Illustrated Physics
physics, drawn
Concept

Variable mass — where it appears

A situation in which the moving part gains or loses matter, so that Newton's second law cannot be applied to it as a fixed set of matter. A chain falling onto a scale reads three times the weight of what has landed, and a chain being lifted off a heap wastes exactly half the kinetic energy handed to it, whatever the speed.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

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 A drop that sweeps up mist settles at a seventh of g. The acceleration of a water drop falling from rest through a still mist that it sweeps up completely, in units of g, against time, for drops starting at radii of 0.2 mm, 0.05 mm and effectively zero, with no air resistance. The mist holds 10 g of water per cubic metre, twenty times a typical cloud, so that the approach takes seconds rather than minutes. Every drop starts at g, because at first it has gathered nothing, and every one settles to g/7 = 0.1429 g: after 2 s the three are at 0.377, 0.172, 0.143 g, and after 8 s at 0.144, 0.143, 0.143 g. The drop that starts from nothing is at g/7 from the start. The final acceleration contains no property of the mist, the drop or the water: a denser mist makes the drop grow faster and reach g/7 sooner, and nothing else.

The raindrop that falls at a seventh of g

A drop of water falling through a still mist, gathering every droplet in its path and with no air to slow it, does not fall at g. It settles to an acceleration of exactly a seventh of g, whatever the mist, whatever the drop, and its radius grows by the same amount for every metre it falls. Of the work gravity does on it, three-sevenths goes into the gathering and never comes back. The same reasoning gives a third of g for a chain paid out from a heap, and a single rule — g/(2n + 1) — for anything that sweeps up mass as the n-th power of the distance it falls.

mechanics · Momentum The spring falls from the top down. Snapshots of the falling spring, every fifth coil, at 0.00, 0.05, 0.09, 0.14, 0.18, 0.23, 0.26, 0.29 seconds after release, for a spring that hangs 1 m; the dashed line is the height of the bottom coil before release. The coils collapse onto one another from the top, and the collapsed block falls on the coils below; everything below the block hangs exactly as it did before the spring was let go, the bottom coil included. Only when the block reaches the bottom, 0.261 s after release, does the bottom begin to fall, and then the whole collapsed spring falls together.

The spring whose bottom does not know it has been dropped

Hang a soft spring from its top, let go, and its bottom end stays exactly where it was — not falling slowly, not falling at all — while the top races down to meet it. A spring that hangs a metre takes about a quarter of a second to collapse, and only then does the bottom start to fall. Nothing holds it up. The centre of mass falls freely from the first instant, as it must, and the bottom simply has not been told: the news that the top has been released travels down the spring as a front of collapsing coils, and nothing below the front can feel anything until the front arrives.

mechanics · Momentum The same energy in less and less whip. The speed of the moving part of a whip 2 m long and weighing 250 g, against how far the loop the arm threw into it still has to go to the tip, on a logarithmic scale, if the 15 J the arm gave it stays in the part still moving. Three whips: uniform, tapering linearly, and tapering faster, each ending in a 0.2 g cracker. At the handle every part moves at 11 m/s. Half a metre from the tip the uniform whip is at 22 m/s and the faster taper at 72; the uniform whip passes the speed of sound (dashed) only within its last millimetre, the tapers 1.6 cm and 1.5 cm from it. All three end at the same ceiling, √(2E/m) = 387 m/s, set by the cracker alone: the taper decides how gradually the speed builds, not how fast the end goes.

The crack at the end of a whip

An arm moves a whip's handle at ten metres a second, and a fifth of a second later its tip passes the speed of sound. Nothing in between is a lever or a spring. The arm throws a loop into the whip, the part behind the loop comes to rest, and the energy stays with the part still moving, which grows lighter as the loop runs out towards the tip. Energy shared by less and less mass means speed, and what caps it is the tip's own mass, the tuft of string at the end. The crack is a sonic boom, made by something the size of a shoelace's end.

mechanics · Momentum

Named alongside it

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

MomentumCentre of massDissipationInelastic collisionKinetic energyImpulseCloud microphysicsConservationConservation of energyEfficiencyFolded chainFree fall

All concepts