Illustrated Physics
physics, drawn
Concept

Non holonomic constraint — where it appears

A restriction on how a system's velocities may be related that cannot be integrated into a restriction on its positions. A wheel rolling without slipping is the standard example: it can reach any position by some path, though at each moment it can only move in certain directions.

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

The band of speeds in which a bicycle rides itself. The growth rates of the small motions of a riderless bicycle against forward speed, from the four roots of the linearised equations for a benchmark machine, with the published eigenvalues reproduced before anything else was drawn. A positive rate means the motion grows. Below 4.29 m/s a pair of roots has positive real part: the weave, a swaying of the steering and lean together, grows. Above 6.02 m/s one real root turns barely positive: the bicycle slowly falls over into a spiral, the capsize. Between these speeds, in the shaded band, all four roots have negative real parts, and a shove dies away on its own with no hand on the bar. Nothing is stabilising the machine except the coupling of lean to steer.

The bicycle that steers itself upright

Give a riderless bicycle a shove and, at the right speed, it wobbles and straightens up, and rides on with nobody on it. People have long credited the gyroscopic wheels or the trail of the fork. A machine built in 2011 had neither, and it stayed up. Four numbers, the roots of a quartic, tell the whole story of when a bicycle rides itself, and they do not contain a simple reason.

mechanics · Harmonic approximation The same stone, and two different answers. Spin against time from the same stone started at plus and minus 1.5 rad/s, with nothing else changed. One direction stops and reverses at 2.8 s. The other takes until 33.8 s, which is 12 times as long and longer than a real stone keeps turning. A mirror would swap the two, and the mirror image of this stone is a different stone.

The stone that turns back against its spin

A polished stone, shaped like half an elongated egg, will spin smoothly one way. Spin it the other way, and within seconds it shudders, stops and turns back, as if its angular momentum had changed its mind. Nothing has broken any law. The trick is a misalignment of about a degree between the axes of its weight and the curve of its base, and with that gone, so is the behaviour.

mechanics · Rotation

Named alongside it

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

Angular momentumBicycle stabilityCapsize modeEigenvalueGyroscopic effectLinear stabilityMoment of inertiaPrincipal axesRolling without slippingTrailWeave mode

All concepts