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:

(C) $T^{2}/R^{3}$ is a constant

JEE Main · 2019 · 09 Apr 2019 · Shift 2 · Gravitation

A test particle is moving in a circular orbit in the gravitational field produced by a mass density

ρ (r) = \frac{K}{r ^{2}}

. Identify the correct relation between the radius

R

of the particle's orbit and its period T:

Answer: (D)

The orbiting test mass feels only the gravity of the matter enclosed within its radius. For

ρ (r) = \frac{K}{r ^{2}}

the enclosed mass is

M (R) = \int_{0}^{R} 4 π r^{2} \cdot \frac{K}{r ^{2}} d r = 4 π K R

, which grows linearly with

R

. Setting gravity equal to the centripetal requirement,

\frac{GM ( R ) m}{R ^{2}} = \frac{m v ^{2}}{R}

, gives

v^{2} = \frac{GM ( R )}{R} = 4 π G K

, a constant independent of

R

. Since the speed is the same on every orbit, the period is

T = \frac{2 π R}{v}

, so

\frac{T}{R} = \frac{2 π}{v}

stays constant.

JEE Main · 2019 · 09 Apr 2019 · Shift 2 · Single correct option