A body revolves with constant speed v in a circular path of radius r. the magnitude of its average acceleration during motion between two points in diametrically opposite direction is

Options

A. Zero
B. `(v^(2))/(r )`
C. `(2v^(2))/(pi r)`
D. `(v^(2))/(2r)`

Answer

To find the magnitude of the average acceleration of a body moving with constant speed \( v \) in a circular path of radius \( r \) between two diametrically opposite points, we can follow these steps:

### Step 1: Understand the motion
The body is moving in a circular path with a constant speed \( v \). We need to analyze the motion between two points that are diametrically opposite on the circle.

### Step 2: Identify the points
Let’s denote the two points as Point 1 and Point 2, which are diametrically opposite to each other. The body moves from Point 1 to Point 2, passing through the center of the circle.

### Step 3: Determine the time taken
The distance traveled from Point 1 to Point 2 is half the circumference of the circle. The circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
Thus, the distance for half the circle (from Point 1 to Point 2) is:
\[ \text{Distance} = \frac{C}{2} = \pi r \]
The time \( t \) taken to travel this distance at constant speed \( v \) is:
\[ t = \frac{\text{Distance}}{v} = \frac{\pi r}{v} \]

### Step 4: Determine the change in velocity
At Point 1, the velocity \( \vec{v}_1 \) can be considered in the horizontal direction (let's say to the right), and at Point 2, the velocity \( \vec{v}_2 \) will be in the opposite direction (to the left). Therefore, we can express these velocities as:
\[ \vec{v}_1 = v \quad \text{(to the right)} \]
\[ \vec{v}_2 = -v \quad \text{(to the left)} \]
The change in velocity \( \Delta \vec{v} \) is given by:
\[ \Delta \vec{v} = \vec{v}_2 - \vec{v}_1 = -v - v = -2v \]

### Step 5: Calculate the average acceleration
Average acceleration \( \vec{a}_{\text{avg}} \) is defined as the change in velocity divided by the time taken:
\[ \vec{a}_{\text{avg}} = \frac{\Delta \vec{v}}{t} = \frac{-2v}{\frac{\pi r}{v}} = \frac{-2v^2}{\pi r} \]
The magnitude of the average acceleration is:
\[ |\vec{a}_{\text{avg}}| = \frac{2v^2}{\pi r} \]

### Final Answer
The magnitude of the average acceleration during the motion between two points in diametrically opposite directions is:
\[ |\vec{a}_{\text{avg}}| = \frac{2v^2}{\pi r} \]