Illustrated Physics
physics, drawn
Concept

Free-body diagram — where it appears

A drawing of one body with every force on it and nothing else, which turns a mechanics problem into an accounting problem. Its discipline is what it excludes: forces the body exerts on other things do not appear, and drawing them is the commonest way of getting the answer wrong.

Named by 8 essays across one field — each of them below, with the objects they name alongside it.

A block on a 27° incline. Free-body diagram of a block resting on an inclined plane: weight straight down, resolved into a component pressing into the surface and one pulling along it, with friction opposing the slide.

The slope, and the two directions that make it easy

An inclined plane looks like a harder problem than a flat one. Split the weight into two components chosen to suit the slope and it becomes an easier one.

mechanics · Free-body Velocity and acceleration in uniform circular motion. Velocity drawn tangent to a circular path and acceleration drawn toward its centre, at eight points around the circle. The speed never changes and the acceleration is never zero; the two vectors are perpendicular everywhere.

Turning is an acceleration, and constant speed does not help

An object going round a circle at unchanging speed is accelerating hard, all the time, toward a point it never reaches. The construction that shows this needs two arrows and no calculus.

mechanics · Circular motion The same 120 N force at 3 different arms. 3 spanners of different lengths, each with the same 120 newton force applied at its end. The torque printed under each is the force times its own moment arm, so it rises with the length while the force does not change.

The same push, further out, and why that is a different quantity

A force is not enough to say whether something turns. What decides is where the line of the force passes, and the distance from the pivot to that line is the whole of the story.

mechanics · Rotation Which failure comes first. The dividing line between a block that slides and one that tips over, for a horizontal push applied at the top. Sliding needs a force of μ_s times the weight; tipping needs the push's moment about the leading bottom edge to beat the weight's, which is the weight times half the width. The weight appears in both and cancels, so the boundary is the curve aspect ratio = 1/2μ_s and nothing else: not the mass, not how hard the block is pushed, and not what it is made of except through μ. At μ_s = 0.5 the dividing shape is as tall as it is wide; at μ_s = 0.2 it is 2.5 times as tall. Of the 5 objects marked, 3 sit above the line and go over rather than sliding: a paperback, standing, a full filing cabinet, a pint glass.

Slide or topple

Push a wardrobe and it goes over; push a brick and it skids. Both are held by the same friction and both are pushed by the same hand, and which of the two failures arrives first has nothing to do with how hard the push is. The floor decides it, by shifting where it pushes back.

mechanics · Free-body Four supports, and a whole line of answers. The same top on four supports at the corners of a square, with the load in the same place. Five sets of reactions are drawn and every one of them satisfies all three equilibrium equations exactly — worst residual 1.1e-16 across the whole family — so statics does not prefer any of them. They differ by a multiple of the pattern plus, minus, plus, minus around the square: pressing one diagonal pair harder and the other pair less adds no net force and no moment about either axis, which is exactly what it means for the problem to have a fourth unknown and only three equations. The rigid-body idealisation has not been applied carelessly here; it has been applied correctly, and the answer it gives is that there is no answer. Every member drawn keeps all four reactions positive, so the requirement that a leg can only push narrows the family without closing it. What decides is left out of the model entirely — how much each leg gives under load.

The table statics cannot settle

A rigid top on three legs has one possible set of reactions and a rigid top on four has infinitely many, all of them balancing every force and every moment exactly. The extra leg does not make the problem harder; it makes it unanswerable, and the answer has to come from somewhere the model deliberately threw away.

mechanics · Free-body Where the thread's line meets the floor. A spool resting on the floor on its rims of radius R, its thread wound round a hub of radius r = 0.5R and leaving the hub's underside tangentially, pulled up and to the right at 20°, 60° and 78° above the horizontal; the dashed lines continue each thread's line back down to the floor. The spool turns about the contact point, and the sense of the turn is set by where the thread's line meets the floor. At 20° it meets the floor behind the contact point, on the far side from the hand, and the spool rolls towards the hand. At 60°, where cos θ = r/R, it passes exactly through the contact point; the pull has no turning effect there, and the spool stays still while the hand pulls. At 78° it meets the floor in front of the contact point, between the spool and the hand, and the spool rolls away from the hand, paying the thread out.

The spool that rolls towards the hand pulling it

Pull gently on the thread of a cotton reel lying on its side on a table, with the thread coming off the underside of the hub, and the reel rolls towards the hand, winding the thread up as it comes. Raise the thread and at one particular angle the reel will not move at all however hard it is pulled; raise it further and the reel rolls away, paying the thread out. Which way it goes is decided by a single line — where the thread's direction, extended, meets the table — and the same line decides which way a bicycle goes when its lower pedal is pulled backwards.

mechanics · Free-body The ladder that leaves the wall. A 4-metre ladder released from rest at 15° from the vertical, its foot on a frictionless floor and its top against a frictionless wall, drawn at equal intervals of time from release to the moment it lies flat, computed by integrating its motion; the dots trace its centre. While both ends touch, the centre moves exactly on a circle of radius L/2 about the corner (dashed). The top leaves the wall when it has come down to 2.58 m, two-thirds of its starting height of 3.86 m; from then on the centre's horizontal speed is fixed, it leaves the circle, and the ladder falls clear of the wall with its foot still sliding. The fall takes 1.34 seconds.

The ladder that lets go of the wall

A ladder whose foot slips on a smooth floor slides down a smooth wall — and partway down, its top comes away from the wall and the ladder falls the rest of the way clear of it. The point where it lets go of the wall is not set by its length, its mass or how steeply it started: it is always two-thirds of the height the top started at. The reason is a wall that can push but cannot pull, and the number falls out of energy alone, without a single force being calculated in advance.

mechanics · Energy The bending a falling column must carry, along its height. The bending moment at each height of a uniform column toppling about its base, in units of its weight times its length, at tilts of 10°, 30°, 60° from upright. Every curve peaks at a third of the height, at mgL·sinθ/27: 0.0064, 0.0185, 0.0321. The base and the top carry none. The same column held still at the same tilt would carry its greatest moment at the base, mgL·sinθ/2 — 0.250 at 30°, 13.5 times as much. Falling relieves the bending by a factor of 13.5 and moves it a third of the way up.

The chimney that breaks a third of the way up

A tall brick chimney felled by demolition rarely reaches the ground in one piece. It snaps partway up, early in its fall, and the top falls behind the rest. Nothing has hit it. The cause is a free-body diagram: as the column rotates about its base, the material near the top must be dragged round faster than its own weight would take it, and the material lower down holds back. The bending this needs is largest a third of the way up, at a thirteenth and a half of what the same column would carry standing still at the same tilt — and brick, which carries no tension, cracks there after only a few degrees of lean.

mechanics · Free-body

Named alongside it

The objects these essays reach for when they reach for this one.

Mechanical advantageMoment of inertiaNormal forceTorqueCentre of massFrictionMoment armResolving vectorsRigid bodyStatic frictionWorkAngular acceleration

All concepts