When a body is spun so fast that its internal forces cannot hold it together, it breaks up into smaller chunks. The criterion for the limiting angular velocity depends on the size of the object and the nature of the forces holding it together.

## Chemical Forces Edit

The expression for chemical forces can be very complicated. However, as a rough estimate, we can assume breakup occurs when the rotational velocity on the edge is the same as the speed of sound.

$ \omega R \approx c^2 $

If the speed of sound is roughly constant, then the limiting angular velocity decreases as the reciprocal of the radius

## Gravity Edit

If the body is held by gravity, but the density is not strong enough to compress matter substantially, then the limiting angular velocity is

$ \omega \approx \sqrt{ \frac{G M}{R} } \approx \sqrt{G \rho} $

so the limiting angular velocity does not depend on the radius.