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.
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:
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.
A beam is held up at both ends. Drag the red load along it, or use the slider.
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).
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.
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.
A 6 m span truss. Red members are in tension, blue in compression; thicker means more force.
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:
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.
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.
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.
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. 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