Question

If the angular momentum of a particle of mass $m$ rotating along a circular path of radius $r$ with uniform speed is L, the centripetal force acting on the particle is

• A. $\frac{L^2}{mr^3}$
• B. $\frac{L^2}{mr}$
• C. $\frac{L}{mr^2}$
• D. $\frac{L^2m}{r}$
• E. $\frac{Lm}{r^2}$

Hint:

This formula is very common in planetary motion and atomic physics (Bohr's model). Memorizing the relationship $F_c = \frac{L^2}{mr^3}$ can save valuable time in entrance exams!

Answer 1

The Correct Option is A

Concept: This question requires linking Circular Motion and Angular Momentum.
• Centripetal Force ($F_c$): The inward force required for circular motion: $F_c = \frac{mv^2}{r}$.
• Angular Momentum ($L$): For a particle in circular motion: $L = mvr$.

Step 1: Express velocity ($v$) in terms of angular momentum ($L$). From $L = mvr$, we can solve for $v$: \[ v = \frac{L}{mr} \]

Step 2: Substitute $v$ into the centripetal force formula. \[ F_c = \frac{m \cdot (\frac{L}{mr})^2}{r} \] \[ F_c = \frac{m \cdot \frac{L^2}{m^2r^2}}{r} \] \[ F_c = \frac{L^2}{m r^2 \cdot r} = \frac{L^2}{m r^3} \]