When a celling fan is switched off, its angular velocity falls to half while it makes `36` rotations. How many more rotations will it make before coming to rest ?

Options

A. 18
B. 12
C. 36
D. 48

Explanation

To solve the problem step by step, we can follow the reasoning outlined in the video transcript:

### Step 1: Define Initial Conditions
Let the initial angular velocity of the ceiling fan be \( \omega \). After making 36 rotations, the angular velocity reduces to half, so the final angular velocity after these rotations is:
\[ \omega_f = \frac{\omega}{2} \]

### Step 2: Calculate Angular Displacement for 36 Rotations
The angular displacement \( \theta \) for 36 rotations can be calculated as:
\[ \theta = 36 \times 2\pi = 72\pi \text{ radians} \]

### Step 3: Use the Equation of Motion
We can use the third equation of motion for rotational motion, which is:
\[ \omega_f^2 = \omega_i^2 + 2\alpha \theta \]
Substituting the known values:
\[ \left(\frac{\omega}{2}\right)^2 = \omega^2 + 2\alpha(72\pi) \]
This simplifies to:
\[ \frac{\omega^2}{4} = \omega^2 + 144\pi\alpha \]

### Step 4: Rearranging the Equation
Rearranging gives:
\[ \frac{\omega^2}{4} - \omega^2 = 144\pi\alpha \]
\[ -\frac{3\omega^2}{4} = 144\pi\alpha \]
Thus, we can express the angular acceleration \( \alpha \):
\[ \alpha = -\frac{3\omega^2}{576\pi} = -\frac{\omega^2}{192\pi} \]

### Step 5: Calculate Total Angular Displacement Until Rest
Now, we need to find the total angular displacement until the fan comes to rest. The initial angular velocity is \( \frac{\omega}{2} \) and the final angular velocity is \( 0 \). Using the same equation of motion:
\[ 0 = \left(\frac{\omega}{2}\right)^2 + 2\alpha \theta' \]
Where \( \theta' \) is the additional angular displacement until rest. Substituting for \( \alpha \):
\[ 0 = \frac{\omega^2}{4} + 2\left(-\frac{\omega^2}{192\pi}\right)\theta' \]
This simplifies to:
\[ 0 = \frac{\omega^2}{4} - \frac{\omega^2}{96\pi}\theta' \]

### Step 6: Solve for \( \theta' \)
Rearranging gives:
\[ \frac{\omega^2}{96\pi}\theta' = \frac{\omega^2}{4} \]
Dividing both sides by \( \omega^2 \) (assuming \( \omega \neq 0 \)):
\[ \frac{\theta'}{96\pi} = \frac{1}{4} \]
Thus,
\[ \theta' = 24\pi \text{ radians} \]

### Step 7: Convert Angular Displacement to Rotations
To find the number of rotations \( n \) corresponding to \( \theta' \):
\[ n = \frac{\theta'}{2\pi} = \frac{24\pi}{2\pi} = 12 \]

### Final Answer
The ceiling fan will make **12 more rotations** before coming to rest.

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