A particle with the constant speed in a circle of radius r and time period T.The centripetal acceleration of a particle is

Options

A. `‘r^2‘ T`
B. `‘T^2‘ r`
C. `(2π/T)^2r`
D. `(2πT)^2/r`

Answer

To find the centripetal acceleration of a particle moving in a circle with a constant speed, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Centripetal Acceleration**:
Centripetal acceleration (\(a_c\)) is the acceleration that keeps a particle moving in a circular path and is directed towards the center of the circle. The formula for centripetal acceleration is given by:
\[ a_c = \frac{v^2}{r} \]
where \(v\) is the linear speed of the particle and \(r\) is the radius of the circular path.

2. **Relating Linear Speed to Angular Velocity**:
The linear speed \(v\) can also be expressed in terms of angular velocity (\(\omega\)) as:
\[ v = r \omega \]
where \(\omega\) is the angular velocity in radians per second.

3. **Substituting for Linear Speed**:
Substituting \(v = r \omega\) into the centripetal acceleration formula gives:
\[ a_c = \frac{(r \omega)^2}{r} = \frac{r^2 \omega^2}{r} = r \omega^2 \]

4. **Relating Angular Velocity to Time Period**:
The angular velocity \(\omega\) is related to the time period \(T\) (the time taken for one complete revolution) by the formula:
\[ \omega = \frac{2\pi}{T} \]

5. **Substituting Angular Velocity into Centripetal Acceleration**:
Now, substituting \(\omega = \frac{2\pi}{T}\) into the centripetal acceleration formula:
\[ a_c = r \left(\frac{2\pi}{T}\right)^2 \]

6. **Simplifying the Expression**:
Simplifying the expression gives:
\[ a_c = r \cdot \frac{4\pi^2}{T^2} = \frac{4\pi^2 r}{T^2} \]

### Final Result:
Thus, the centripetal acceleration of the particle is:
\[ a_c = \frac{4\pi^2 r}{T^2} \]