JEE PYQ: Gravitation Question 37
Question 37 - 2019 (09 Apr Shift 2)
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 $T$:
(1) $T/R$ is a constant
(2) $T^2/R^3$ is a constant
(3) $T/R^2$ is a constant
(4) $TR$ is a constant
Show Answer
Answer: (1)
Solution
$M(R) = \int_0^R \frac{K}{r^2} 4\pi r^2 dr = 4\pi KR$. $\frac{GMm}{R^2} = \frac{mv^2}{R}$: $v^2 = \frac{4\pi GKR}{R} = 4\pi GK$. $v = $ const. $T = \frac{2\pi R}{v} \propto R$. So $T/R$ is constant.
BETA
