Question

A hollow vertical cylinder of radius $R$, with block $M$ on its inner wall, is rotated with angular velocity $\omega$ about an axis through its centre. What is the minimum coefficient of static friction between block $M$ and cylinder wall necessary to keep the block suspended on the inside of the cylinder?

• A. $\mu =\frac{gR}{{\omega }^2}$
• B. $\mu =\frac{{\omega }^{2}g}{R}$
• C. $\mu =\frac{{\omega }^{2}R}{g}$
• D. $\mu =\frac{g}{{\omega }^{2}R}$

Answer 1

The correct option is D $\mu =\frac{g}{{\omega }^{2}R}$

FBD of block $M$:

Normal reaction force provides the necessary centripetal force. The force of friction balances the weight of the body.
i.e $N=mR{\omega }^2$ & $f_s=mg$
$\because f_s\le {\mu }_{s}N$
We get, $mg\le {\mu }_s\left(mR{\omega }^2\right)$
$\Rightarrow {\mu }_s\ge \frac{g}{R{\omega }^2}$
i.e minimum coefficient of friction is $\frac{g}{R{\omega }^2}$