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A particle with mass m is influenced by the gravitational force of a much larger mass M (M >> m). The particle orbits in a circular path of radius r₀ with a time period T₀. An additional central force is introduced, arising from the potential energy V(r) = αmα/r³, where α is a positive constant with appropriate units and r is the orbital radius. If the particle continues to move in the same circular orbit of radius r₀ under the combined gravitational and additional potential, but its time period changes to T₁, what is the value of (T₁² - T₀²)/T₀²? [G represents the gravitational constant.]
- 3α/(GMr₀²)
- α/(2GMr₀²)
- α/(GMr₀²)
- 2α/(GMr₀²)
Correct answer: 3α/(GMr₀²)
Solution
The introduction of an additional central force affects the time period of the particle's orbit, and by analyzing the combined gravitational and additional potential, we can determine the change in the time period and find the value of (T₁² - T₀²)/T₀².
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