Question

When a ceiling fan is switched off, its angular velocity reduces to half its initial value after it completes 36 rotations. The number of rotations it will make further before coming to rest is (Assuming angular retardation to be uniform)

• A. 10
• B. 20
• C. 18
• D. 12

Answer 1

The Correct Option is D

$\omega ^2 =\omega ^2_0+ 2\alpha \theta_n $
where $\omega $ is final angular velocity, $\omega_0$ is initial angular velocity, $\alpha$ is angular retardation and $\theta_n$ is the angle turned in number of rotations.
Here initially $\omega =\frac{\omega_0}{2},\theta =36 \times 2 \pi$
hence $\frac{\omega^2_0}{4},\theta =36 \times 2 \pi$
or $\frac{3\omega ^2_0}{4}=144 \pi \alpha $ or $\alpha =-\frac{3 \omega ^2_3}{4\times 144 \pi}$
Negative sign signifies angular retardation. Again, for the IInd case
$0=\frac{\omega^2_0}{4}-2 \times \frac{3 \omega^2_0}{4 \times 144 \pi}\times \theta'$
or$\theta =\frac{\omega^2_0 \times 4 \times 144 \pi}{4 \times 2\times 3\omega^2_0}$
$ 24 \pi$
Hence, number of rotations it makes before coming to rest
$=\frac{24 \pi}{2 \pi}=12$