Question

A particle of mass \(m\) is rotating in a circle of radius \(r\) having angular momentum \(L\). Then the centripetal force will be

• A. \( \dfrac{L^2}{mr} \)
• B. \( \dfrac{L^2 m}{r} \)
• C. \( \dfrac{L^2}{mr^3} \)
• D. \( \dfrac{L^2}{mr^2} \)

Hint:

Always express velocity in terms of angular momentum when force is asked in terms of \(L\).

Answer 1

The Correct Option is C

Step 1: Write the expression for angular momentum.
For circular motion, angular momentum is given by:
\[ L = mvr \]
Step 2: Express velocity in terms of angular momentum.
\[ v = \frac{L}{mr} \]
Step 3: Use centripetal force formula.
Centripetal force is:
\[ F = \frac{mv^2}{r} \]
Step 4: Substitute the value of velocity.
\[ F = \frac{m}{r} \left(\frac{L}{mr}\right)^2 \] \[ F = \frac{L^2}{mr^3} \]
Step 5: Conclusion.
The centripetal force is given by \( \dfrac{L^2}{mr^3} \), hence option (C) is correct.