The frequency of a particle performing circular motion changes from 60 rpm to 180 rpm in 20 s . Then the angular acceleration is

Options

A. `0.1 pi rad //s^(2)`
B. `0.2 pi rad //s^(2)`
C. `0.3 pi rad //s^(2)`
D. `0.4 pi rad //s^(2)`

Explanation

To find the angular acceleration of a particle performing circular motion when its frequency changes from 60 rpm to 180 rpm in 20 seconds, we can follow these steps:

### Step 1: Convert frequencies from rpm to rad/s
The angular velocity (ω) in radians per second can be calculated from the frequency in revolutions per minute (rpm) using the formula:
\[ \omega = 2\pi \times \left(\frac{\text{frequency in rpm}}{60}\right) \]

For the initial frequency (60 rpm):
\[ \omega_1 = 2\pi \times \left(\frac{60}{60}\right) = 2\pi \, \text{rad/s} \]

For the final frequency (180 rpm):
\[ \omega_2 = 2\pi \times \left(\frac{180}{60}\right) = 6\pi \, \text{rad/s} \]

### Step 2: Calculate the change in angular velocity
The change in angular velocity (Δω) is given by:
\[ \Delta \omega = \omega_2 - \omega_1 \]
Substituting the values:
\[ \Delta \omega = 6\pi - 2\pi = 4\pi \, \text{rad/s} \]

### Step 3: Calculate the angular acceleration
Angular acceleration (α) is defined as the change in angular velocity over time:
\[ \alpha = \frac{\Delta \omega}{\Delta t} \]
Given that the time interval (Δt) is 20 seconds:
\[ \alpha = \frac{4\pi}{20} = \frac{\pi}{5} \, \text{rad/s}^2 \]

### Final Answer
The angular acceleration is:
\[ \alpha = \frac{\pi}{5} \, \text{rad/s}^2 \]

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