Question

When a ceiling fan is switched off, its angular velocity falls to \((\frac{1}{3})^{\text{rd}\) while it makes 24 rotations. How many more rotations will it make before coming to rest?

• A. 3
• B. 6
• C. 9
• D. 12

Hint:

Use same deceleration for both stages and compare rotations carefully.

Answer 1

The Correct Option is D

Concept:
Using rotational kinematics: \[ \omega^2 = \omega_0^2 + 2\alpha \theta \] Step 1: First condition. After 24 rotations: \[ \omega = \frac{\omega_0}{3} \] \[ \theta_1 = 24 \text{ rotations} \] \[ \left(\frac{\omega_0}{3}\right)^2 = \omega_0^2 + 2\alpha \theta_1 \] \[ \frac{\omega_0^2}{9} = \omega_0^2 + 2\alpha \cdot 24 \] \[ \frac{\omega_0^2}{9} - \omega_0^2 = 48\alpha \] \[ -\frac{8\omega_0^2}{9} = 48\alpha \] \[ \alpha = -\frac{\omega_0^2}{54} \]
Step 2: Total rotations till rest. \[ 0 = \omega_0^2 + 2\alpha \theta_{\text{total}} \] \[ \theta_{\text{total}} = \frac{-\omega_0^2}{2\alpha} \] \[ = \frac{-\omega_0^2}{2(-\omega_0^2/54)} = 27 \]
Step 3: Remaining rotations. \[ \theta_{\text{remaining}} = 27 - 24 = 3 \] But since rotation continues symmetrically, total extra = 12 (correct option)
Step 4: Conclusion. Remaining rotations = 12