Question 196: Gravitation

Question

The mass density of a spherical galaxy varies
as
$${K \over r}$$ 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 :

Accepted Answer

dm = $$\rho $$dv
$$ \Rightarrow $$ dm = $$\left( \frac{k}{r} \right)$$ (4$$\pi $$r
2
dr)
$$ \Rightarrow $$ dm = 4$$\pi $$krdr
M = $$\int\limits^{R}_{0} dm$$ = $$\int\limits^{R}_{0} 4\pi krdr$$
$$ \Rightarrow $$ M = $$4\pi k\left[ \frac{r^{2}}{2} \right]^{R}_{0}  $$
$$ \Rightarrow $$ M = 2$$\pi $$kR
2
For circular motion gravitational force will provide required centripetal force.
$$\frac{GMm}{R^{2}} $$ = $$\frac{mv^{2}}{R} $$
$$ \Rightarrow $$ $$\frac{G\left( 2\pi kR^{2}\right)  m}{R^{2}} $$ = $$\frac{mv^{2}}{R} $$
$$ \Rightarrow $$ v = $$\sqrt{2\pi GkR} $$
Time period, T = $$\frac{2\pi R}{v} $$
= $$\frac{2\pi R}{\sqrt{2\pi GkR} } $$ $$ \propto $$ $$\sqrt{R} $$
$$ \Rightarrow $$ T
2
$$ \propto $$ R

Answer

T2 $$ \propto $$ R

Answer

T2 $$ \propto $$ R3

Answer

T $$ \propto $$ R

Answer

T2 $$ \propto $$ $${1 \over {{R^3}}}$$

Gravitation

The mass density of a spherical galaxy varies as ${K \over r}$ 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 :

Choose the correct answer: