**Instability of a rigid body**

**spinning freely in space**

Hugh Hunt, Cambridge University

(movies on this subject are here)

A rigid body spinning in space conserves kinetic energy T
and moment of momentum **H**.
Expressions for the two are T=½(A* w*_{1}** ^{2}** + B

Here is an example in which the momental ellipsoid is shown green/yellow/orange
and the energy ellipsoid is blue/white.
The black line represents the intersection of the two ellipsoids, *ie*
the allowable states, showing that spin near the ‘B’ axis is a ‘saddle’ and is
unstable. You can check this by
spinning a book, a tennis racquet, a cellphone … anything really, and noticing
that it will not spin stably about the axis of its intermediate moment of
inertia. Spin about the other two axes
is stable.

To get a more complete picture, the momental ellipsoid shown below is marked with the intersection lines of the energy ellipsoid for all possible spins, the blue line being for spin about the ‘A’ axis (or close thereto), the white being for spin about the ‘C’ axis – and note that both of these are stable. The saddle for spin about ‘B’ is the red line. (note: the colour coding of the ellipsoid is not important – it is only intended to help give a 3D effect)

But spin about the ‘A’ axis is not *really *stable
because of energy dissipation effects*.
*For a given moment of momentum H,
the highest energy state is spin about the ‘A’ axis (the blue
line). Let us suppose that we spin the
body about axis ‘A’. It is possible to
envisage that as energy is lost within a spinning body (due to internal
friction, say) then the trajectory will gradually ‘spiral’ around the surface
of the ellipsoid, all the while conserving angular momentum H but losing energy
T. The spin will appear rather ‘wobbly’
as the body gradually settles down to spin about the lowest-energy state, about
the ‘C’ axis. There is no more energy
to lose now.

You can try this too. Take an object that will dissipate energy internally, for instance a key wallet or a bag of frozen peas. Spin this about any axis and it will quickly make its way to spinning about the axis with largest moment of inertia – ie the minimum-energy state.

Home: www.eng.cam.ac.uk/~hemh