The angular speed of the wheel of a vehicle is increased from 360 rpm to 1200 rpm in 14 second Its angular acceleration is

Options

A. `2pi rad/s^2`
B. `28 pi rad/s^2`
C. `120 pi rad/s^2`
D. `1 rad/s^2`

Answer

To find the angular acceleration of the wheel, we can use the formula for angular acceleration, which is defined as the change in angular velocity over time. The steps to solve the problem are as follows:

### Step-by-Step Solution:

1. **Convert Initial Angular Speed from RPM to Radians per Second:**
- The initial angular speed \( \omega_0 \) is given as 360 RPM.
- To convert RPM to rotations per second, divide by 60:
\[ \omega_0 = \frac{360 \text{ RPM}}{60} = 6 \text{ rotations per second} \]
- Since there are \( 2\pi \) radians in one rotation, we convert to radians per second:
\[ \omega_0 = 6 \times 2\pi = 12\pi \text{ radians per second} \]

2. **Convert Final Angular Speed from RPM to Radians per Second:**
- The final angular speed \( \omega_t \) is given as 1200 RPM.
- Convert to rotations per second:
\[ \omega_t = \frac{1200 \text{ RPM}}{60} = 20 \text{ rotations per second} \]
- Convert to radians per second:
\[ \omega_t = 20 \times 2\pi = 40\pi \text{ radians per second} \]

3. **Calculate the Change in Angular Velocity:**
- The change in angular velocity \( \Delta \omega \) is:
\[ \Delta \omega = \omega_t - \omega_0 = 40\pi - 12\pi = 28\pi \text{ radians per second} \]

4. **Use the Formula for Angular Acceleration:**
- Angular acceleration \( \alpha \) is defined as:
\[ \alpha = \frac{\Delta \omega}{\Delta t} \]
- Here, \( \Delta t = 14 \text{ seconds} \):
\[ \alpha = \frac{28\pi}{14} = 2\pi \text{ radians per second squared} \]

5. **Final Result:**
- The angular acceleration of the wheel is:
\[ \alpha = 2\pi \text{ radians per second squared} \]