A fan makes 2400 rpm. If after it is switched off, it comes to rest in 10 s, then find the number of times it will rotate before it comes to rest after it is switched off.

Options

A. 400
B. 100
C. 200
D. 50

Explanation

To solve the problem step by step, we need to find out how many times the fan rotates before it comes to rest after being switched off.

### Step 1: Convert RPM to Angular Velocity
The fan operates at 2400 revolutions per minute (rpm). We first convert this to revolutions per second (rps) and then to angular velocity in radians per second.

\[ \text{Revolutions per second} = \frac{2400 \text{ rpm}}{60} = 40 \text{ rps} \]

Now, we convert revolutions per second to radians per second using the formula:

\[ \text{Angular velocity} (\omega) = 2\pi \times \text{frequency} = 2\pi \times 40 = 80\pi \text{ rad/s} \]

### Step 2: Use the Equation of Motion
When the fan is switched off, it decelerates uniformly until it comes to rest. We can use the angular displacement formula:

\[ \theta = \omega_{\text{initial}} \cdot t + \frac{1}{2} \cdot \alpha \cdot t^2 \]

Where:
- \(\theta\) is the angular displacement in radians,
- \(\omega_{\text{initial}} = 80\pi \text{ rad/s}\),
- \(t = 10 \text{ s}\),
- \(\alpha\) is the angular acceleration.

Since the fan comes to rest, \(\omega_{\text{final}} = 0\). We can find the angular acceleration (\(\alpha\)) using:

\[ \alpha = \frac{\omega_{\text{final}} - \omega_{\text{initial}}}{t} = \frac{0 - 80\pi}{10} = -8\pi \text{ rad/s}^2 \]

### Step 3: Calculate Angular Displacement
Now, substituting the values into the angular displacement formula:

\[ \theta = (80\pi) \cdot (10) + \frac{1}{2} \cdot (-8\pi) \cdot (10^2) \]

Calculating each term:

\[ \theta = 800\pi - \frac{1}{2} \cdot 8\pi \cdot 100 \]
\[ = 800\pi - 400\pi = 400\pi \text{ radians} \]

### Step 4: Convert Angular Displacement to Number of Rotations
To find the number of rotations, we divide the angular displacement by \(2\pi\):

\[ \text{Number of rotations} (n) = \frac{\theta}{2\pi} = \frac{400\pi}{2\pi} = 200 \]

### Final Answer
The fan will rotate **200 times** before it comes to rest after it is switched off.

---