Illustrated Physics
physics, drawn
Concept

Centripetal force — where it appears

The net force directed toward the centre that keeps a body on a curved path, equal to mv²/r for speed v on radius r. It is not a separate kind of force but a role played by tension, gravity, friction or a normal force, whichever is pulling inward.

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

Arms in: 4.33× the rate, and 4.33× the energy. A body of 1.2 kg m² carrying two 4 kg masses on arms, spinning freely at 60 revolutions a minute with the arms out at 0.75 m, as the arms are pulled in to 0.12 m. The axis runs right to left, in the direction the arms move. Angular momentum is flat — nothing exerts a torque about the axis, and pulling inward is a force along a radius, which has no moment about the centre. The rate rises as the inverse of the moment of inertia, by a factor of 4.33 here, and the kinetic energy L²/2I rises by exactly the same factor, which is where the usual account stops and where the question starts. The fourth curve is the work done by whoever pulled the arms in, integrated from the force needed to hold each mass on its circle. It lies on the energy curve, to 1.6e-7 joules. Nothing is unaccounted for and nothing is created: the energy is bought, at full price, by pulling against the force that would otherwise fling the arms out.

The quantity that survives a change of shape

A skater pulls her arms in and spins four times faster. Angular momentum is conserved, which is the usual explanation, and it accounts for only half of what happened — because the kinetic energy has gone up by the same factor, and something had to pay for it.

mechanics · Rotation A governor that hunts. The speed of an engine with a centrifugal governor after its load is suddenly increased, against time in units of √(ℓ/g), for three amounts of friction in the governor's linkage: damping 0, 0.25, 0.6, against Vyshnegradsky's minimum for this governor, 0.397. With no friction the speed oscillates about its new, slightly lower value with a growing swing — the governor hunts, opening and closing the throttle out of step with the engine — until the arms hit their stops and the curve ends. At damping 0.25, below the minimum, it still hunts, more slowly. At damping 0.6, above the minimum, the oscillation dies away and the engine settles at its new speed. The friction that seems to spoil the mechanism is what makes it work.

The governor that needed friction to work

James Watt's steam engines were kept at a steady speed by two heavy balls on hinged arms, spun by the engine itself: run faster and the balls fly out and close the steam valve; run slower and they drop and open it. The balls stand at a height set by the speed alone. As engines grew and governors were made more sensitive, many began to hunt — speeding and slowing in a growing rhythm, the governor always too late. In 1868 James Clerk Maxwell wrote the equations down and showed why: a governor with no friction cannot settle at all, and the more precisely it is meant to hold the speed, the more friction it needs.

mechanics · Circular motion

Named alongside it

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

Angular momentumCentral forceCircular motionConical pendulumConservation lawsControl theoryDampingFeedbackKinetic energyMoment of inertiaReference framesRotation

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