Question

When a rotating disc suddenly shrinks in radius, what happens to the angular velocity \( \omega \)?

• A. \( \omega \) decreases
• B. \( \omega \) increases
• C. \( \omega \) remains constant
• D. None of the above

Hint:

When a rotating body shrinks in size, its moment of inertia decreases, causing the angular velocity to increase to conserve angular momentum.

Answer 1

The Correct Option is B

When a rotating disc shrinks in radius, its moment of inertia decreases. According to the **conservation of angular momentum**, which states that the angular momentum \( L \) of a system is conserved if no external torque acts on it, we have: \[ L = I \omega \] Where: - \( I \) is the moment of inertia, - \( \omega \) is the angular velocity. Since no external torque is applied, angular momentum is conserved: \[ I_1 \omega_1 = I_2 \omega_2 \] Where: - \( I_1 \) and \( \omega_1 \) are the moment of inertia and angular velocity initially, - \( I_2 \) and \( \omega_2 \) are the moment of inertia and angular velocity after the radius shrinks. Since the moment of inertia \( I \) for a disc is given by: \[ I = \frac{1}{2} m r^2 \] If the radius \( r \) decreases, the moment of inertia decreases. To conserve angular momentum, the angular velocity \( \omega \) must increase. Thus, the correct answer is: \[ \boxed{(B) \, \omega \, \text{increases}} \]