The tangential velocity of a particle making p rotations along a circle of radius `pi` in t seconds is

Options

A. `(2pip)/(t^2)`
B. `(2pip^2)/(t)`
C. `(pi^2p)/(2t)`
D. `(2pi^2p)/(t)`

Answer

To find the tangential velocity of a particle making \( P \) rotations along a circle of radius \( \pi \) in \( t \) seconds, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Radius \( R = \pi \)
- Number of rotations \( P \)
- Time \( t \) seconds

2. **Understand the Relationship Between Linear and Angular Velocity**:
- The tangential velocity \( V \) is related to angular velocity \( \omega \) by the formula:
\[ V = R \cdot \omega \]

3. **Calculate Angular Velocity \( \omega \)**:
- Angular velocity \( \omega \) is defined as the number of radians rotated per second.
- Each rotation corresponds to \( 2\pi \) radians. Therefore, for \( P \) rotations, the total radians is:
\[ \text{Total radians} = P \cdot 2\pi \]
- To find the angular velocity, we divide the total radians by the time \( t \):
\[ \omega = \frac{P \cdot 2\pi}{t} \]

4. **Substitute \( \omega \) into the Tangential Velocity Formula**:
- Now substitute \( \omega \) back into the equation for \( V \):
\[ V = R \cdot \omega = \pi \cdot \left(\frac{P \cdot 2\pi}{t}\right) \]

5. **Simplify the Expression**:
- Simplifying the above expression gives:
\[ V = \frac{2\pi^2 P}{t} \]

6. **Final Result**:
- Therefore, the tangential velocity \( V \) of the particle is:
\[ V = \frac{2\pi^2 P}{t} \]

### Conclusion:
The tangential velocity of the particle making \( P \) rotations along a circle of radius \( \pi \) in \( t \) seconds is \( \frac{2\pi^2 P}{t} \).