Centripetal Forces

To solve the problem, consider the forces and time periods involved in circular motion under combined potentials.

1. Initial circular orbit under gravitational force alone:
Gravitational force: Time period of orbit: Angular velocity:

2. Additional central force from potential:
Given potential: Force from this potential (radial inward, since ):

3. Total radial force for circular motion at :
This force provides the centripetal force for circular motion:

4. Relate new time period to :
Square both sides: Similarly,

5. Substitute :
Rearranged: Divide both sides by :

6. Express :

Using , substitute:

Final Answer: