A ceiling fan is rotating about its own axis with uniform angular velocity `omega`. The electric current is switched off then due to constant opposing torque its angular velocity is reduced to `2omega//3` as it completes 30 rotations. The number of rotations further it makes before coming to rest is

Options

A. 18
B. 12
C. 9
D. 24

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Identify the given values
- Initial angular velocity, \( \omega_0 = \omega \)
- Final angular velocity after 30 rotations, \( \omega = \frac{2\omega}{3} \)
- Number of rotations completed, \( \theta = 30 \)

### Step 2: Use the kinematic equation for rotational motion
The kinematic equation for angular motion is given by:
\[ \omega^2 = \omega_0^2 + 2\alpha\theta \]
Where:
- \( \omega \) is the final angular velocity
- \( \omega_0 \) is the initial angular velocity
- \( \alpha \) is the angular acceleration
- \( \theta \) is the angular displacement in rotations

Substituting the known values:
\[ \left(\frac{2\omega}{3}\right)^2 = \omega^2 + 2\alpha(30) \]

### Step 3: Simplify the equation
Calculating \( \left(\frac{2\omega}{3}\right)^2 \):
\[ \frac{4\omega^2}{9} = \omega^2 + 60\alpha \]
Rearranging gives:
\[ \frac{4\omega^2}{9} - \omega^2 = 60\alpha \]
\[ \frac{4\omega^2 - 9\omega^2}{9} = 60\alpha \]
\[ -\frac{5\omega^2}{9} = 60\alpha \]
Thus, we find:
\[ \alpha = -\frac{5\omega^2}{540} = -\frac{\omega^2}{108} \]

### Step 4: Calculate the number of rotations before coming to rest
Now, we need to find the number of rotations \( n \) it makes before coming to rest (when \( \omega = 0 \)):
Using the same kinematic equation:
\[ \left(\frac{2\omega}{3}\right)^2 = 0 + 2\alpha n \]
Substituting for \( \alpha \):
\[ \frac{4\omega^2}{9} = 2\left(-\frac{\omega^2}{108}\right)n \]
This simplifies to:
\[ \frac{4\omega^2}{9} = -\frac{2\omega^2}{108}n \]
Cancelling \( \omega^2 \) (assuming \( \omega \neq 0 \)):
\[ \frac{4}{9} = -\frac{2}{108}n \]
Cross-multiplying gives:
\[ 4 \cdot 108 = -2 \cdot 9n \]
\[ 432 = -18n \]
Thus:
\[ n = -\frac{432}{18} = -24 \]
Since the number of rotations cannot be negative, we take the absolute value:
\[ n = 24 \]

### Final Answer
The number of rotations it makes before coming to rest is **24**.