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. `60pi" rad s"^(-2)`
B. `90pi" rad s"^(-2)`
C. `2pi" rad s"^(-2)`
D. `40pi" 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 1: Convert angular speeds from rpm to radians per second
We know that the angular speed in radians per second (ω) can be calculated using the formula:
\[ \omega = 2\pi n \]
where \( n \) is the speed in revolutions per minute (rpm).

For the initial angular speed (\( \omega_1 \)):
\[ n_1 = 1800 \, \text{rpm} \]
\[ \omega_1 = 2\pi \times \frac{1800}{60} = 2\pi \times 30 = 60\pi \, \text{radians/second} \]

For the final angular speed (\( \omega_2 \)):
\[ n_2 = 3000 \, \text{rpm} \]
\[ \omega_2 = 2\pi \times \frac{3000}{60} = 2\pi \times 50 = 100\pi \, \text{radians/second} \]

### Step 2: Calculate the change in angular velocity
The change in angular velocity (\( \Delta \omega \)) is given by:
\[ \Delta \omega = \omega_2 - \omega_1 \]
Substituting the values:
\[ \Delta \omega = 100\pi - 60\pi = 40\pi \, \text{radians/second} \]

### Step 3: Calculate angular acceleration
Angular acceleration (\( \alpha \)) is defined as the change in angular velocity divided by the time taken (\( \Delta t \)):
\[ \alpha = \frac{\Delta \omega}{\Delta t} \]
Given that \( \Delta t = 20 \, \text{seconds} \):
\[ \alpha = \frac{40\pi}{20} = 2\pi \, \text{radians/second}^2 \]

### Final Answer
The angular acceleration of the wheel is:
\[ \alpha = 2\pi \, \text{radians/second}^2 \]

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