Friction, you make my head right round!

A sphere of mass \(\displaystyle\text{m}\) and radius \(\displaystyle R\) is placed on a rough plank of mass \(\displaystyle \text{2m}\). This system is placed on a rough inclined plane, with an incline of \(\displaystyle \theta\).

The friction coefficient between the plank and the incline is \(\displaystyle\mu_1\), and that between sphere and plank is \(\displaystyle \mu_2\).

Find the maximum value of \(\displaystyle \frac{\mu_1}{\mu_2}\), such that the sphere is always pure rolling on the plank.

Details and Assumptions:
\(\bullet\) \(\displaystyle m = 2kg\)
\(\bullet\) \(\displaystyle \theta = 30^o\)
\(\bullet\) \(\displaystyle R = 15cm\)
\(\bullet\) \(\displaystyle g = 9.8m/s^2\)


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