Concept

Rotational energy — where it appears

The kinetic energy stored in a body's spin, equal to half its moment of inertia times the square of its angular velocity. Measured about the centre of mass it is the same for every observer, which is why it survives a change of frame when the translational part does not.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The staircase equipartition cannot climb. The heat capacity of hydrogen at constant volume, in units of R, against temperature on a logarithmic axis. The three translational directions contribute 3/2 at every temperature. The two rotations switch on near 85.4 K — the temperature at which kT matches the first rotational step — taking the total to 5/2, computed here by summing the rigid rotor's partition function over two hundred levels rather than by assuming the plateau. The vibration switches on near 6332 K, taking it to 7/2. At 50 K the value is 2.469; at 300 K the value is 2.502; at 5000 K the value is 3.376. Classical equipartition predicts 7/2 at every temperature, including at four kelvin, and the size of the steps is set by ħ — which is how a heat capacity measures a quantum constant.

Half a kT for every way of moving

A heat capacity ought to be a count. Every quadratic term in a system's energy carries half a kT of it, so warming a gas is a matter of enumerating the ways its molecules can move — and the fact that the count comes out wrong for hydrogen is how a thermometer measured Planck's constant.

thermodynamics · Equipartition

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 top, let go four ways. The path traced by the top of the axis, seen from directly above, over 1.2 precession periods. The dashed circle is the tilt the top was released at and the outer circle is 46.8° from the vertical. Released from rest the axis falls, and the fall is what generates the sideways motion: the path comes to a cusp each time it returns to the starting tilt, because at that instant the precession rate is momentarily zero. Launched at exactly the steady rate the path is a circle and the nutation is absent. Launched slower it waves; launched faster it loops: at 0× the steady rate the path comes to cusps, at 0.45× the steady rate the path waves, at 1× the steady rate the path stays a circle, at 1.9× the steady rate the path waves. Every one of these is the same equation with the same top and the same spin.

The top that nods before it settles

A spinning top let go from rest does not begin to precess. It falls, catches itself, and comes back up, over and over, at a frequency that has nothing to do with gravity — and the steady precession every textbook draws is what is left after friction has removed the nod.

mechanics · Rotation

Named alongside it

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

Angular momentum Moment of inertia Torque Conservation laws Degrees of freedom Dissipation Equipartition Exponential sensitivity Heat capacity Initial conditions Instability Kinetic theory

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