A particle with radius R is moving in a circular path with constant speed. The time period of the particle is T.

Calculate the tiime for the following after t = T/6

What is the average velocity of the particle?

The correct option is 6R/T.

Step-by-step Explanation:

1. Understand the circular motion parameters:
Let a particle move in a circular path of radius R with a constant speed. The time period (time taken to complete one full revolution of 2π radians) is T.
The angular velocity ($\omega$) of the particle is given by:
$$\omega =\frac{2\pi }{T}$$

2. Calculate the angular displacement:
We need to find the average velocity of the particle during the time interval from t = 0 to t = T/6. The time duration is:
$$\Delta t=\frac{T}{6}$$
The angular displacement ($\theta$) swept by the particle in this time interval is:
$$\theta =\omega \cdot \Delta t=\left(\frac{2\pi }{T}\right)\cdot \left(\frac{T}{6}\right)=\frac{\pi }{3}\text{radians}$$
Since $\frac{\pi }{3}$ radians is equal to 60°, the particle rotates through an angle of 60° around the center of the circular path.

3. Calculate the linear displacement:
The straight-line distance (displacement, d) between the initial and final positions of a particle moving along a circle of radius R through an angle $\theta$ is given by the formula:
$$d=2R\sin \left(\frac{\theta }{2}\right)$$
Substituting $\theta ={60}^{\circ }$ into the equation:
$$d=2R\sin \left({30}^{\circ }\right)$$
Since $\sin \left({30}^{\circ }\right)=\frac{1}{2}$:
$$d=2R\cdot \frac{1}{2}=R$$
Thus, the magnitude of the displacement is equal to the radius R.

4. Calculate the average velocity:
Average velocity is defined as the total displacement divided by the total time interval:
$$v_{\text{avg}}=\frac{\text{Displacement (d)}}{\text{Time interval (}}$$
Substituting the values of displacement and time:
$$v_{\text{avg}}=\frac{R}{T/6}=\frac{6R}{T}$$
Therefore, the average velocity of the particle is 6R/T.