Question

A very long (length L) cylindrical galaxy is made of uniformly distributed mass and has radius $R (R << L)$. A star outside the galaxy is orbiting the galaxy in a plane perpendicular to the galaxy and passing through its centre. If the time period of star is $T$ and its distance from the galaxy's axis is $r$, then :

• A. $T^{2} \,\propto\,r^{3}$
• B. $T \,\propto\,r^{2}$
• C. $T \,\propto\,r$
• D. $T \,\propto\,\sqrt{r}$

Answer 1

The Correct Option is C

Let the linear mass density of the cylindrical galaxy be $\lambda\, kg / m$.
Gravitational field $=\frac{2 G \lambda}{r}=E_{r}$
Therefore, gravitational force $F=m E_{g}=m \omega^{2} r$
Hence, $m\left(\frac{2 G \lambda}{r}\right)=m \cdot\left(\frac{2 \pi}{T}\right)^{2} r$
$\Rightarrow T^{2} \propto r^{2} \Rightarrow T \propto r$