A particle is moving along a circular path of radius 5 m with a uniform speed `5ms^(-1)`. What will be the average acceleration when the particle completes half revolution?

Options

A. Zero
B. `10ms^(-2)`
C. `10pims^(-2)`
D. `(10)/(pi)ms^(-2)`

Explanation

To solve the problem of finding the average acceleration of a particle moving along a circular path of radius 5 m with a uniform speed of 5 m/s after completing half a revolution, we can follow these steps:

### Step 1: Understand the Motion
The particle is moving in a circular path with a radius of 5 m and a uniform speed of 5 m/s. We need to analyze the motion after the particle completes half a revolution.

### Step 2: Determine Initial and Final Velocity
- **Initial Velocity (v_initial)**: At the starting point of the half revolution, the particle's velocity is directed tangentially to the circle. Let's assume it starts at point A (0 degrees) and moves to point B (180 degrees). The initial velocity vector can be represented as:
\[ \vec{v}_{initial} = 5 \, \text{m/s} \, \hat{i} \]
- **Final Velocity (v_final)**: After half a revolution, the particle reaches point B. The velocity at this point is also directed tangentially to the circle but in the opposite direction. Thus, it can be represented as:
\[ \vec{v}_{final} = -5 \, \text{m/s} \, \hat{i} \]

### Step 3: Calculate Change in Velocity
The change in velocity (\(\Delta \vec{v}\)) is given by:
\[ \Delta \vec{v} = \vec{v}_{final} - \vec{v}_{initial} \]
Substituting the values:
\[ \Delta \vec{v} = (-5 \, \hat{i}) - (5 \, \hat{i}) = -10 \, \hat{i} \, \text{m/s} \]

### Step 4: Determine the Time Taken for Half Revolution
The time taken for one complete revolution (T) can be calculated using the formula:
\[ T = \frac{2\pi r}{v} \]
Substituting the values:
\[ T = \frac{2\pi \times 5 \, \text{m}}{5 \, \text{m/s}} = 2\pi \, \text{s} \]
For half a revolution, the time taken (\(t\)) is:
\[ t = \frac{T}{2} = \frac{2\pi}{2} = \pi \, \text{s} \]

### Step 5: Calculate Average Acceleration
The average acceleration (\(a_{avg}\)) is defined as the change in velocity divided by the time taken:
\[ a_{avg} = \frac{\Delta \vec{v}}{t} \]
Substituting the values:
\[ a_{avg} = \frac{-10 \, \hat{i} \, \text{m/s}}{\pi \, \text{s}} = -\frac{10}{\pi} \, \hat{i} \, \text{m/s}^2 \]
The magnitude of the average acceleration is:
\[ |a_{avg}| = \frac{10}{\pi} \, \text{m/s}^2 \]

### Final Answer
The average acceleration when the particle completes half a revolution is:
\[ \boxed{\frac{10}{\pi} \, \text{m/s}^2} \]