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. 24
B. 36
C. 18
D. 12

Answer

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

### Step 1: Understand the problem
When the ceiling fan is switched off, its angular velocity reduces to half while it completes 36 rotations. We need to find out how many more rotations it will make before coming to rest.

### Step 2: Define the initial parameters
Let the initial angular velocity of the fan be \( \omega \). After making 36 rotations, the final angular velocity becomes \( \frac{\omega}{2} \).

### Step 3: Convert rotations to radians
The total angle covered in 36 rotations can be calculated as follows:
\[ \theta = 36 \text{ rotations} \times 2\pi \text{ radians/rotation} = 72\pi \text{ radians} \]

### Step 4: Use the equation of motion
We will use the third equation of rotational motion:
\[ \omega_f^2 = \omega_i^2 + 2\alpha\theta \]
Where:
- \( \omega_f = \frac{\omega}{2} \)
- \( \omega_i = \omega \)
- \( \theta = 72\pi \)

Substituting the values into the equation:
\[ \left(\frac{\omega}{2}\right)^2 = \omega^2 + 2\alpha(72\pi) \]
This simplifies to:
\[ \frac{\omega^2}{4} = \omega^2 + 144\pi\alpha \]

### Step 5: Rearranging the equation
Rearranging the equation gives:
\[ \frac{\omega^2}{4} - \omega^2 = 144\pi\alpha \]
\[ -\frac{3\omega^2}{4} = 144\pi\alpha \]
Thus, we can express \( \alpha \) as:
\[ \alpha = -\frac{3\omega^2}{576\pi} = -\frac{\omega^2}{192\pi} \]

### Step 6: Analyze the next phase of motion
Now, we need to find how many more rotations the fan will make before coming to rest. In this phase:
- Initial angular velocity \( \omega_i = \frac{\omega}{2} \)
- Final angular velocity \( \omega_f = 0 \)
- Angular acceleration \( \alpha = -\frac{\omega^2}{192\pi} \)

### Step 7: Use the equation of motion again
Using the same equation:
\[ \omega_f^2 = \omega_i^2 + 2\alpha\theta \]
Substituting the known values:
\[ 0 = \left(\frac{\omega}{2}\right)^2 + 2\left(-\frac{\omega^2}{192\pi}\right)\theta \]
This simplifies to:
\[ 0 = \frac{\omega^2}{4} - \frac{\omega^2}{96\pi}\theta \]

### Step 8: Solve for \( \theta \)
Rearranging gives:
\[ \frac{\omega^2}{96\pi}\theta = \frac{\omega^2}{4} \]
Cancelling \( \omega^2 \) from both sides (assuming \( \omega \neq 0 \)):
\[ \theta = 24\pi \]

### Step 9: Convert radians to rotations
To find the number of rotations:
\[ \text{Number of rotations} = \frac{\theta}{2\pi} = \frac{24\pi}{2\pi} = 12 \]

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

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