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)`
C. `(L^(2))/(mr^(2))`
D. `(L^(2))/(mr^(3))`

Answer

To find the centripetal force acting on a particle of mass `m` rotating in a circular path of radius `r` with angular momentum `L`, 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:
\[ L = mvr \]
where:
- \( m \) is the mass of the particle,
- \( v \) is the linear velocity,
- \( r \) is the radius of the circular path.

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

### 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:
\[ F_c = \frac{mv^2}{r} \]

### Step 4: Substitute the expression for linear velocity into the centripetal force formula
Now, we will substitute the expression for \( v \) from Step 2 into the centripetal force formula:
\[ F_c = \frac{m\left(\frac{L}{mr}\right)^2}{r} \]

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

### Conclusion
Thus, the centripetal force acting on the particle is:
\[ F_c = \frac{L^2}{mr^3} \]

### Final Answer
The centripetal force acting on the particle is \( \frac{L^2}{mr^3} \).

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