A particle moves with constant speed v along a circular path of radius r and completes the circle in time T. The acceleration of the particle is

Options

A. `(2piv)/(T)`
B. `(2pir)/(T)`
C. `(2pir^(2))/(T)`
D. `(2piv^(2))/(T)`

Explanation

To solve the problem, we need to find the acceleration of a particle moving with constant speed \( v \) along a circular path of radius \( r \) and completing the circle in time \( T \).

### Step-by-Step Solution:

**Step 1: Understand the type of acceleration in circular motion.**
- In circular motion, when a particle moves along a circular path at a constant speed, it experiences centripetal acceleration directed towards the center of the circle.

**Step 2: Write the formula for centripetal acceleration.**
- The formula for centripetal acceleration \( a \) is given by:
\[ a = \frac{v^2}{r} \]
where \( v \) is the constant speed of the particle, and \( r \) is the radius of the circular path.

**Step 3: Relate speed to the time period.**
- The speed \( v \) can also be expressed in terms of the time period \( T \) and the radius \( r \). The relationship is:
\[ v = \frac{2\pi r}{T} \]
This equation comes from the fact that the distance traveled in one complete revolution (the circumference of the circle) is \( 2\pi r \), and it takes time \( T \) to complete this distance.

**Step 4: Substitute the expression for speed into the acceleration formula.**
- Now, substituting \( v = \frac{2\pi r}{T} \) into the centripetal acceleration formula:
\[ a = \frac{(\frac{2\pi r}{T})^2}{r} \]

**Step 5: Simplify the expression.**
- Simplifying the equation:
\[ a = \frac{4\pi^2 r^2}{T^2} \cdot \frac{1}{r} = \frac{4\pi^2 r}{T^2} \]

**Step 6: Final result.**
- Therefore, the acceleration of the particle is:
\[ a = \frac{4\pi^2 r}{T^2} \]

### Conclusion:
The acceleration of the particle moving in a circular path with constant speed \( v \) is \( \frac{4\pi^2 r}{T^2} \).