Question

A block ($P$) is rotating in contact with the vertical wall of a rotor as shown in figures A, B, C. The relation between angular velocities $\omega_A, \omega_B, \omega_C$ so that the block does not slide down. (Given: $R_A < R_B < R_C$, where $R$ denotes radius) 

• A. $\omega_A<\omega_B<\omega_C$
• B. $\omega_A = \omega_B = \omega_C$
• C. $\omega_C<\omega_B<\omega_A$
• D. $\omega_C = \omega_A + \omega_B$

Hint:

In problems involving rotational motion and centripetal force, remember that the centripetal force is proportional to the square of the angular velocity and the radius. For the same force, larger radii require lower angular velocities.

Answer 1

The Correct Option is C

To prevent the block from sliding down due to gravity, the centripetal force needs to be provided by the rotation of the rotor. The centripetal force is given by $F_c = mR\omega^2$, where $R$ is the radius of the rotor, and $\omega$ is the angular velocity. The relationship between the angular velocity and the radius is inverse, meaning a larger radius requires a smaller angular velocity to provide the same centripetal force. Therefore, for the block to not slide down, we need: $$\omega_C<\omega_B<\omega_A$$ This relationship ensures that the block stays in place on the rotor. Larger radii (like $R_C$) correspond to smaller angular velocities, and smaller radii (like $R_A$) require higher angular velocities to maintain the same centripetal force.