Question 207: Work, Energy and Power & Circular Motion

Question

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

Accepted Answer

Let the initial angular velocity be $\omega_0$.
Final angular velocity after 24 rotations is $\omega_1 = \frac{\omega_0}{3}$.
Let the angular retardation (deceleration) be $\alpha$ (negative value, since fan is slowing down).
Number of rotations made during retardation is $n$.
Step 1: Use rotational motion equation
The relation between angular velocities, angular acceleration, and angular displacement is:
$$ \omega_1^2 = \omega_0^2 + 2\alpha 	heta $$
where $	heta$ is the angle rotated (in radians).
1 rotation = $2\pi$ radians
After 24 rotations:
$$ 	heta_1 = 24 	imes 2\pi = 48\pi~	ext{radians} $$
So,
$$ \left(\frac{\omega_0}{3}ight)^2 = \omega_0^2 + 2\alpha 	imes 48\pi $$
$$ \frac{\omega_0^2}{9} = \omega_0^2 + 96\pi\alpha $$
Bring like terms together:
$$ \frac{\omega_0^2}{9} - \omega_0^2 = 96\pi\alpha $$
$$ \left( \frac{1}{9} - 1 ight)\omega_0^2 = 96\pi\alpha $$
$$ - \frac{8}{9}\omega_0^2 = 96\pi\alpha $$
$$ \alpha = -\frac{8\omega_0^2}{9 	imes 96\pi} $$
$$ \alpha = -\frac{\omega_0^2}{108\pi} $$
Step 2: Find total rotations till rest
Let total number of rotations before coming to rest be $N$.
At rest, final angular velocity $\omega = 0$.
Apply same formula:
$$ \omega^2 = \omega_0^2 + 2\alpha	heta_{	ext{total}} $$
$$ 0 = \omega_0^2 + 2\alpha	heta_{	ext{total}} $$
$$ 2\alpha	heta_{	ext{total}} = -\omega_0^2 $$
$$ 	heta_{	ext{total}} = \frac{-\omega_0^2}{2\alpha} $$
Substitute $\alpha$ from earlier:
$$ 	heta_{	ext{total}} = \frac{-\omega_0^2}{2 	imes \left(-\frac{\omega_0^2}{108\pi}ight)} $$
$$ 	heta_{	ext{total}} = \frac{-\omega_0^2}{-\frac{2\omega_0^2}{108\pi}} $$
$$ 	heta_{	ext{total}} = \frac{-\omega_0^2 	imes 108\pi}{-2\omega_0^2} $$
$$ 	heta_{	ext{total}} = \frac{108\pi}{2} $$
$$ 	heta_{	ext{total}} = 54\pi~	ext{radians} $$
Number of total rotations:
$$ N = \frac{	heta_{	ext{total}}}{2\pi} = \frac{54\pi}{2\pi} = 27 $$
Step 3: Find required answer
Rotations still to be made after first $24$ rotations:
$$ N - 24 = 27 - 24 = 3 $$
Final Answer:
$ \boxed{3} $
Option A is correct.

Answer

3

Answer

6

Answer

9

Answer

12

Work, Energy and Power & Circular Motion

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

Choose the correct answer: