A body rotating with uniform angular acceleration covers `100 pi` (radian) in the first 5 s after the start. Its angular speed at the end of 5 s (in rad/s) is

Options

A. `40 pi`
B. `30 pi`
C. `20 pi`
D. `10 pi`

Explanation

To solve the problem, we need to find the angular speed at the end of 5 seconds for a body rotating with uniform angular acceleration. We know the angular distance covered in the first 5 seconds is \(100\pi\) radians.

### Step-by-Step Solution:

1. **Identify the given values**:
- Angular distance (\(\theta\)) = \(100\pi\) radians
- Time (\(t\)) = 5 seconds
- Initial angular speed (\(\omega_0\)) = 0 rad/s (since the body starts from rest)

2. **Use the angular displacement formula**:
The formula for angular displacement when starting from rest is:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
Since \(\omega_0 = 0\), this simplifies to:
\[ \theta = \frac{1}{2} \alpha t^2 \]

3. **Substitute the known values into the equation**:
\[ 100\pi = \frac{1}{2} \alpha (5^2) \]
\[ 100\pi = \frac{1}{2} \alpha (25) \]
\[ 100\pi = 12.5\alpha \]

4. **Solve for angular acceleration (\(\alpha\))**:
\[ \alpha = \frac{100\pi}{12.5} = 8\pi \text{ rad/s}^2 \]

5. **Use the angular speed formula**:
The formula for angular speed at time \(t\) is:
\[ \omega = \omega_0 + \alpha t \]
Again, since \(\omega_0 = 0\):
\[ \omega = \alpha t \]

6. **Substitute the values of \(\alpha\) and \(t\)**:
\[ \omega = (8\pi)(5) = 40\pi \text{ rad/s} \]

7. **Conclusion**:
The angular speed at the end of 5 seconds is \(40\pi\) rad/s.

### Final Answer:
\(\omega = 40\pi\) rad/s

---