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, we will follow these steps:

### Step 1: Convert RPM to Radians per Second
The fan makes 2400 revolutions per minute (rpm). We need to convert this to radians per second (rad/s).

1. **Convert rpm to revolutions per second (rps)**:
\[ \text{Revolutions per second} = \frac{2400 \text{ rpm}}{60 \text{ seconds}} = 40 \text{ rps} \]

2. **Convert revolutions to radians**:
\[ \text{Angular velocity} (\omega_1) = 40 \text{ rps} \times 2\pi \text{ rad/revolution} = 80\pi \text{ rad/s} \]

### Step 2: Use the Angular Deceleration
When the fan is switched off, it comes to rest in 10 seconds. We can find the angular deceleration (\(\alpha\)) using the formula:
\[ \alpha = \frac{\Delta \omega}{\Delta t} = \frac{\omega_2 - \omega_1}{t} \]
Where:
- \(\omega_2 = 0\) (final angular velocity)
- \(\omega_1 = 80\pi\) rad/s (initial angular velocity)
- \(t = 10\) s

Substituting the values:
\[ \alpha = \frac{0 - 80\pi}{10} = -8\pi \text{ rad/s}^2 \]

### Step 3: Calculate the Total Angle Rotated
We can use the angular motion equation to find the total angle (\(\theta\)) rotated before coming to rest:
\[ \theta = \omega_1 t + \frac{1}{2} \alpha t^2 \]
Substituting the known values:
\[ \theta = (80\pi)(10) + \frac{1}{2}(-8\pi)(10^2) \]
Calculating each term:
\[ \theta = 800\pi + \frac{1}{2}(-8\pi)(100) = 800\pi - 400\pi = 400\pi \text{ radians} \]

### Step 4: Calculate the Number of Rotations
To find the number of rotations, we divide the total angle by \(2\pi\) (the angle for one complete revolution):
\[ n = \frac{\theta}{2\pi} = \frac{400\pi}{2\pi} = 200 \]

### Final Answer
The number of times the fan will rotate before it comes to rest is **200 revolutions**.

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