Question

What is the average velocity of the particle A moving with radius in a circular path with constant speed The time period of the particle is T Calculate the time for the following after t=T/3.

• A. 2R/T
• B. 3R/T
• C. 6R/T
• D. 4R/T

Answer 1

Answer: (D)

4R/T

Answer 2

The question asks to find the average velocity of a particle moving in a circular path with constant speed. The radius of the path is R and the time period is T. The phrase "Calculate the time for the following after t=T/3" is grammatically incorrect and confusing. Given the options are in units of velocity (R/T), it is highly probable that the question is asking for the average velocity itself, and the latter part of the sentence is either a distractor or a poorly phrased condition.

Average velocity is defined as the total displacement divided by the total time taken:

$\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}$

For a particle moving in a circular path, the most common non-zero average velocity calculation is for half a revolution.

1. Displacement: When the particle completes half a revolution, it moves from one point on the circle to the diametrically opposite point. If the particle starts at position $(R, 0)$, after half a revolution, its position will be $(-R, 0)$. The displacement vector $\Delta \vec{r}$ is the final position minus the initial position: $\Delta \vec{r} = (-R\hat{i}) - (R\hat{i}) = -2R\hat{i}$ The magnitude of the displacement is $|\Delta \vec{r}| = 2R$.

2. Time Taken: The time taken to complete half a revolution is half of the time period T, i.e., $\Delta t = T/2$.

3. Average Velocity (Magnitude):

$|\vec{v}_{avg}| = \frac{|\Delta \vec{r}|}{\Delta t} = \frac{2R}{T/2} = \frac{4R}{T}$

This result, $4R/T$, matches one of the given options. The phrase "after t=T/3" is likely irrelevant to the calculation of the magnitude of average velocity over a standard interval, as the average velocity for a fixed duration in uniform circular motion depends only on the duration and not on the starting point.