Question

A man carrying a monkey on his shoulder does cycling smoothly on a circular track of radius \( 9 \, \text{m} \) and completes \( 120 \) revolutions in \( 3 \) minutes. The magnitude of centripetal acceleration of the monkey is (in \( \text{m/s}^2 \)):

• A. zero
• B. \( 16 \pi^2 \, \text{m/s}^2 \)
• C. \( 4 \pi^2 \, \text{m/s}^2 \)
• D. \( 57600 \pi^2 \, \text{m/s}^2 \)

Answer 1

The Correct Option is B

Given:
- Radius of the circular track: \( R = 9 \, \text{m} \)
- Number of revolutions completed: 120 revolutions
- Time taken: 3 minutes

Step 1: Calculate the Angular Velocity
The angular velocity \( \omega \) is given by:

\[ \omega = \frac{\text{Total revolutions}}{\text{Time taken}} \times 2\pi \, \text{rad/s}. \]

Substituting the given values:

\[ \omega = \frac{120 \, \text{revolutions}}{3 \, \text{minutes}} \times \frac{2\pi \, \text{rad}}{1 \, \text{revolution}}. \]

Converting time to seconds:

\[ \omega = \frac{120 \times 2\pi}{3 \times 60} \, \text{rad/s} = \frac{4\pi}{3} \, \text{rad/s}. \]

Step 2: Calculate the Centripetal Acceleration
The centripetal acceleration \( a_{\text{centripetal}} \) is given by:

\[ a_{\text{centripetal}} = \omega^2 R. \]

Substituting the values of \( \omega \) and \( R \):

\[ a_{\text{centripetal}} = \left(\frac{4\pi}{3}\right)^2 \times 9. \]

Simplifying:

\[ a_{\text{centripetal}} = \frac{16\pi^2}{9} \times 9 = 16\pi^2 \, \text{m/s}^2. \]

Therefore, the magnitude of the centripetal acceleration of the monkey is \( 16\pi^2 \, \text{m/s}^2 \).

Answer 2

Step 1: Given data.
Radius of circular track, r = 9 m
Number of revolutions = 120
Time taken = 3 minutes = 180 seconds

Step 2: Calculate angular velocity (ω).
Angular velocity is given by:
\[ \omega = \frac{\text{Total angle covered}}{\text{Time}} = \frac{2\pi \times \text{Number of revolutions}}{\text{Time}} \]
\[ \omega = \frac{2\pi \times 120}{180} = \frac{240\pi}{180} = \frac{4\pi}{3} \, \text{rad/s} \]

Step 3: Calculate centripetal acceleration (a_c).
Centripetal acceleration is given by:
\[ a_c = \omega^2 r \]
Substitute the values:
\[ a_c = \left(\frac{4\pi}{3}\right)^2 \times 9 = \frac{16\pi^2}{9} \times 9 = 16\pi^2 \]

Step 4: Final result.
The magnitude of centripetal acceleration of the monkey is:
\[ \boxed{16\pi^2 \, \text{m/s}^2} \]

Final Answer: \( 16\pi^2 \, \text{m/s}^2 \)