Question

If there is a tunnel made from North Pole to South Pole. Then, a ball is dropped from North Pole , How will it go to South Pole?

Answer 1

Assuming no friction or other dissipative losses, neglecting the fact that the density of the Earth is non-uniform, neglecting the centrifugal force that acts on the mass due to Earth's spin, and neglecting other sundry stuff like Coriolis forces (if all this sounds excessive, remember that it's not as excessive as digging a friggin' tunnel through the Earth!), the ball will keep shuttling back and forth in Simple Harmonic Motion.

As you move inside the Earth, the Force acting on you will reduce linearly. This is often counter-intuitive for many people. The formula F=Gm1m2r2F=Gm1m2r2 is applicable only if you are outside the Earth's surface.

This will work inside the Earth too, but you'd need to use the integral form (Gauss' Law for Gravity). Suffice it to know (unless you are interested in the details) that, inside the sphere,

F=G(4πrρm)/3F=G(4πrρm)/3

ρ=M/(4π/3R3e)ρ=M/(4π/3Re3)

Putting all the constants together as 'k'
F=krF=kr (attractive)

If the force acting on a body is directly proportional to the displacement (from a reference point) and acts opposite to the direction of motion, the body executes what is called SHM (Simple Harmonic Motion). Other examples are a mass attached to a spring and a simple pendulum (for small angular displacements)