Illustrated Physics
physics, drawn
Concept

Impulse — where it appears

The integral of a force over the time it acts, which equals the change of momentum it produces. Collision calculations use it in place of a force, because the total is fixed by the endpoints however violent and unknown the middle is.

Named by 9 essays across one field — each of them below, with the objects they name alongside it.

A collision with restitution 0.6. Two bodies before and after a head-on collision. Momentum is the same on both rows by construction; kinetic energy is only preserved when the collision is elastic.

Collisions are easier than forces, and momentum is the reason

Nobody knows what happens inside a collision. Momentum conservation makes that ignorance irrelevant, which is the whole trick — and energy, deliberately, is not conserved.

mechanics · Momentum Thrust against the air it is supposed to be pushing on. The thrust of one F-1 engine against ambient pressure, in atmospheres. It is 6.78 meganewtons at sea level and 7.77 in vacuum — 14.6 per cent more with the air taken away. The line falls at exactly 9.787 newtons per pascal, which is the nozzle's exit area, because the term is (p_e − p_a)·A_e: the ambient pressure pushes on the exit plane from outside and there is nothing to push back on it. The account in which the exhaust shoves against the atmosphere makes the opposite prediction — thrust falling with the pressure and vanishing in vacuum — and it is not a small disagreement about a coefficient. It has the sign wrong. A rocket works better in vacuum than in air, and every engine ever fired has said so.

The push that needs nothing to push against

A rocket engine produces more thrust in vacuum than at sea level — fifteen per cent more, for the engine drawn here, and the extra is exactly the ambient pressure times the nozzle's exit area. The account in which the exhaust shoves against the atmosphere does not merely overstate a coefficient. It has the sign wrong, and every engine ever fired has said so.

mechanics · Momentum What a scale reads while a chain falls onto it. The reading of a scale, in units of the whole chain's weight, against the length of chain that has already landed, for two ways of putting the same chain down. Lowered gently, the scale reads the weight of what is resting on it and nothing else, so the reading climbs along the diagonal to one and stops. Dropped from rest with its lower end just touching, the scale reads three times that at every instant of the fall: one part is the pile's weight and two parts is the force needed to stop the links that are arriving, which is λv² with v² = 2gx and is therefore exactly twice λgx however far the fall has got. The peak, read off the drawn curve, is 3.00 chain weights. It is reached at the instant the last link lands, and the reading then falls discontinuously to one, because the momentum flux stops all at once. The discontinuity is the part a real experiment does not show — a real chain has links of a finite size and a scale has a response time — and it is the reason a chain dropped into a bucket on a kitchen scale reads high and then settles.

The pile that lands heavier than it weighs

Drop a chain onto a scale and the reading is three times the weight of the part that has landed — not approximately, exactly, all the way through the fall. The extra two parts are the force needed to stop links that are still arriving, and the same arithmetic run backwards says that picking a chain up wastes exactly half the energy it takes to get it moving.

mechanics · Momentum 5 balls whose contact force goes as the overlap to the three halves, touching. The velocity of every ball in a line of 5, against time in units of one binary contact, with the first arriving at unit speed. Nothing about collisions is assumed: neighbours push on each other with k times their overlap raised to the power 1.5, and the equations of motion are integrated. With the balls touching there is no such separation — several overlaps are non-zero at once and the disturbance crosses the line as a single compression wave. The far ball leaves at 0.989 of the striking speed and the others keep 0.011 between them, which is why a real cradle's balls do not quite come to rest. Momentum and energy are conserved to 4.4e-16 and 4.1e-8, so the difference between the two cases is the contact law and not the bookkeeping.

Five balls, and the law that does not choose

The usual account of a Newton's cradle says that momentum and energy conservation force one ball out at the striking speed. For three balls or more they do no such thing: the two laws leave a whole curve of possible outcomes, and what picks one is the shape of the force between two touching spheres.

mechanics · Momentum A ball dropped from 1 m with e = 0.8, bouncing for ever and stopping anyway. The height of a ball dropped from 1 m onto a floor that returns 0.8 of its impact speed, against time. The first fall takes 0.452 s. Every flight after it is 0.8 times as long as the one before, so the flight times are a geometric series and their sum is finite: the ball has made infinitely many bounces by 4.064 s, t₁(1 + e)/(1 − e), and is at rest from then on. The stepped sum of the first 60 flights plus the geometric remainder agrees with the closed form to better than a part in a billion. The dashed line marks the moment it stops; the hops just before it are too small to draw.

The bounces that add up to a stop

A ball dropped on a hard floor bounces, and bounces again, each time a fixed fraction lower. The bounces never run out — there is always another, smaller one — and yet the ball is lying still four seconds later. The flight times form a geometric series with a finite sum, the floor's infinitely many kicks add up to exactly the ball's weight times that time, and the series is only broken where each flight becomes shorter than the impact that launched it.

mechanics · Momentum Where to strike so the pivot feels nothing. The impulse the pivot has to supply, as a fraction of the blow, against where along the body the blow lands, measured from the pivot, for a uniform rod pivoted at one end and for a 0.84 m wooden bat held 15 cm from its knob, with the blow's position as a fraction of the length beyond the grip. For the rod the reaction is 1 − 3q/2L, positive for blows nearer the pivot, negative beyond, and zero at 0.6667 of the length. For the bat it is zero 51.9 cm from the grip, 17.1 cm from the end of the barrel. A blow there sets the body turning about the pivot as though the pivot were not needed.

The point that takes the blow without a jolt

Hit a stick held at one end and the hand feels a jolt — forwards if the blow lands near the hand, backwards if it lands near the tip, and nothing at all if it lands two-thirds of the way along. That point is the centre of percussion, and it is exactly where a simple pendulum with the same period would end: two different questions about a swinging body with one answer.

mechanics · Pendulum 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 Where a kick lands decides whether the clock gains. A pendulum's state as a point on a circle: horizontally its angle, vertically its angular speed, both scaled by the swing, so that undisturbed it goes round the grey circle clockwise once a period and its phase is the angle round the circle from the top. An escapement's kick raises its speed: an upward jump. Kicked at the bottom of the swing (top of the circle), the point jumps straight outward — a bigger swing, phase unchanged. Kicked 42° of phase before the bottom, the jump also carries it forward round the circle, by 9.8°, and the pendulum arrives early; kicked the same amount after the bottom, it is set back by 9.8° and arrives late. The kick is exaggerated here to 0.32 of the top speed to make the angles visible.

The kick that does not change the beat

A pendulum clock has to push its pendulum, or friction would stop it within minutes, and every push risks disturbing the very period the clock exists to keep. In 1826 George Airy worked out exactly when a push does no harm: at the bottom of the swing, where the pendulum is moving fastest, an impulse changes how far it swings and leaves the timing untouched. A push before the bottom makes the clock gain; after it, lose. Three centuries of escapement design are a search for a mechanism that pushes where Airy said — or that pushes in the wrong place in a way that cancels some other error.

mechanics · Pendulum A fountain in proportion to the drop. The height of the arch above the jar against the drop below it, for three values of α, the share of each link's momentum that the pile supplies. A steady chain has one tension at the top of the arch, equal to its momentum flux λv². Going down either leg the tension falls by the weight of chain below. At the jar the tension must be (1 − α)λv²; at the floor, with the floor taking all of the arriving momentum, it is zero. That gives v²α = gh₁ and h₁ = h₂ α/(1 − α), a straight line through the origin, and no arch at all if α = 0. A measured α near 0.12 gives 27 cm of fountain at a 2 m drop, and this proportion, about a seventh of the drop, is what the experiments found.

The chain that climbs out of the jar

Pull the end of a long bead chain over the rim of a jar and let it fall to the floor, and the rest follows, as a siphon does. That much is unsurprising. What a video showed in 2013 was stranger: the chain does not run over the rim. It leaps up from the jar in a standing arch, tens of centimetres above it, with nothing supporting it. The explanation took a year and turned on something a string does not have: a link is stiff, so the pile has to push back.

mechanics · Momentum

Named alongside it

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

Centre of massMomentumContact mechanicsDissipationElastic collisionEnergy conservationIsochronismMoment of inertiaMomentum conservationMomentum fluxRadius of gyrationStatically indeterminate

All concepts