A particle of mass `m` is rotating in a plane in circular path of radius `r`. Its angular momentum is `L`. The centripetal force acting on the particle is

Options

A. `L^(2)//mr`
B. `L^(2)m//r^(2)`
C. `L^(2)//m^(2)r^(2)`
D. `L^(2)//mr^(3)`

Explanation

To find the centripetal force acting on a particle of mass `m` rotating in a circular path of radius `r` in terms of its angular momentum `L`, we can follow these steps:

### Step-by-Step Solution:

1. **Define Angular Momentum (L)**:
The angular momentum `L` of a particle moving in a circular path is given by the formula:
\[ L = m \cdot v \cdot r \]
where `m` is the mass of the particle, `v` is its linear velocity, and `r` is the radius of the circular path.

2. **Centripetal Force (F)**:
The centripetal force required to keep the particle moving in a circular path is given by:
\[ F = \frac{m \cdot v^2}{r} \]

3. **Express Velocity (v) in terms of Angular Momentum (L)**:
From the angular momentum equation, we can express `v` as:
\[ v = \frac{L}{m \cdot r} \]

4. **Substitute v in the Centripetal Force Formula**:
Now, substitute the expression for `v` into the centripetal force equation:
\[ F = \frac{m \cdot \left(\frac{L}{m \cdot r}\right)^2}{r} \]

5. **Simplify the Expression**:
Simplifying the equation:
\[ F = \frac{m \cdot \frac{L^2}{m^2 \cdot r^2}}{r} \]
\[ F = \frac{L^2}{m \cdot r^3} \]

6. **Final Expression for Centripetal Force**:
Thus, the centripetal force acting on the particle in terms of its angular momentum `L` is:
\[ F = \frac{L^2}{m \cdot r^3} \]

### Final Answer:
The centripetal force acting on the particle is given by:
\[ F = \frac{L^2}{m \cdot r^3} \]