JEE PYQ: Gravitation Question 19
Question 19 - 2020 (02 Sep Shift 1)
The mass density of a spherical galaxy varies as $\frac{K}{r}$ over a large distance $r$ from its centre. In that region, a small star is in a circular orbit of radius $R$. Then the period of revolution, $T$ depends on $R$ as:
(1) $T^2 \propto R$
(2) $T^2 \propto R^3$
(3) $T^2 \propto \frac{1}{R^3}$
(4) $T \propto R$
Show Answer
Answer: (1)
Solution
Mass enclosed $M(R) = \int_0^R \frac{K}{r} 4\pi r^2 dr = 4\pi K \int_0^R r,dr = 2\pi K R^2$. $\frac{GMm}{R^2} = \frac{mv^2}{R} \Rightarrow v^2 = \frac{2\pi KGR}{1}$. $T = \frac{2\pi R}{v} \propto \sqrt{R}$. So $T^2 \propto R$.
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