Question

When a ceiling fan is switched off. It's angular velocity reduces by 50% while it makes 36rotations .how many more rotations will it make before coming to rest?

Answer 1

Answer:

12 rotations

Explanation:

Let the initial angular velocity of the fan is $\omega$

Since after 36 rotations the velocity is reduced by 50%

Therefore, if the angular acceleration is $\alpha$

Then using the third equation of rotational motion

$\boxed{\omega'^2=\omega^2+2\alpha \times \theta}}$

Here angular displacement

$\theta=2\pi n$

$\implies \theta=2\pi\times 36$

$\implies \theta=72\pi$

Thus,

$(\frac{\omega}{2})^2=\omega^2-2\alpha\times 72\pi$

$\implies \omega^2/4=\omega^2-144\pi\alpha$

$\implies \alpha=\frac{3\omega^2}{4\times 144\pi}$

When the fan will stop, its final angular velocity will be zero

Let the rotations be N

Then

$0^2=\omega^2/4-2\alpha(2\pi N)$

$\implies\frac{\omega^2}{4}=4\times\frac{3\omega^2}{4\times 144\pi}\times\pi N$

$\implies \frac{1}{4}=\frac{12N}{4\times 144}$

$\implies N=\frac{144}{12}$

$\implies N=12$

Therefore, the fan will make 12 more rotations before coming to stop.

Hope this helps.