When a ceiling fan is switched on, it makes $10$ rotations in the first $3$ seconds. How many rotations will it make in the next $3$ seconds? (Assume uniform angular acceleration.)

The correct option is D $30$
Given that,
at $t=3\text{s}$
${\theta }_1=10$ rotations
$=20\pi$ rad
Using equations of motion,
${\theta }_1=\frac{1}{2}\alpha t^2$ [As fan starts from rest]
$\Rightarrow 20\pi =\frac{9\alpha }{2}$
$\Rightarrow \alpha =\frac{40\pi }{9}{\text{rad/s}}^2$
For total of $6$ secs, equation of motion is
${\theta }_1+{\theta }_2=\frac{1}{2}\alpha t^2$
$20\pi +{\theta }_2=\frac{1}{2}\times \frac{40\pi }{9}\times 36$
${\theta }_2=60\pi$
${\theta }_2=30\left(2\pi \right)=30$ rotations
Hence, option $\left(C\right)$ is correct.