A 0.5 kg mass is in contact against the inner wall of a cylindrical drum of radius 4 m rotating about its vertical axis. The minimum rotational speed of the drum to enable the mass to remain stuck to the wall (without falling) is 5 rad/s. The coefficient of friction between the drum’s inner wall surface and mass is _________. (Take $g = 10\ \mathrm{m/s^2}$)

A particle is rotating in a circular path and at any instant its motion can be described as

$\theta = \frac{5t^4}{40} - \frac{t^3}{3}$.

The angular acceleration of the particle after 10 seconds is _________ rad/s².

In case of vertical circular motion of a particle by a thread of length $r$ if the tension in the thread is zero at an angle $30^{\circ}$ shown in figure, the velocity at the bottom point $(A)$ of the circular path is (g = gravitational acceleration)

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 :