When a ceiling fan is switched off, its angular velocity reduces to 50% while it makes 36 rotations. How many more rotations will it make before coming to rest?(Assume uniform angular retardation)

Options

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

Answer

To solve the problem step by step, we will use the equations of motion for rotational motion.

### Step 1: Understand the initial conditions
When the ceiling fan is switched off, its initial angular velocity (\( \omega_0 \)) is reduced to 50%. Therefore, the final angular velocity (\( \omega \)) after making 36 rotations is:
\[ \omega = \frac{\omega_0}{2} \]

### Step 2: Calculate the angular displacement for the first part
The fan makes 36 rotations. We convert this to radians:
\[ \theta_1 = 36 \text{ rotations} \times 2\pi \text{ radians/rotation} = 72\pi \text{ radians} \]

### Step 3: Apply the equation of motion
Using the equation of motion for rotational motion:
\[ \omega^2 = \omega_0^2 + 2\alpha\theta \]
Substituting the known values:
\[ \left(\frac{\omega_0}{2}\right)^2 = \omega_0^2 + 2\alpha(72\pi) \]
This simplifies to:
\[ \frac{\omega_0^2}{4} = \omega_0^2 + 144\pi\alpha \]

### Step 4: Rearranging the equation
Rearranging gives:
\[ \frac{\omega_0^2}{4} - \omega_0^2 = 144\pi\alpha \]
\[ -\frac{3\omega_0^2}{4} = 144\pi\alpha \]
Thus,
\[ \alpha = -\frac{3\omega_0^2}{576\pi} = -\frac{\omega_0^2}{192\pi} \]

### Step 5: Calculate the additional rotations before coming to rest
Now, we need to find out how many more rotations the fan makes before coming to rest. When the fan comes to rest, \( \omega = 0 \). Using the equation of motion again:
\[ 0 = \left(\frac{\omega_0}{2}\right)^2 + 2\alpha\theta_2 \]
Substituting for \( \alpha \):
\[ 0 = \frac{\omega_0^2}{4} + 2\left(-\frac{\omega_0^2}{192\pi}\right)\theta_2 \]
This simplifies to:
\[ 0 = \frac{\omega_0^2}{4} - \frac{\omega_0^2}{96\pi}\theta_2 \]

### Step 6: Solve for \( \theta_2 \)
Rearranging gives:
\[ \frac{\omega_0^2}{96\pi}\theta_2 = \frac{\omega_0^2}{4} \]
Dividing both sides by \( \frac{\omega_0^2}{96\pi} \):
\[ \theta_2 = \frac{4 \times 96\pi}{1} = 384\pi \]

### Step 7: Convert angular displacement to rotations
To find the number of rotations:
\[ \text{Rotations} = \frac{\theta_2}{2\pi} = \frac{384\pi}{2\pi} = 192 \]

### Step 8: Total rotations
The total rotations before coming to rest is the sum of the initial rotations and the additional rotations:
\[ \text{Total Rotations} = 36 + 192 = 228 \]

### Final Answer
The number of additional rotations before coming to rest is:
\[ \text{Additional Rotations} = 192 \]