Illustrated Physics
physics, drawn
Concept

The small-angle approximation — where it appears

Replacing a sine by its argument, exact only in the limit and adequate over a range each problem has to state for itself. The error is a sixth of the cube of the angle, so it is one part in a thousand at ten degrees and one in a hundred at twenty-five.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The pendulum's phase portrait. Angle plotted against angular velocity. Closed loops are swinging back and forth; the open curves above and below are rotating all the way round; the dashed curve between them is the separatrix.

The pendulum, and the small lie that makes it simple

A pendulum's period does not depend on how far it swings. This is one of the most useful false statements in physics, and it is worth knowing exactly how false.

mechanics · Pendulum The pendulum's period against its amplitude. The exact period of a simple pendulum divided by the small-angle period, plotted against amplitude. The small-angle formula is the horizontal line at one; the exact curve leaves it slowly and then climbs without limit as the amplitude approaches a half turn.

The period that depends on the swing, computed exactly

A pendulum's period is not independent of amplitude. The exact answer is an elliptic integral, it has no elementary form, and plotting it shows precisely where the famous approximation earns its keep and where it collapses.

mechanics · Pendulum Reflected upside down. A pulse arriving at a join where the impedance rises by a factor of 3, drawn at three moments. The amplitudes are read off the marched wave: the reflected pulse is -0.500 of the incident one and the transmitted pulse is 0.500, against (1−Z₂/Z₁)/(1+Z₂/Z₁) = -0.500 and 2/(1+Z₂/Z₁) = 0.500 from the two matching conditions. The reflection is inverted, which is the same fact as a pulse on a string flipping when it reaches a wall: a wall is a medium of infinite impedance, and the inversion is what keeps the displacement at the join equal to zero. Note that the transmitted amplitude exceeds one where the second medium is lighter, and that this is not a violation of anything: amplitude is not energy.

The equation that lets a shape travel

Newton's second law applied to a piece of string a millimetre long gives T·y″ = µ·ÿ, and the derivation never once asks what the string is made of. Two things fall out immediately: the speed is √(T/µ) and belongs to the medium, and the general solution holds two arbitrary functions rather than one. The second of them is the reflection, which is why a boundary condition can be met at all.

waves · Wave motion 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

Named alongside it

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

Restoring forceSeparatrixSimple harmonic motionAmplitude dependenceThe arithmetic–geometric meanBoundary conditionsCentre of massCurvatureDispersionDispersion relationThe elliptic integralEquivalent length

All concepts