First slide

Motion of planets and satellites

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

Moderate

A

$T \propto R$
B

$T^{2} \propto \frac{1}{R^{3}}$
C

$T^{2} \propto R^{3}$
D

$T^{2} \propto R$

Solution

Gravitational force on star
$\begin{array}{l} F_{G} = \frac{G m M}{r^{2}}, w h e r e M i s m a s s o f g a l a x y w i t h i n r a d i u s r \\ S o, M =_{0}^{r} 4 π r^{2} d r \frac{K}{r} = \frac{4 π^{2} k r^{2}}{2} \\ \Rightarrow F_{G} = m ω^{2} r \\ \Rightarrow F_{G} = m . r \frac{4 π^{2}}{T^{2}} \\ \Rightarrow T^{2} = \frac{4 π^{2} m r}{F_{G}} = \frac{4 π^{2} m r}{G m M / r^{2}} = \frac{4 π^{2} r^{3}}{G (\frac{4 π^{2} k r^{2}}{2})} \\ \Rightarrow T^{2} \propto r \end{array}$

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