The angular speed of a motor wheel is increased from 1200 rpm to 3120 rpm in 16 seconds. The angular acceleration of the motor wheel is a) 2 π rads−2 b) 4 π rads−2 c) 6 π rads−2 d) 8 π rads−2

Options

A. `2 pi rad s^(-2)`
B. `4pi rad s^(-2)`
C. `6pi rad s^(-2)`
D. `8pi rad s^(-2)`

Explanation

To find the angular acceleration of the motor wheel, we can follow these steps:

### Step 1: Identify the given values
- Initial angular speed, \( \omega_0 = 1200 \, \text{rpm} \)
- Final angular speed, \( \omega = 3120 \, \text{rpm} \)
- Time, \( t = 16 \, \text{s} \)

### Step 2: Convert angular speeds from rpm to rad/s
To convert from revolutions per minute (rpm) to radians per second (rad/s), we use the conversion factor:
\[ 1 \, \text{rpm} = \frac{2\pi \, \text{rad}}{60 \, \text{s}} \]
Thus, we convert \( \omega_0 \) and \( \omega \):
\[ \omega_0 = 1200 \, \text{rpm} \times \frac{2\pi \, \text{rad}}{60 \, \text{s}} = 1200 \times \frac{2\pi}{60} = 40\pi \, \text{rad/s} \]
\[ \omega = 3120 \, \text{rpm} \times \frac{2\pi \, \text{rad}}{60 \, \text{s}} = 3120 \times \frac{2\pi}{60} = 104\pi \, \text{rad/s} \]

### Step 3: Use the formula for angular acceleration
The formula for angular acceleration \( \alpha \) is given by:
\[ \alpha = \frac{\Delta \omega}{t} = \frac{\omega - \omega_0}{t} \]
Substituting the values we have:
\[ \alpha = \frac{104\pi - 40\pi}{16} = \frac{64\pi}{16} = 4\pi \, \text{rad/s}^2 \]

### Step 4: Conclusion
The angular acceleration of the motor wheel is:
\[ \alpha = 4\pi \, \text{rad/s}^2 \]
Thus, the correct answer is **(b) \( 4\pi \, \text{rads}^{-2} \)**.

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