Satellites - Orbital Velocity & Period question from JEE Main 2020 (2 Sep, Shift 1)

Answer: (a)

1. Mass inside radius $r$: $M(r) = \int_0^r \frac{K}{r} 4\pi r^2 dr = 2\pi K r^2$.
2. Field at $R$: $g = \frac{GM(R)}{R^2} = 2\pi G K$, a constant.
3. $\frac{v^2}{R} = 2\pi G K \Rightarrow v \propto \sqrt{R}$.
4. $T = \frac{2\pi R}{v} \propto \sqrt{R} \Rightarrow T^2 \propto R$.
5. So the answer is (a) $T^2 \propto R$.