# Disk falling while rotating round a point

1. Aug 22, 2015

### Karol

1. The problem statement, all variables and given/known data
The disk of mass m and radius R is at position (a), upwards. it is free to rotate round point O on it's circumference.
If falls and reaches state (b), horizontal position.
What are the velocities and accelerations in position (b)

2. Relevant equations
Moment of inertia of a disk: $I=\frac{1}{2}mR^2$
Torque of rigid body: $M=I\alpha$
Energy of a rigid body: $E=\frac{1}{2}I\omega^2$
Centripetal acceleration: $a_{cen}=\frac{v^2}{R}=\omega^2 R$

3. The attempt at a solution
The angular velocity ω is from energy conservation:
$$mgR=mR^2\left( \frac{1}{2}+1 \right)\omega^2\; \rightarrow \omega^2=\frac{4g}{3R}$$
COM's velocity is $v_{cm}=\omega R=\sqrt{\frac{4g}{3R}}R=2\sqrt{\frac{gR}{3}}$
The radial velocity from the circular motion:
$$a_{cen}=\omega^2 R=\frac{4g}{3}$$
Tangential acceleration is from the gravity:
$$M=I\alpha\rightarrow mgR=\frac{3}{2}mR^2\alpha\rightarrow\alpha=\frac{2g}{3R}$$
$$a_{tan}=R\alpha=\frac{2g}{3}$$

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2. Aug 22, 2015

### Staff: Mentor

Looks right.

3. Aug 22, 2015