A particle is moving along a circular 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, we need to find the average acceleration of a particle moving in a circular path after it completes half a revolution. Let's break down the solution step by step.

### Step 1: Understand the Motion
The particle is moving in a circular path with a radius \( r = 5 \, m \) and a uniform speed of \( v = 5 \, m/s \).

### Step 2: Identify Initial and Final Velocities
- At the starting point (let's call it point A), the particle has an initial velocity \( \vec{v_i} \) directed tangentially to the circle.
- After completing half a revolution, the particle reaches point B, where its final velocity \( \vec{v_f} \) is also directed tangentially to the circle but in the opposite direction.

### Step 3: Calculate the Change in Velocity
The change in velocity \( \Delta \vec{V} \) can be calculated as:
\[ \Delta \vec{V} = \vec{v_f} - \vec{v_i} \]
Since the magnitudes of both velocities are the same (5 m/s), but their directions are opposite, we can represent them as:
- \( \vec{v_i} = 5 \, m/s \) (in the positive direction)
- \( \vec{v_f} = -5 \, m/s \) (in the negative direction)

Thus,
\[ \Delta \vec{V} = -5 \, m/s - 5 \, m/s = -10 \, m/s \]

### Step 4: Calculate the Time Taken for Half Revolution
The time taken to complete half a revolution can be calculated using the formula for the period of circular motion:
\[ T = \frac{2\pi r}{v} \]
Substituting the values:
\[ T = \frac{2\pi \times 5 \, m}{5 \, m/s} = 2\pi \, s \]
The time for half a revolution is:
\[ \Delta t = \frac{T}{2} = \frac{2\pi}{2} = \pi \, s \]

### Step 5: Calculate Average Acceleration
Average acceleration \( \vec{A}_{avg} \) is given by:
\[ \vec{A}_{avg} = \frac{\Delta \vec{V}}{\Delta t} \]
Substituting the values we found:
\[ \vec{A}_{avg} = \frac{-10 \, m/s}{\pi \, s} = -\frac{10}{\pi} \, m/s^2 \]

### Final Answer
The average acceleration when the particle completes half a revolution is:
\[ \vec{A}_{avg} = -\frac{10}{\pi} \, m/s^2 \]

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