Illustrated Physics
physics, drawn
Concept

Principal axes — where it appears

The three perpendicular directions in a rigid body about which its inertia tensor has no cross terms. Spin about one of them keeps the angular momentum along the spin; spin about any other axis does not, so the body wobbles or exerts a torque on its bearings.

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

Every path the body can take, at one angular momentum. The angular momentum vector, drawn in the body's own frame on the sphere its length confines it to, for a body whose principal moments are 3.068e-3, 6.817e-3 and 9.750e-3 kg m². Each closed curve is one motion, traced by integrating Euler's equations rather than by solving for the intersection of the sphere with the energy ellipsoid, so a curve closes only if the physics closes it. The low-energy curves circle the greatest-moment axis and the high-energy ones circle the least; both sets are small loops that stay near their axis, which is what stability looks like. Between them is the one curve that is not a loop at all — four arcs, drawn heavier, meeting at the intermediate axis and leaving it again. A body spun about that axis is balanced on the crossing point of paths that go somewhere else, which is the whole of why it does not stay.

The axis that will not hold

A book spun about its long edge keeps spinning about it. Spun about the axis through its covers, it keeps spinning about that. Spun about the third axis, it flips end over end, again and again, with nothing touching it. Three numbers decide, and what matters is only their order.

mechanics · Rotation 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 momentumMoment of inertiaConservation lawsExponential sensitivityInstabilityNon holonomic constraintPhase portraitRollingRolling without slippingRotating frameRotational energySeparatrix

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