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 galaxys axis is r, then :

  • Option 1)

     T^{2}\alpha r^{3}

  • Option 2)

    T\alpha r^{2}

  • Option 3)

    T\alpha r

  • Option 4)

    T\alpha \sqrt{r}

 

Answers (1)

As we learnt in

Time period of satellite -

T=2\pi\sqrt{\frac{r^{3}}{GM}} 

r= radius of orbit

T\rightarrow Time period

M\rightarrow Mass of planet

 

- wherein

T=\frac{Circumference\: of\: orbit}{orbital\: velocity}

 

 F=\frac{2GM}{Lr}m,                F=(\frac{k}{r})m,

Here K is some constant

(\frac{mv^2}{r})=\frac{km}{r}\Rightarrow v=\:constant

T=\frac{2\pi r}{v}                    

\therefore T\propto r


Option 1)

 T^{2}\alpha r^{3}

Incorrect

Option 2)

T\alpha r^{2}

Incorrect

Option 3)

T\alpha r

Correct

Option 4)

T\alpha \sqrt{r}

Incorrect

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