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*

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.