Question

A bar of length l carrying a small mass m at one of its ends rotates with a uniform angular speed $\omega$ in a vertical plane about the mid-point of the bar. During the rotation, at some instant of time when the bar is horizontal, the mass is detached from the bar but the bar continues to rotate with same $\omega$. The mass moves vertically up, comes back and reches the bar at the same point. At that place, the acceleration due to gravity is g.

• A. This is possible if the quantity $\frac{{\omega }^2\ell }{2\pi g}$ is an integer
• B. The total time of flight of the mass is proportional to ${\omega }^2$
• C. The total distance travelled by the mass n air is proportional to ${\omega }^2$
• D. The total distance travelled by the mass in air and its total time of flight are both independent on its mass.

Answer 1

Answer: (A), (C), (D)

The correct options are
A This is possible if the quantity $\frac{{\omega }^2\ell }{2\pi g}$ is an integer
C The total distance travelled by the mass n air is proportional to ${\omega }^2$
D The total distance travelled by the mass in air and its total time of flight are both independent on its mass.

The whole system is possible only if $\frac{{\omega }^{2}L}{2mg}$ is an integer, where the total distance travelled by the mass is proportional to ${\omega }^2$ and the distance travelled by the mass in air and its total time of flight are both independent on its mass.

The law of conservation of angular momentum states that '' When the net external torque acting on a system about a given axis is zero, the total angular momentum of the system about the axis remains constant''.

Hence, this question has multiple solutions.