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.
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.
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.
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:
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.
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.
Cart 1 moves at 10 m/s and hits cart 2, which is at rest.
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:
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.
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:
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.
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.
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.
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.
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. 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