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 directly proportional to r b) T directly proportional to root r c) T directly proportional to r square d) T square directly proportional to r cube
Answers
Answer:
T∝r
The gravitational potential due to cylindrical mass distribution can be found by comparison with electric field due to cylindrical charge distribution.
Applying gauss law for cylindrical charge distribution, consider a cylindrical Gaussian surface of length L:
∮
E
.d
A
=
ϵ
Q
en
E2πrL=
ϵ
λL
E=
2πϵr
λ
Similarly gravitational field =
E=
r
2Gλ
where
4πϵ
1
=G from similarity
As the Gravitational field is inversely proportional to distance from center, The centripetal force is
F
centripetal
=mE=m
r
2Gλ
...(i)
F
centripetal
=mω
2
r...(ii)
substituting value of centripetal force in equation (i) from equation (ii)
mω
2
r=m
r
2Gλ
ω=
r
1
2Gλ
T=
ω
2π
=
2Gλ
r
⇒Tαr hence correct answer is option B
Explanation: