Gravitation revision...........

These two questions are good for revision as I think ---- took me a few days to solve them --

1> What is the energy change assosiated with the making of the earth from tiny particles assuming std. values ?

2> A and B are the centers of two spherical bodies of mass m and j .Show that if a particle placed at the pt. where gravitational intensity vanishes is given a very small displacement in the direction perpendicular to AB , then it will execute simple harmonic motion .Find the time period of that motion.

Yeah absolutely correct i am just furnishing the proof ----
Consider the planet earth is made by assembling spherical layers of radii ranging between zero to R. Suppose that at any instant during the process of assembling the radius of the spherical core of the earth is x.If p denotes the density of earth , then the mass of the spherical core of the earth of radius x is
M = ( 4x³pÎ ) / 3
The mass of the material of the earth in the form of spherical shell , having inner and outer radii x and x + dx respectively is
dm = ( Î 4 x² dx p )
Suppose , that the mass dm is brought from infinity and put around the core of the earth of radius x . Then amount of work done to bring the mass from infinity up to the surface of the core of radius x is given by
dW = P.E of the system of the core of mass m and the mass dm at the distance x apart
= - ( G m dm ) / x
= - ( Î G 4 p x³ ) X ( 4 Î p x² dx) / 3x
= - 16 ( Î ²G p² x⁴ dx ) / 3
Therefore the amount of work done to assemble the earth from zero to its present radius R.
W
= dW
= - 16 ( G p² Î ²) ∫₀^R x⁴ dx / 3
= - 16 ( G p² R⁵ Î ² ) / 15
Now p X ( 4Î R³ ) / 3 = M
So p = 3M / ( 4Î R³ )
So W = - 3 ( G M² ) / 5R
Putting the standard values , we get W = - 1.5 X 10³² J So the energy change = 0 - ( W )
= 1.5 X 10 ³² J