Question

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

• A. $T \propto R$
• B. $T ^{2} \propto \frac{1}{ R ^{3}}$
• C. $T ^{2} \propto R$
• D. $T ^{2} \propto R ^{3}$

Answer 1

Answer: (C)

$dm =\rho d v$
$dm =\left(\frac{ k }{ r }\right)\left(4 \pi r ^{2} dr \right)$
$dm =4 \pi krdr$
$M =\int\limits_{0}^{ R } dm =\int\limits_{0}^{ R } 4 \pi krdr$
$M =\left.4 \pi k \frac{ r ^{2}}{2}\right|_{0} ^{ R }$
$M =2 \pi k \left( R ^{2}-0\right)$
$M =2 \pi kR ^{2}$
for circular motion gravitational force will provide required centripital force or
$\frac{ GMm }{ R ^{2}}=\frac{ mv ^{2}}{ R }$
$\frac{ G \left(2 \pi kR ^{2}\right) m }{ R ^{2}}=\frac{ mv ^{2}}{ R }$
$ \Rightarrow v =\sqrt{2 \pi GkR }$
Time period $T =\frac{2 \pi R }{ v }$
$T =\frac{2 \pi R }{\sqrt{2 \pi GkR }} \propto \sqrt{ R }$
or $T ^{2} \propto R$