Question

A particle moves around a circular path of radius 'r' with uniform speed 'V'. After moving half the circle, the average acceleration of the particle is

• A. $\frac{V^{2}}{r}$
• B. $\frac{2V^{2}}{r}$
• C. $\frac{2V^{2}}{\pi r}$
• D. $\frac{V^{2}}{\pi r}$

Hint:

Physics Tip: While the instantaneous centripetal acceleration is $V^2/r$, the average acceleration over a specific interval must consider the vector change in velocity over that specific time period.

Answer 1

The Correct Option is C

Concept: Physics (Circular Motion) - Average Acceleration.

Step 1: Determine the change in velocity. At the end points of a half revolution, the magnitude of the velocity remains the same ($V$), but its direction is exactly opposite. Taking one direction as positive and the other as negative: $$\Delta V = V - (-V) = 2V \text{ }$$

Step 2: Calculate the time taken. The distance covered in half a circle is $\pi r$. Since the speed is uniform ($V$), the time taken is: $$t = \frac{\text{distance}}{\text{speed}} = \frac{\pi r}{V} \text{ }$$

Step 3: Calculate average acceleration. Average acceleration ($a$) is the change in velocity divided by the time taken: $$a = \frac{\Delta V}{t} = \frac{2V}{\frac{\pi r}{V}} \text{ }$$ $$a = \frac{2V^{2}}{\pi r} \text{ }$$ $$ \therefore \text{The average acceleration is } \frac{2V^{2}}{\pi r}. \text{ } $$