Try this beautiful problem, useful for Physics Olympiad, based on Button on a Rotating Platform.

The Problem:

A small button placed on a horizontal rotating platform with diameter (0.320m) will revolve with the plate when it is brought up to a speed of (40rev/min), provided the button is no more than (0.150m) from the axis. What is the coefficient of static friction between button and platform?


The button moves in a circle so it has acceleration (a_{rad}).
The coefficient of static friction $$ \mu_s=\frac{v^2}{Rg}$$ Expressing (v) in terms of the period (T), $$ v=\frac{2\pi R}{T}$$ so, $$\mu_s=\frac{4\pi^2R}{T^2g}$$.A platform speed of (40.0rev/min) corresponds to a period of (1.50s) so $$\mu_s=0.269$$