Question

Suppose the gravitational force varies inversely as the nth power of distance the time period of planet in circular orbit of radius R around the sun will be proportional to 

(A) R^{[(n+1) / 2]} 

(B) R^{[(n – 1)/2]} 

(C) Rⁿ 

(D) R^{[(n – 1)/2]}

Answer 1

Correct option: (A) R^{[(n+1) / 2]}  

Explanation:

Gravitational force = F

distance = R

Given :  F ∝ (1 / Rⁿ)

for circular orbit, F = [(mV²) / R]

as V = Rw

hence F = mRw² 

i.e.    mRw² ∝ [1 / (Rⁿ)]

i.e.  mw² ∝ [1 / (Rⁿ⁺¹)]

as w = (2π / T),

m(2π/T)² ∝ [1 / (Rⁿ⁺¹)]

i.e.   (1 / T²) ∝[1 / (Rⁿ⁺¹)]

i.e. T² ∝ Rⁿ⁺¹       

hence T ∝ R^[(n+1)/2]