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  4. A test particle is moving in a circular orbit in the gravitational field produced by a mass density ρ(r) = (K/r2). Identify the correct relation between the radius R of the particle's orbit and its period T :

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Q. A test particle is moving in a circular orbit in the gravitational field produced by a mass density $\rho(r) = \frac{K}{r^2}$. Identify the correct relation between the radius $R$ of the particle's orbit and its period $T$ :

JEE MainJEE Main 2019Gravitation

Solution:

$ m = \int^{R}_{0} \rho4\pi r^{2}dr $
$ m = 4\pi KR$
$ v \propto \sqrt{4\pi K} $
$ \frac{T}{R} = \frac{2\pi}{\sqrt{4\pi K}}$