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 \( \text{m/s}^2 \)):
- zero
- \( 16 \pi^2 \, \text{m/s}^2 \)
- \( 4 \pi^2 \, \text{m/s}^2 \)
- \( 57600 \pi^2 \, \text{m/s}^2 \)
The Correct Option is B
Approach Solution - 1
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 \( 16\pi^2 \, \text{m/s}^2 \).
Approach Solution -2
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 (ac).
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: \( 16\pi^2 \, \text{m/s}^2 \)
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