Question

The angular speed of motor wheel is increased from 1200 rpm to 3120 rpm in 16 sec. The angular acceleration of the motor wheel is 

• A. 6π rad/s2
• B. 2π rad/s²
• C. 8π rad/s²
• D. 4π rad/s²

Answer 1

The Correct Option is D

Angular acceleration \( \alpha = \frac{\omega_f - \omega_i}{t} \)

Given: 

• Initial angular velocity \( \omega_i = 1200 \, \text{rpm} \)
• Final angular velocity \( \omega_f = 3120 \, \text{rpm} \)
• Time taken \( t = 16 \, \text{seconds} \)

Step 1: Convert the angular velocities from rpm to radians per second (rad/s):

We know that:

• 1 revolution = \( 2\pi \) radians
• 1 minute = 60 seconds

For the initial angular velocity \( \omega_i \):

\[ \omega_i = 1200 \, \text{rpm} \times \frac{2\pi \, \text{radians}}{1 \, \text{revolution}} \times \frac{1 \, \text{minute}}{60 \, \text{seconds}} = 40\pi \, \text{rad/s} \]

For the final angular velocity \( \omega_f \):

\[ \omega_f = 3120 \, \text{rpm} \times \frac{2\pi \, \text{radians}}{1 \, \text{revolution}} \times \frac{1 \, \text{minute}}{60 \, \text{seconds}} = 104\pi \, \text{rad/s} \]

Step 2: Calculate the angular acceleration \( \alpha \):

\[ \alpha = \frac{\omega_f - \omega_i}{t} = \frac{104\pi - 40\pi}{16} = \frac{64\pi}{16} = 4\pi \, \text{rad/s}^2 \]

Therefore, the angular acceleration of the motor wheel is: \( 4\pi \, \text{rad/s}^2 \), which corresponds to option (D).

Answer 2

We shall use the equation:

\( \omega = \omega_0 + \alpha t \) 

Where \( \omega_0 \) is the initial angular speed in radians per second:

\( \omega_0 = 2\pi \times \text{Angular speed in revs}^{-1} = \frac{60}{\text{s/min}} \times 2\pi \times \text{Angular speed in rev/min} = \frac{60}{2\pi \times 1200} = 40\pi \, \text{rads}^{-1} \)

Similarly, \( \omega \) is the final angular speed in radians per second:

\( \omega = \frac{60}{2\pi} \times 3120 = 2\pi \times 52 = 104\pi \, \text{rads}^{-1} \)

To calculate the angular acceleration, we use:

\( \alpha = \frac{\omega - \omega_0}{t} = \frac{104\pi - 40\pi}{16} = 4\pi \, \text{rads}^{-2} \)

The angular acceleration of the motor is \( 4\pi \, \text{rads}^{-2} \).