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

Provided:
- Circular track radius: \( R = 9 \, \text{m} \)
- Revolutions completed: 120
- Time taken: 3 minutes

Step 1: Determine Angular Velocity
Angular velocity \( \omega \) is calculated as:

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

Using given values:

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

Convert 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 Centripetal Acceleration
Centripetal acceleration \( a_{\text{centripetal}} \) is given by:

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

Substitute \( \omega \) and \( R \) values:

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

Simplify:

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

The centripetal acceleration magnitude is \( 16\pi^2 \, \text{m/s}^2 \).