A large drum having radius $R$ is spinning around its axis with angular velocity $\omega$, as shown

Question

A large drum having radius $R$ is spinning around its axis with angular velocity $\omega$, as shown in figure. The minimum value of $\omega$ so that a body of mass $M$ remains stuck to the inner wall of the drum, taking the coefficient of friction between the drum surface and mass $M$ as $\mu$, is :

Professor PC Thomas Classes · Answer

For the mass to remain stuck to the wall without falling, the static frictional force must balance the gravitational force, so $f = Mg$. The normal force provided by the wall gives the required centripetal acceleration, $N = MR\omega^2$. Since friction $f \le \mu N$, we have $Mg \le \mu MR\omega^2$, which simplifies to $\omega^2 \ge \frac{g}{\mu R}$ or $\omega \ge \sqrt{\frac{g}{\mu R}}$.