A very long (length L) cylindrical galaxy is made of uniformly distributed mass and has radius R `(R lt lt 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-

Options

A. `T prop r`
B. ` T prop sqrt(r)`
C. `T prop r^(2)`
D. `T^(2) prop r^(3)`

Answer

A

Explanation

F = `(2GM)/(Lr) m or, (mv^(2))/(r) = (2GM)/(Lr) =` m
mr `((2pi)/(T))^(2) = (2"GMm")/("Lr") [ because v = r omega and omega = (2pi)/(T) ] `
`rArr " " T prop r `