what will happen if we dig a hole from north pole of earth to the south pole and then throw a body in that ?

Assuming the following:

The whole could be dug all the way through with no obstruction.

Earth has a uniform mass density.

Any fictional energy loss, such as air resistance, is neglected.

Then the body will execute simple harmonic motion. That is, it will oscillate back and forth like a pendulum. More like up and down.

If you include frictional loss, then the motion would be like a damped harmonic oscillator.

If you assume nonuniform mass distribution then the motion would be different.

To get an idea of the motion, we would use Guass's Law as applied to a gravitational field. We could plug in different mass distributions and get different results.

Hole In North Pole

The ball will ultimately attains equilibrium at the center of the earth and stay there!!

Reason:

We know that as the depth from the surface of the earth increases the gravitation decreases!

This is represented by the formula G = g(1 - d/R)

Where 'd' is the depth from the surface and R is the radius of the earth!

If depth is 0 then G = g (where g is the acceleration due to gravity)

If depth is increases G decreases as you can see that from the formula!

If the depth reaches the center i.e d = R

Then G becomes 0 as you can see that from the formula!

So, when you throw a ball in to the hole, the gravitational force continuously decreases and becomes 0 at the center of the earth!!

But the ball won't stop there due to the inertia of motion of the ball!!

After reaching the center of the earth, the depth is again decreases when you see from other side, and the gravitation increases opposite to the motion of the ball and ultimately the ball stays at the center of the earth!!

Thank you for asking this interesting question!!

Assuming no air resistance, the body will fall to the center, then rise to the surface on the other side, come back, and repeat.

With air resistance (and bumping along the sides of the hole), the body will eventually settle at the center of the Earth, hanging there weightless because the gravitational pull will be equal in all directions.

Well after digging through the crust you'd meet the lava, which basically means you created an artificial volcano. Lets say you do manage to block all the lava and you do manage to somehow dig a hole through the earth. The body will fall down and stop in the core, and get crushed there and maybe turn into a diamond. It won't continue on down to the South Pole because the center of gravity is the core

this is practically impossible but theoretically the body will perform simple harmonic motion i.e the to and fro moment of the body from north to south for infinite times.

It would eventually end up in the center, since all matter is attracted to the center of the earth.