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Question:

A particle is moving with a uniform speed in a circular orbit of radius R in a central force inversely proportional to the nth power of R. If the period of rotation of the particle is T, then what is the relationship between T, R, and n?

T∝Rn2+1

T∝R(n+1)/2

T∝Rn/2

T∝R3/2 for any n

Solution:

The correct option is A: T∝R(n+1)/2
Central force = Fc = mv²/R = k/Rn
mv² = k/Rn
v = √(k/mRⁿ)
Now period of rotation
T = 2πR/v
T ∝ R/√(1/Rⁿ)
T ∝ R^(n+1)/2
Hence option A is the correct one.