A very basic question here; it's related to this one, but not quite the same.
If a rotating rigid body (a sphere for the sake of discussion) with mass $m$, radius $r$ and inertial tensor $I$ has initial linear velocity $v_0=0$ and non-zero, initial angular velocity $\omega_0$, what are the idealized final angular and linear velocities if the body finds itself instantaneously on a surface with infinite friction and mass, assuming $\omega_0$ is parallel to the surface (and the surface doesn't absorb any of the energy, nothing is lost to heat)?
The body should end up rolling along the surface at some constant velocity. So I think we know that $v_f = r\times \omega_f$ with $r$ anti-parallel to the surface normal. Clearly the inertial tensor must come into play since if all of the mass is concentrated on the outside of the body, it has a lot more angular momentum than if the mass is mostly at the center of the body. I tried using conservation of momentum here, but the units don't match up between angular momentum and linear momentum.
Answer
Angular momentum is not conserved since there is an external force applied to the body - the friction.
According to the condition there is no energy loss, so you can use energy conservation law. Since the friction is infinite the body will instantly start rolling without any sliding. The initial rotational energy $$ E_0 = \frac{I \omega_0^2}{2} $$ will be distributed between translation movement with velocity $v_1$
and rotation with new angular velocity $\omega_1$: $$ \frac{I \omega_0^2}{2} = \frac{I \omega_1^2}{2} + \frac{m v_1^2}{2} $$
The velocities $v_1$ and $\omega_1$ should fit the condition of rolling without sliding: $$ \omega r = v $$
No comments:
Post a Comment