Suppose that a tunnel could be dug along diameter.Show that the motion of a particle dropped into the tunnel is simple harmonic motion.Neglect all frictional forces assume that the earth has a uniform density.I need the answer as a numerical solved problem.

Let`s assume that we`re able to dig a hole that starts from one side of the earth, passes through the center of the earth and comes out from the other side. Then we dropped a ball in that hole. Initially ball would fall towards the center due to gravitational pull of the earth and its speed would keep increasing. However, its acceleration would decrease i.e. its rate of speed gain would decrease. This happens until it reaches the center. At this point its acceleration is 0 as no force is acting on it (for it is already at the center). Once it overshoots the center, ball is still attracted towards the center however now it is going in the opposite direction of the pull. So it starts losing its speed and the rate of speed lose keeps increasing away from the center. This happens till the ball stops (which would happen at a distance from the center exactly equal to the distance from which it was released on the other side).Once the ball has stopped, it repeats the whole process again and again - it is trapped in an infinite oscillatory motion. This motion is in fact a special oscillatory motion formally called as Simple Harmonic Motion.