Disc Rolling on Disc

A disc or radius r on top of another disc of radius 4r is made to go around it via pure rolling motion. How many maximum full rotations will the small disc make in this process?

Solution

Pure rolling motion = Pure translational motion + Pure rotational motion

Consider pure rotational motion,

Time for 1 rotation $T = \frac {2\pi}{\omega}$

In the same time T the distance $d_{CM} = v_{CM}\times T$

In pure rolling motion, $v_{CM} = r\omega$

$\therefore {d_{CM}} = r\omega \times \frac{{2\pi }}{\omega } = 2\pi r$

While $2\pi r$ is the distance covered by the centre of the disc in one rotation, the total distance to be covered by the centre of the disc =$2\pi (r+4r)$

Let there be n number of full rotations,

Then, $n.(d_{CM}=2\pi r) \le 2\pi (r + 4r)$

$\therefore{n_{\max }} = 5$

Reference: Study material file on, "Rotational Motion" version 5.1 or higher.