Illustrated Physics
physics, drawn
Concept

Rigid body — where it appears

A body whose parts keep fixed distances from one another, which relativity forbids exactly. A push travels along a real rod at its speed of sound — 600 microseconds for three metres of steel, against ten nanoseconds for light — so rigidity is a signal at infinite speed rather than a limit a hard material approaches.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

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 The far end that has not been told yet. A rod 3 metres long, pushed at one end at time zero. On the left, position across and time upwards, for the two fastest disturbance speeds drawn here, with the light cone beside them: nothing may lean further to the right than that line, which crosses the rod in 10.0 nanoseconds. A rigid rod would be the vertical dashed line — the far end moving at the same instant as the near one — and it is not a limit that a hard material approaches. It is a signal at infinite speed. On the right, how long the far end actually waits, against how fast the disturbance travels, both logarithmic, with every material on it: 8.8 ms at 3.4e+2 m/s, 600.0 μs at 5.0e+3 m/s, 250.0 μs at 1.2e+4 m/s, 100.0 ns at 3.0e+7 m/s, 20.0 ns at 1.5e+8 m/s. The line has slope −1 and the light cone is a hard floor beneath it. Ordinary materials sit four to five decades above that floor, which is why rigidity is such a good approximation and why it is still not a limit: steel's delay is not small compared with light's, it is 6e+4 times larger. Everything usually derived from rigid bodies survives, because the delay is beneath notice in ordinary circumstances. What does not survive is the use of rigidity in an argument about simultaneity, which is where it does real damage: a rod pushed at one end is compressed for as long as the wave takes to cross it, and there is a frame in which its far end is still at rest while its near end is moving.

Nothing is allowed to be rigid

A rigid body would move its far end at the instant its near end was pushed, which is a signal at infinite speed. Relativity forbids it — not approximately, and not as a limit that a hard enough material approaches. What follows is a ceiling on how stiff matter may be, and that ceiling caps the mass of every neutron star.

relativity · Relativistic dynamics The same top, let go four ways. The path traced by the top of the axis, seen from directly above, over 1.2 precession periods. The dashed circle is the tilt the top was released at and the outer circle is 46.8° from the vertical. Released from rest the axis falls, and the fall is what generates the sideways motion: the path comes to a cusp each time it returns to the starting tilt, because at that instant the precession rate is momentarily zero. Launched at exactly the steady rate the path is a circle and the nutation is absent. Launched slower it waves; launched faster it loops: at 0× the steady rate the path comes to cusps, at 0.45× the steady rate the path waves, at 1× the steady rate the path stays a circle, at 1.9× the steady rate the path waves. Every one of these is the same equation with the same top and the same spin.

The top that nods before it settles

A spinning top let go from rest does not begin to precess. It falls, catches itself, and comes back up, over and over, at a frequency that has nothing to do with gravity — and the steady precession every textbook draws is what is left after friction has removed the nod.

mechanics · Rotation The point that does not notice the collision. Two bodies of rest mass 1 and 2, approaching at 0.8c and -0.3c, colliding elastically and leaving at -0.6168c and 0.5969c — speeds obtained by reversing the motion in the zero-momentum frame, with the total energy and momentum checked to a part in a million million. The third line is the energy-weighted centre. It runs straight through the collision at 0.1872c, which is the total momentum divided by the total energy, and it has no kink — verified at two hundred instants. Nothing here is the centre of mass: the rest masses are unchanged by the collision but the energies are redistributed, and it is the energies that do the weighting.

The centre that is not a place

The centre of mass is replaced in relativity by the centre of energy, which moves uniformly and does everything the old point did — except be the same point for everybody. Boost a spinning body and its centre moves, so a spinning object has no centre at all.

relativity · Relativistic dynamics The energy a spin is allowed, and the only direction it can go. A body with principal moments 1.000, 3.000, 4.000 spinning with a fixed angular momentum. Every possible motion has an energy somewhere in the band drawn here, and the three marks are the three principal-axis spins: energy L²/2I, so the largest moment of inertia gives the smallest energy. The band runs from 0.1250 to 0.5000 in units where the momentum is one — a ratio of 4.0. Anything inside the body that flexes and warms takes energy out and leaves the momentum untouched, so the state can only move leftwards along this band, and there is exactly one place for it to stop. A spin about the axis of least inertia is at the far right: it is a perfectly good solution of the equations of motion, stable against small disturbances in a perfectly rigid body, and it sits at the top of a hill the smallest leak will roll it off.

The axis a leak of energy chooses

A body spinning with nothing pushing on it keeps its angular momentum exactly, and keeps its kinetic energy only while nothing inside it flexes. At fixed momentum the energy is least for a spin about the axis of greatest inertia — so any leak, however small, has a destination. The first American satellite found this out in orbit.

mechanics · Rotation A floor that has to hit a rod nobody dropped. The normal force on the lower end of a uniform rod at 63° with μ = 1.6, sliding outward at √(gl) and not yet turning, against time on a logarithmic axis, for a floor whose contact stiffness is 10⁴, 10⁵ and 10⁶ times mg/l. The rigid equations have no solution here, and the compliant floor shows why: the end digs in, the deeper it digs the harder friction twists it further down, and the force climbs until the sliding stops. The stiffer the floor, the higher and briefer the spike — 20, 56, 163 times the rod's weight, ending after 41, 18, 7 thousandths of √(l/g) — a collision with no approach.

The rod that friction cannot decide about

Newton's laws, a rigid rod and Coulomb's law of friction are the first things a mechanics course teaches, and in 1895 Paul Painlevé showed that together they can fail to give any answer at all. A rod sliding on a rough enough floor, at the right angle, has no motion consistent with the three — or has two. The way out is not a new law but a forgotten fact: real contacts give a little. With that, the rod jams, and the floor delivers a sudden blow to a rod that nothing struck — an impact without a collision, which is how a stick of chalk comes to hop.

mechanics · Friction 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 Stopped for half a second, a raw egg starts again by itself. Two eggs spinning at 3 revolutions a second, each touched with a finger at t = 4 s so that its shell is held still for 0.5 s and then released. The cooked egg (solid) stops and stays stopped. The raw egg's shell (dashed) stops too, but its contents (dotted) are still turning — slowed to 2.15 revolutions a second during the hold, by viscosity working against the held shell — and once the finger lifts they drag the shell back into rotation, reaching 1.81 revolutions a second, 60 per cent of its speed before the touch. The table's friction is left out here, so that nothing else slows either egg. It is the oldest kitchen test of whether an egg is cooked, and the whole of the physics is a liquid's reluctance to stop turning.

The egg that keeps turning after it is stopped

Spin an egg on a table, touch it for a moment to stop it, and let go. A cooked egg stays still. A raw one starts turning again by itself, as if something inside had kept going — which it has. A raw egg is two bodies, a thin shell and a liquid that the shell can turn only through the slow grip of viscosity, and the kitchen test that tells them apart is a measurement of how long that grip takes. The same two-body arithmetic says that a raw egg wastes most of the energy of a flick, starts out spinning six times faster than a cooked one given the same twist, and keeps a record of its past rotation in its interior for a second or two after its shell has forgotten.

mechanics · Rotation

Named alongside it

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

Moment of inertiaAngular momentumDissipationFree-body diagramSimultaneityStabilityStatically indeterminateStiffnessTorqueAngular accelerationBending momentBorn rigidity

All concepts