- #1
Karol
- 1,380
- 22
Homework Statement
A ball of mass m and radius r starts off on top of a fixed ball of radius R. it rotates down without slipping.
What's the ball's center of mass velocity as a function of θ.
The angular velocity is ω.
Homework Equations
Moment of inertia: ##I=\frac{2}{5}mr^2##
Potential energy: ##E_p=mgh##
Kinetic energy due to rotation: ##E_ω=\frac{1}{2}I\dot\theta^2##
The Attempt at a Solution
Kinematics: the path the small ball makes is equal to the path on the big one, and so are the angular velocities
$$R\dot\theta=r\omega\rightarrow\omega=\frac{r}{R}\dot\theta$$
The height of the ball's center during the fall is ##(R+r)\cos\theta##
Total energy at the top, relative to the center, is equal during the fall:
$$mg(R+r)=\frac{1}{2}\left( \frac{2}{5}mr^2 \right)\omega^2+mg(R+r)\cos\theta$$
$$mg(R+r)=\frac{mr^2}{5}\frac{R^2}{r^2}\dot\theta^2+mg(R+r)\cos\theta$$
The center of the ball's velocity is ##v=(R+r)\dot\theta##
I can't solve for v, and the answer should be an isolated v