A rotating wheel changes angular speed from 1800 rpm to 3000 rpm in 20 s. What is the angular acceleration assuming to be uniform?

Options

A. `60 pi rad s^(-2)`
B. `90 pi rad s^(-2)`
C. `2 pi rad s^(-2)`
D. `40 pi rad s^(-2)`

Explanation

To solve the problem of finding the angular acceleration of a rotating wheel that changes its angular speed from 1800 rpm to 3000 rpm in 20 seconds, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Initial angular speed, \( \omega_0 = 1800 \, \text{rpm} \)
- Final angular speed, \( \omega = 3000 \, \text{rpm} \)
- Time interval, \( t = 20 \, \text{s} \)

2. **Convert Angular Speeds from RPM to RPS:**
- To convert revolutions per minute (rpm) to revolutions per second (rps), divide by 60.
- Initial angular speed:
\[ \omega_0 = \frac{1800}{60} = 30 \, \text{rps} \]
- Final angular speed:
\[ \omega = \frac{3000}{60} = 50 \, \text{rps} \]

3. **Use the Angular Acceleration Formula:**
- The formula for angular acceleration (\( \alpha \)) when the acceleration is uniform is given by:
\[ \omega = \omega_0 + \alpha t \]
- Rearranging this formula to solve for \( \alpha \):
\[ \alpha = \frac{\omega - \omega_0}{t} \]

4. **Substitute the Known Values:**
- Plugging in the values we have:
\[ \alpha = \frac{50 \, \text{rps} - 30 \, \text{rps}}{20 \, \text{s}} = \frac{20 \, \text{rps}}{20 \, \text{s}} = 1 \, \text{rps}^2 \]

5. **Convert Angular Acceleration to Radians per Second Square:**
- Since the options are in radians per second square, we need to convert revolutions per second square to radians per second square.
- We know that \( 1 \, \text{revolution} = 2\pi \, \text{radians} \), so:
\[ \alpha = 1 \, \text{rps}^2 \times 2\pi \, \text{radians/revolution} = 2\pi \, \text{radians/s}^2 \]

6. **Final Answer:**
- Therefore, the angular acceleration \( \alpha \) is:
\[ \alpha = 2\pi \, \text{radians/s}^2 \]

### Conclusion:
The correct answer is \( 2\pi \, \text{radians/s}^2 \), which corresponds to option 3.