Illustrated Physics
physics, drawn
Concept

Radius of gyration — where it appears

The distance from an axis at which a body's whole mass could be concentrated and have the same moment of inertia. It turns an awkward distribution into one number, sets the period of a swinging body and where a struck body pivots, and measures the size of a polymer coil.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Two pivots with one period, and the length between them. The period of a uniform bar 1 m long swung about a pivot, against the distance of that pivot from its centre of mass. Close to the centre the period runs away, because there is almost no restoring torque; far from it the bar behaves more and more like a simple pendulum. In between is a minimum at h = k = 0.2887 m, the radius of gyration, and because there is a minimum every period above it belongs to two pivots at once — here 0.4 m and 0.2083 m, whose product is k² and whose periods agree to 2.2e-16 seconds. Suspend the bar from those two knife edges in turn, adjust until the periods match, and the equivalent simple pendulum is the distance between them: 0.6083 m. Then g = 4π²(h+h′)/T² gives 9.8100 m/s², recovered from the drawing to 1.8e-16. The mass of the bar, the distribution of that mass, and the position of the centre of mass appear nowhere in the answer.

The length nobody has to measure

A pendulum's period depends on its length, so a pendulum will measure gravity — except that a real swinging bar has no single length, and finding where its mass effectively sits is far harder than timing it. Kater's answer was to build the instrument so that the awkward quantity cancels, leaving a distance between two knife edges that a rule can read to five figures.

mechanics · Pendulum 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

Named alongside it

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

Moment of inertiaCentre of massImpulseAngular momentumCentre of percussionCompound pendulumConjugate pointsContact mechanicsEquivalent lengthFrictionMeasurementMetrology

All concepts