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 find 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 1: Identify the given values
- Initial angular speed (\( \omega_0 \)) = 1800 rpm
- Final angular speed (\( \omega \)) = 3000 rpm
- Time interval (\( t \)) = 20 seconds

### Step 2: Convert angular speeds from rpm to rps
Since the angular speeds are given in revolutions per minute (rpm) and the time is in seconds, we need to convert the angular speeds to revolutions per second (rps).

\[ \omega_0 = \frac{1800 \text{ rpm}}{60} = 30 \text{ rps} \]
\[ \omega = \frac{3000 \text{ rpm}}{60} = 50 \text{ rps} \]

### Step 3: Use the formula for angular acceleration
The formula relating angular acceleration (\( \alpha \)), initial angular speed (\( \omega_0 \)), final angular speed (\( \omega \)), and time (\( t \)) is:

\[ \omega = \omega_0 + \alpha t \]

### Step 4: Rearrange the formula to solve for angular acceleration (\( \alpha \))
Rearranging the formula gives us:

\[ \alpha = \frac{\omega - \omega_0}{t} \]

### Step 5: Substitute the values into the equation
Now, substituting the values we have:

\[ \alpha = \frac{50 \text{ rps} - 30 \text{ rps}}{20 \text{ s}} = \frac{20 \text{ rps}}{20 \text{ s}} = 1 \text{ rps}^2 \]

### Step 6: Convert angular acceleration to radians per second squared
Since the problem asks for the answer in radians per second squared, we need to convert revolutions per second squared to radians per second squared. We know that:

\[ 1 \text{ revolution} = 2\pi \text{ radians} \]

Thus, we can convert:

\[ \alpha = 1 \text{ rps}^2 \times 2\pi = 2\pi \text{ rad/s}^2 \]

### Final Answer
Therefore, the angular acceleration is:

\[ \alpha = 2\pi \text{ rad/s}^2 \]

### Conclusion
Looking at the options provided, the correct answer is option 3: \( 2\pi \) rad/s².

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