Question

A hollow straight tube length and mass m can turn freely about its center on a smooth horizontal table. Another smooth uniform rod of the same and mass is fitted into the tube so that their center collides. The system is set in motion with an initial angular velocity \[{\omega _0}\]. Find the angular velocity of the tube at the instant when the rod slips out of the tube:
A. \[\dfrac{{{\omega _0}}}{3}\]
B. \[\dfrac{{{\omega _0}}}{2}\]
C. \[\dfrac{{{\omega _0}}}{4}\]
D. \[\dfrac{{{\omega _0}}}{7}\]

Answer 1

Hint: Conservation of angular momentum is a physical property of a spinning system such that its spin remains constant unless it is acted upon by an external torque.

Complete step by step solution:
Moment of Inertia of the hollow tube is equal to,
\[\dfrac{{M{I^2}}}{{12}} = I\]
Moment of inertia of the smooth uniform rod is equal to:
\[\dfrac{{M{I^2}}}{{12}} = {I_2}\]
Initial angular momentum is equal to:
\[ = {I_1}{w_0}\]
Final angular momentum is equal to:
\[ = \left( {{I_1} + {I_2}} \right)w\]
By angular momentum conservation:
\[{I_1}{w_0} = \left( {{I_1} + {I_2}} \right)w\]
Therefore, after simplification we get,
\[w = \dfrac{{{w_0}}}{2}\]

Therefore, the correct answer is option (B) i.e. \[w = \dfrac{{{w_0}}}{2}\]

Additional information:
Conservation of angular momentum is a physical property of a spinning system such that its spin remains constant unless it is acted upon by an external torque. In other words, the speed of rotation is constant if net torque is zero.

Angular momentum, also known as spin, is the velocity of rotation of an object around an axis. Gyroscopes are simple devices that exploit the conservation of angular momentum to stabilize, guide or measure the rotational movement in many types of systems. The law of conservation of angular momentum explains why a gyroscope or a spinning top remains upright.

Note:
Before solving the sum students need to understand the meaning of the law of conservation and energy and also its relevance in sums. Students need to understand that under the Law momentum is neither created nor destroyed. It only changes from one state to another.