Find a ratio of velocities.

Question

The following image shows a circular disk rolling on a surface. If the velocity of a point on the edge of the circular disk is $V{p}$ and the velocity of the center of the disk is $V_{cm}$ then find $\frac{V_p}{V_{cm}}$.

Answer 1

$$\frac{V_p}{V_{cm}}=2$$ It holds true because from the rule of addition of velocities in classical mechanics follows that $V_p=V_{cm}+v$, where $v=\omega r \rightarrow v=\frac{2\pi r}{T}$ and $V_{cm}=\frac{2\pi r}{T} \rightarrow V_p=2V_{cm}$.

Note that the velocity of the point of touching the ground equals zero, because we have a sum of two velocities with the same value like at the top point, but with exactly opposite directions.