A particle is moving with uniform speed along the circumference of a circle of radius R under the action of a central fictitious force F which is inversely proportional to R³. Its time period of revolution will be given by :

Solution

Physics sir · 3 steps

Let the given force supply the centripetal force, find how v depends on R, then build T from it.

1. Force provides centripetal pull. For uniform circular motion the net inward force must equal mv²/R. Here that force is F = k/R³, where k is a constant.

   mv²/R = k/R³ ⇒ v² = k/mR² ⇒ v ∝ 1/R

2. Time for one round. One revolution covers 2π R, so T = (2π R)/v. With v ∝ R⁻¹, this gives T ∝ R × R = R².
3. So. T ∝ R², which is option (d).

The idea to keep. If the central force goes as F ∝ R⁻ⁿ, equate it to mv²/R to get v ∝ R^((1-n)/2), and then T = (2π R)/v ∝ R^((n+1)/2).

Answer from the NTA final answer key. Solution by PrepWiser.