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

Options

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

Explanation

To find the centripetal force acting on a particle of mass `m` moving in a circular path of radius `r`, we can follow these steps:

### Step 1: Understand the relationship between angular momentum and linear velocity.
The angular momentum \( L \) of a particle moving in a circular path is given by the formula:
\[ L = m \cdot v \cdot r \]
where \( v \) is the linear velocity of the particle.

### Step 2: Solve for linear velocity \( v \).
From the equation for angular momentum, we can express the linear velocity \( v \) as:
\[ v = \frac{L}{m \cdot r} \]

### Step 3: Write the formula for centripetal force.
The centripetal force \( F_c \) acting on a particle moving in a circular path is given by:
\[ F_c = \frac{m \cdot v^2}{r} \]

### Step 4: Substitute the expression for \( v \) into the centripetal force formula.
Now, substituting the expression for \( v \) from Step 2 into the centripetal force formula:
\[ F_c = \frac{m \cdot \left(\frac{L}{m \cdot r}\right)^2}{r} \]

### Step 5: Simplify the expression.
Now we simplify the expression:
\[ F_c = \frac{m \cdot \frac{L^2}{m^2 \cdot r^2}}{r} \]
\[ F_c = \frac{L^2}{m \cdot r^3} \]

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