The linear velocity of a body , moving on the circumference of a circle of radius r, equal to the velocity acquired by a freely falling body in covering a distance to half the radius of the . Then the centripetal acceleration of the body is

Options

A. `(g)/(4)`
B. `(g)/(2)`
C. `(g)/(3)`
D. `g`

Explanation

To solve the problem, we need to find the centripetal acceleration of a body moving on the circumference of a circle of radius \( r \). The linear velocity of the body is equal to the velocity acquired by a freely falling body that covers a distance equal to half the radius of the circle.

### Step-by-Step Solution:

1. **Identify the distance fallen**:
The distance fallen by the freely falling body is given as \( \frac{r}{2} \), where \( r \) is the radius of the circle.

2. **Use the equation of motion**:
The equation of motion for a freely falling body is given by:
\[ V^2 = U^2 + 2AS \]
where \( V \) is the final velocity, \( U \) is the initial velocity (which is 0 for a freely falling body), \( A \) is the acceleration due to gravity \( g \), and \( S \) is the distance fallen.

3. **Substituting values**:
Since the initial velocity \( U = 0 \), the equation simplifies to:
\[ V^2 = 0 + 2g\left(\frac{r}{2}\right) \]
Simplifying this gives:
\[ V^2 = g r \]

4. **Finding linear velocity**:
Taking the square root of both sides, we find the linear velocity \( V \):
\[ V = \sqrt{g r} \]

5. **Centripetal acceleration formula**:
The centripetal acceleration \( a_c \) for an object moving in a circle is given by:
\[ a_c = \frac{V^2}{r} \]

6. **Substituting the value of \( V \)**:
Now substituting \( V^2 = g r \) into the centripetal acceleration formula:
\[ a_c = \frac{g r}{r} = g \]

7. **Final answer**:
Therefore, the centripetal acceleration of the body is:
\[ a_c = g \, \text{m/s}^2 \]