Due to failure of electricity a fan executing 10 rotations per sec comes to rest in 10 second. Teh number of rotations executed by it, before it stops, will be:

Options

A. 10
B. 20
C. 50
D. 100

Answer

To solve the problem of how many rotations a fan executes before coming to rest, we can follow these steps:

### Step 1: Identify the given values
- Initial angular velocity (ω₀) = 10 rotations per second
- Time (t) = 10 seconds
- Final angular velocity (ω) = 0 rotations per second (since the fan comes to rest)

### Step 2: Convert rotations per second to radians per second
Since angular motion is often expressed in radians, we need to convert the initial angular velocity from rotations per second to radians per second.
- 1 rotation = 2π radians
- Therefore, ω₀ = 10 rotations/second × 2π radians/rotation = 20π radians/second

### Step 3: Determine angular acceleration (α)
Using the formula for angular motion:
\[ \omega = \omega_0 + \alpha t \]
Substituting the known values:
\[ 0 = 20\pi + \alpha(10) \]
Rearranging gives:
\[ \alpha = -\frac{20\pi}{10} = -2\pi \text{ radians/second}^2 \]
(The negative sign indicates that it is a deceleration.)

### Step 4: Calculate the angular displacement (θ)
Using the angular displacement formula:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
Substituting the known values:
\[ \theta = (20\pi)(10) + \frac{1}{2}(-2\pi)(10^2) \]
Calculating each term:
- First term: \( 20\pi \times 10 = 200\pi \)
- Second term: \( \frac{1}{2} \times (-2\pi) \times 100 = -100\pi \)

Now, combine the two terms:
\[ \theta = 200\pi - 100\pi = 100\pi \text{ radians} \]

### Step 5: Convert angular displacement to rotations
To find the number of rotations, we convert radians back to rotations:
- Number of rotations = \( \frac{\theta}{2\pi} \)
\[ \text{Number of rotations} = \frac{100\pi}{2\pi} = 50 \text{ rotations} \]

### Final Answer
The number of rotations executed by the fan before it stops is **50 rotations**.

---