Rocket Propulsion 📘 Mattingly & Boyer, Elements of Propulsion 📘 Clark, Ignition! 📘 Sutton & Biblarz, Rocket Propulsion Elements Step 21. Systems Engineering & the Capstone: Many Parts, One Product -------------------------------------------------------------------- Why: A product is many subsystems agreeing with each other; that's the capstone. A product is many subsystems agreeing with each other. Most famous failures happened on the lines between them. A phone in your pocket does several things at once. The radio talks to a tower, the screen glows, the processor thinks, the battery pushes current through all of it, and the whole slab has to stay cool enough to hold. Every one of those parts was designed by a different team. Yet when you press a button, they agree. Now imagine one of them quietly disagrees. One team counts in one unit, another in a different one. Nothing is wrong with any single part. Everything is wrong with the product. This subject is about that gap: not how to design a part, but how to make a hundred parts mean the same thing. Engineers who have taken it carry a habit for life. When someone proudly shows a clever component, they ask three questions before admiring it: what does it plug into, what does it assume about its neighbours, and what happens when it is wrong? The course in one box Who takes it: final-year mechanical and aerospace engineers, usually as a capstone (a year-long team project), and in many universities every engineering major has a version of it. What it is for: turning a pile of well-designed parts into one product that meets what someone actually needed, on a budget and a schedule. Comes after: Machine Design (34/19), Control Systems (34/20) and Software Engineering (34/31), which is also a Computer Science subject. Leads to: nothing on the map; it is the last stop of the Mechanical & Aerospace path. What it leads to is the job, where most of the work is this. What this piece gives you, and what it doesn't: you won't be able to run a real programme or write a requirements document an auditor would accept. You will see why a spacecraft is weighed more often than it is admired, and why the famous failures were almost all about neighbours disagreeing. A product is many subsystems agreeing A system is a set of parts whose combined behaviour is something none of them can do alone. A car is not an engine plus wheels. It is an engine that must fit in the space the body left, make heat the radiator can reject, and shake at frequencies the mounts can absorb. Each sentence in that list is an agreement between two subsystems. Systems engineers write those agreements down in two forms. A requirement says what the whole thing, or one part, must do, in a sentence a test can settle: "The device shall have a mass of no more than 12 kg." A good requirement can be checked by measuring; "lightweight" cannot. An interface says exactly what passes between two parts: the bolt pattern, the voltage and connector, the message format, the units. A document that holds these agreements is called an interface control document, and it is the most boring and most valuable thing in the project. Most of a system's trouble is not inside the parts. It is on the lines between them. The craft is a loop: break the whole into subsystems, give each a budget and an interface, let teams work, then bring the results back and see whether the whole still works. The V: build down, prove up The classic picture is a letter V. Down the left side you go from need to detail: what the customer wants, then system requirements, then the design of each subsystem, then the parts. At the bottom you build. Up the right side you test, and the key idea is that each level of testing answers to the level on the left at the same height. You test the part against the part's design, the assembled subsystem against its requirements, the whole product against the system requirements, and finally the product in real use against the original need. That gives the two words every systems engineer uses precisely. Verification asks whether you built it right: does it meet the requirements as written? Validation asks whether you built the right thing: does it actually solve the problem the person had? A product can pass every verification and still fail validation, if the requirements were wrong. The V-model has several roots: it appeared in the late 1980s in Germany (for defence procurement) and in the United States, and it is a teaching picture, not a law. Real projects loop back and forth across it, and agile software teams (see Software Engineering, 34/31) spiral through small versions of it. Interactive · Walk the V Now classify. Is each of these verification or validation? Worked example: a requirement says the part "shall have a mass of no more than 12 kg". You put it on a scale and read 11.6 kg. That is verification, and the requirement is met with 0.4 kg (about 3%) to spare. Whether anyone can lift a 12 kg part into place is a validation question, and the scale cannot answer it. Trade studies: choosing without a perfect answer Early in a project there are always several ways to do a job, and none is best at everything. A trade study scores each option against the criteria that matter, weights the criteria by how much they matter, and adds it up. The sum is not the point. The point is that the weights make the argument visible: two people who disagree about the answer are usually disagreeing about the weights, and now they can say so. score = (w1s1 + w2s2 + …) ÷ (w1 + w2 + …) Interactive · Wheels, tracks or legs? A small delivery robot. The scores (1 = poor, 10 = excellent) are invented for the exercise; the point is what the weights do. Drag a weight and watch the winner change. Try it: with the starting weights, plain wheels win comfortably. Put nearly all the weight on performance over rough ground and legs edge ahead. Nobody's arithmetic changed, only what they cared about. A mature team also checks sensitivity: how far can a weight move before the winner flips? If a tiny change flips it, the study has not decided anything yet. Budgets and margins: mass, power and heat Hardware obeys sums. A spacecraft has a mass budget (everything added up against what the rocket can lift), a power budget (everything it draws against what the panels and battery supply) and a thermal budget (everything it dissipates against what it can radiate away). Each subsystem gets an allowance. The system engineer's job is to add them up, often, and to be suspicious of the total. The suspicion is earned. Designs get heavier. NASA's systems engineering handbook, in an example, says spacecraft dry mass can grow by as much as 25 to 30 percent between detailed design and launch. So teams hold a margin: they add a deliberate percentage to the current best estimate, larger when the design is young and shrinking as parts are actually built and weighed. A margin is not padding. It is an honest admission of what you don't know yet. Interactive · A mass budget with margin An invented small satellite with a 500 kg launch limit. Its best estimate is the sum of the parts below. Choose how much margin you hold, then how much the design really grows. Margin you hold Growth the design really shows The best estimate here is 415 kg. With 20% margin the budget reads 498 kg, comfortably under 500. Then the design grows by 25%: 415 × 1.25 = 518.75 kg, and the satellite no longer fits. The margin was not too generous; the growth was bigger than the margin. That is how real programmes learn their margins were thin, usually late. The budgets are coupled, which is the deeper lesson. Take mass in a rocket. The rocket equation says that the ratio of liftoff mass to final mass is e raised to (Δv divided by exhaust speed). One extra kilogram of dead weight at the end therefore costs several kilograms of propellant at the start. mstart ÷ mend = eΔv ÷ (Isp × g0) Interactive · The snowball A stage with 1,000 kg of structure and payload that must change its speed by 4,000 m/s. The engine's specific impulse is 320 s (a good chemical engine). Add dead weight and see what it costs at the start. Extra dead weight Velocity change needed At 4,000 m/s the ratio is 3.58, so each extra kilogram at the end costs 2.58 kg more propellant, and 3.58 kg of extra liftoff mass in all. Ten kilograms of "just a bracket" becomes about 36 kg on the pad. This simple model ignores that the bigger tanks weigh more too, so reality is worse. This is why aerospace engineers fight for grams. Heat has a vacuum-sized version of the same coupling. In space a spacecraft can only get rid of heat by radiating it. The hotter and larger its radiating surface, the more it sheds, following the fourth power of temperature: P = ε σ A T4  →  A = P ÷ (ε σ T4) Interactive · Where does the heat go? Every watt the electronics use comes out as heat. With a radiator at 300 K (27 °C), emissivity 0.85, and ignoring sunlight, how much area must it have? Heat to reject Radiator temperature At 300 K, 100 W needs about 0.26 m² of radiator. So a power decision, a few more watts for a faster computer, becomes a thermal decision (more area or a hotter radiator) and a mass decision (more structure to carry it). One number changed and three subsystems moved. That chain is what "system" means. Interfaces are where the failures live Two famous losses show this better than any lecture, and both are now textbook cases. On 23 September 1999, NASA's Mars Climate Orbiter lost contact during its orbit-insertion burn. The investigation found the root cause was a failure to use metric units in a piece of ground software. A ground file that described the effect of the spacecraft's small thruster firings reported it in pound-force seconds. The navigation software that used them expected newton-seconds, as the specification said. One pound-force is 4.448 newtons, so every number was read about 4.45 times too small. The board noted that the project's own checks did not catch it: it cited a lack of complete end-to-end verification of the navigation software and models. It was meant to pass Mars at about 226 km. Calculations afterwards put it as low as about 57 km, below the roughly 80 km thought survivable. On 4 June 1996, the first Ariane 5 rocket, flight 501, veered and broke up about 37 seconds after the launch sequence began, roughly 30 seconds after liftoff. The inquiry, chaired by Jacques-Louis Lions, traced it to the inertial reference system, which had reused software from Ariane 4. A number describing horizontal motion, held as a 64-bit floating-point value, was converted to a 16-bit signed integer. Ariane 5 flew a faster early trajectory than Ariane 4, the value no longer fitted, and the conversion raised an error that was not handled. The two redundant units had identical hardware and software, so both shut down together, and the rocket's computer then read diagnostic data as if it were real guidance. The board's lesson was not "a programmer blundered". It was that an assumption valid for the old rocket was never tested against the new one: the board reported that the new rocket's trajectory data was left out of the unit's specification, and the units were not part of the full closed-loop simulation. Three of the seven vulnerable conversions had been left unguarded on the belief that the values were physically limited, and that belief proved wrong for this one. Interactive · The two interface lab 1. Units. Slide the real impulse. The ground file gives a number in pound-force seconds; navigation reads it as newton-seconds. Real impulse of a burst 2. Range. A 16-bit signed integer holds -32,768 to 32,767. Slide the value being converted, and choose whether the conversion is guarded. Value to convert Both stories share a shape. Every part worked for the conditions it was designed for. The failure sat in an assumption between parts. Systems engineers answer this with the same few habits: write the interface down, put units and ranges in it, test the parts together as early as possible, and test with real data rather than the data you expect. FMEA: asking how it breaks before it does Failure mode and effects analysis, or FMEA, is a structured way of asking "how could this break, and what happens then?" It was first described in a US military procedure in 1949, and by the early 1960s NASA contractors were using versions of it on programmes such as Apollo. For each part, a team lists the ways it can fail, the effect of each, and how it could be noticed. Then they rate three things from 1 to 10: how severe the effect is, how often it will happen (occurrence), and how hard it is to detect before it hurts someone. Multiply the three and you get a risk priority number (RPN), used to rank what to fix first. Interactive · Rank the risks An electric scooter. The ratings are invented for the exercise. Pick a failure and move its three ratings (for detection, 10 means hardest to detect). Severity Occurrence Detection (10 = hardest) The starting numbers show the catch. Battery overheating while charging has the worst possible consequence (severity 10), yet its RPN of 60 ranks it third, below a loosening bolt that scores 192. That is not an arithmetic error: multiplying scales hides a very bad but rare event behind a mild but common one. Critics have said exactly this, and the 2019 joint automotive handbook of the AIAG and VDA replaced the RPN with an "action priority" table that looks at severity first. The honest use of FMEA is the conversation it forces, not the number. Integration, test and what the capstone teaches The right-hand side of the V is where the plan meets the world. Integration means putting the parts together a little at a time, not all at once on the last week, so a surprise is found when it is cheap. Test is the evidence: vibration tables for launch, thermal-vacuum chambers for space, wind tunnels, fatigue rigs, and for software, automated suites (see 34/31). The rule teams repeat: test as you fly, and fly as you tested. The senior capstone is the one place most students get to feel all of this at small scale. In the US, accreditation criteria from ABET require engineering programmes to include a culminating major design experience that draws on earlier coursework and works within multiple realistic constraints, which is why almost every programme has one. A typical capstone gives a team of four to six students a real or realistic customer, a budget of a few thousand dollars, and two semesters. The lessons are rarely the equations. They are these: The customer's first sentence is not the requirement; asking the second question is. The part you assumed would be easy is the one that is late. The interface you did not write down is the one that does not fit. Your design is only done when somebody else can build, test and run it from your documents. What a graduate just sees The hard part of a product is on the lines between its parts, so read the interfaces first. A requirement you cannot test is a wish; every requirement should say how it will be verified. Verification is "built it right"; validation is "built the right thing". They fail separately. Everything gets heavier, hotter and later than the first estimate, so hold margin and shrink it only with evidence. In aerospace a kilogram late in the stack costs several at the start, and a watt costs heat, area and mass. Redundancy that runs the same software fails the same way twice. Ask "how could this break?" before it does, and rank by consequence, not only by a product of scores. Disagreements about a design are usually disagreements about weights; make them explicit. Where it shows up next Machine Design (34/19) gives the parts and their loads; this is how they become a product. Mechanics of Materials (34/11) supplies the mass you can afford to remove. Control Systems (34/20): a controller is a subsystem whose interface is a sensor, an actuator and a rate. A wrong sign or a wrong unit there is the Mars Climate Orbiter story with a motor. Heat Transfer (34/16) and Thermodynamics (34/14) are the thermal budget. Vibrations (34/17) is why launch loads decide what survives. Software Engineering (34/31) is the other half of nearly every modern product; the Ariane story sits exactly between them. Computer Security (34/33) is the same discipline applied to an adversary. In the real world: spacecraft, aircraft, cars, medical devices and phones are built this way, and in companies the job title is often "systems engineer" or "integration and test engineer". If you want to go deeper 📘 Systems Engineering: Principles and Practice, Kossiakoff, Seymour, Flanigan and Biemer (Wiley, 3rd ed., 2020): the standard university textbook for the whole subject. 📘 Product Design and Development, Ulrich, Eppinger and Yang (McGraw-Hill, 7th ed., 2020): a common capstone text on turning a need into a product. 🌐 NASA Systems Engineering Handbook: free to download from NASA, and the best-known practical reference. 📘 To Engineer Is Human: The Role of Failure in Successful Design, Henry Petroski (1985): a gentle, readable book on why things fail and what designers learn. 🌐 MIT OpenCourseWare 16.842, Fundamentals of Systems Engineering (Prof. Olivier de Weck): free and complete, though a graduate-level course, so expect it to be dense. The takeaway A product is many subsystems agreeing. Systems engineering is the discipline of writing the agreements down (requirements and interfaces), proving them level by level (the V, verification and validation), spending scarce mass, power and heat honestly (budgets and margins), and hunting the ways it can break (FMEA). The big failures were not exotic physics. They were two neighbours with different assumptions. Try this: pick something in your home with two or more parts that must work together, such as a phone and its charger, or a door and its lock. Write down three things the one part assumes about the other. Which assumption would be most expensive if it were wrong? Sources & Further Reading Sources. Mars Climate Orbiter: NASA press release of 10 November 1999 on the Mishap Investigation Board's Phase I report; Wikipedia, "Mars Climate Orbiter". Ariane 5 flight 501: the Inquiry Board report of 19 July 1996 (chair Jacques-Louis Lions); Wikipedia, "Ariane flight V88". V-model and verification versus validation: Wikipedia, "V-model". FMEA history and the move to action priority: Wikipedia, "Failure mode and effects analysis". Dry-mass growth of 25 to 30 percent: NASA Systems Engineering Handbook SP-6105 (1995), p. 62, as quoted in a project document. Accreditation: ABET Criterion 5. Computed in node: rocket-equation ratio 3.577, radiator area 0.256 m², the mass-budget sums, trade-study scores and RPN products. 🌐 NASA Systems Engineering Handbook 🌐 MIT OpenCourseWare 16.842: Fundamentals of Systems Engineering 🌐 NASA: Mars Climate Orbiter Phase I report press release 🌐 Ariane 501 Inquiry Board report (mirror) 📘 To Engineer Is Human, Henry Petroski