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The scenario I'm considering is a solid sphere (of uniform density) rotating with constant angular velocity when it abruptly splits into two hemispheres along a cut which contains the rotation axis. The hemispheres will begin to separate; if, for example, we consider the rotation to be counterclockwise, then the upper hemisphere will seem to slide to the right (against the rotation) until they separate.

My question is, which of these (if either) is the correct differential equation to describe the motion? Force considerations (in an angularly-accelerating, co-rotating frame) show that the only forces parallel to the direction of sliding are the fictitious centrifugal, Coriolis, and Euler forces. Computing these carefully, we have

[tex]\frac{d^2\ell}{dt^2}=\ell(t)\left(\frac{\frac{2}{5}R^2\omega_0}{\ell(t)^2+\frac{83}{320}R^2}-\frac{\frac{3}{4}R(\frac{83}{320}R^2+\frac{3}{4}\ell(t)^2)}{\ell(t)(\ell(t)^2+\frac{83}{320}R^2)\sqrt{\ell(t)^2-(\frac{3}{4}R)^2}}\frac{d\ell}{dt}\right)^2.[/tex]

Here [itex]\ell(t)[/itex] is the separation of the CM of the hemispheres.

If I assume kinetic energy is conserved (which it appears it must be, by symmetry and the fact that the normal here is an internal force), then we instead obtain the first-order differential equation

[tex]\frac{dr}{dt}=\frac{\sqrt\frac{2}{5}Rr\omega_0}{\sqrt{\frac{83}{320}R^2+r^2}},[/tex]

where [itex]r[/itex] is the distance between the centers (not CM) of the hemispheres.

The latter equation is decidedly simpler, but does not seem to produce the right results, even in limiting cases. The former is qualitatively accurate, which is reassuring, but can only be solved numerically (I think - I'd be very glad if anyone can prove me wrong!).

If the energy conservation approach is wrong, could someone please explain to me why? The normal force can't transfer energy from one hemisphere to the other, because the system has order 2 rotational symmetry. There is nowhere else that the energy can go, so it just transfers between the two types of kinetic energy. (I'm assuming also that the total kinetic energy is the sum of the rotational kinetic energy about the CM and the translational kinetic energy of the CM, which I've seen elsewhere is valid.)

Any help anyone can give me in unconundruming this conundrum will be greatly appreciated!QM