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 step by step, we will follow these steps:

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

### Step 2: Calculate the angular acceleration (\( \alpha \))
We can use the equation of motion for angular velocity:
\[ \omega_f = \omega_i + \alpha t \]
Substituting the known values:
\[ 0 = 10 + \alpha \cdot 10 \]
Rearranging the equation to solve for \( \alpha \):
\[ \alpha \cdot 10 = -10 \]
\[ \alpha = -1 \text{ rotation per second}^2 \]

### Step 3: Calculate the number of rotations (\( \theta \))
We will use the equation for angular displacement:
\[ \theta = \omega_i t + \frac{1}{2} \alpha t^2 \]
Substituting the known values:
\[ \theta = 10 \cdot 10 + \frac{1}{2} \cdot (-1) \cdot (10^2) \]
Calculating each term:
\[ \theta = 100 + \frac{1}{2} \cdot (-1) \cdot 100 \]
\[ \theta = 100 - 50 \]
\[ \theta = 50 \text{ rotations} \]

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

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