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CGPET 2016 Solved Paper

Section: Physics

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Question : 3 of 150

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A very long (length $L$ ) cylindrical galaxy ismade of uniformly distributed mass and hasradius $R (R \ll L)$ . A star outside the galaxy isorbiting the galaxy in a plane perpendicularto the galaxy and passing through its center.If the time period of star is $T$ and its distancefrom the galaxy’s axis is $r$ , then

$T^2 \propto r^3$
$T \propto r^2$
$T \propto r$
$T \propto \sqrt{r}$

Solution:

Length of the galaxy

= L

Mass is uni-formally distributed.

Distance of star from axis of galaxy

= r

Star is orbiting the galaxy. So, centripetal force must equal to the force experience by gravitation of galaxy

∴

F_c = \frac{2GM}{L \cdot r} m

F_c = \frac{mv^2}{r}

[ \because v = \omega r = m\omega^2 r]

m\omega^2 r = \frac{2GMm}{L \cdot r}

but,

\omega = \frac{2\pi}{T}

where,

T =

time period

m\left[\frac{2\pi}{T}\right]^2 \cdot r = \frac{2GMm}{L \cdot r}

∴

T \propto r

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