A spiral galaxy can be approximated as an infinitesimally thin disc of a uniform surface mass density (mass per unit area) located at z = 0 , Two stars A and B start from rest from heights 2 z 0 and z 0 ( z 0 < < radial extent of the disc), respectively and fall towards the disc, cross over to the other side and execute periodic oscillations. The ratio of time periods of A and B is

Step-by-step solution

Spiral galaxies consists of flat rotating disc containing stars, gas and dust. Now, we thought of spiral galaxy in form of a disc. Gravitational intensity at distance r r over axis is I g = 2 G M a 2 1 - r r 2 + a 2 1 / 2 = 2 G M a 2 1 - 1 1 + a 2 r 2 1 / 2 = 2 G M a 2 1 - 1 + a 2 r 2 - 1 2 Using binomial approximation, we have I g ≈ 2 G M a 2 1 - 1 + 1 2 a 2 r 2 ≈ 2 G M r 2 So, forces on planet A and B are F A = 2 G M m A 4 z 0 2 , F B = 2 G M m B z 0 2 Accelerations of A and B respectively, are a A = G M 2 z 0 2