A paricle is describing the circular path of radius 20 m in every 2 s. The average angular speed of the particle during 4 s is

Options

A. 20 `pi`rad`s^-1`
B. 4 `pi`rad`s^-1`
C. `pi`rad`s^-1`
D. 2`pi`rad`s^-1`

Answer

To find the average angular speed of a particle moving in a circular path, we can follow these steps:

### Step 1: Understand the Problem
The particle is moving in a circular path with a radius of 20 m and completes one full revolution in 2 seconds. We need to find the average angular speed over a time period of 4 seconds.

### Step 2: Determine the Angle Rotated
In one complete revolution, the angle rotated is \(2\pi\) radians. Since the particle completes one revolution in 2 seconds, in 4 seconds, it will complete:
\[ \text{Number of revolutions in 4 seconds} = \frac{4 \text{ seconds}}{2 \text{ seconds/revolution}} = 2 \text{ revolutions} \]
Thus, the total angle rotated in 4 seconds is:
\[ \text{Total angle} = 2 \text{ revolutions} \times 2\pi \text{ radians/revolution} = 4\pi \text{ radians} \]

### Step 3: Calculate the Average Angular Speed
The average angular speed (\(\omega_{\text{avg}}\)) is defined as the total angle rotated divided by the time taken. Therefore, we can calculate it as follows:
\[ \omega_{\text{avg}} = \frac{\text{Total angle}}{\text{Time}} = \frac{4\pi \text{ radians}}{4 \text{ seconds}} = \pi \text{ radians/second} \]

### Step 4: Final Answer
The average angular speed of the particle during 4 seconds is:
\[ \omega_{\text{avg}} = \pi \text{ radians/second} \]

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