Question

why is the gravity 0 at the centre of the earth?
please explain briefly.

Answer 1

At the very center, the gravitational force is zero because there's equal mass pulling on you from all sides, and it all cancels. If you built a room there, you could float around freely. That's what it means to say that gravity is zero at the center of Earth. However, the gravitational well is a different story.

Answer 2

There are two regions to consider; the region outside the Earth and the region inside.

The solution inside (non-zero mass density) and the solution outside (zero mass density) are different but must give the same value at the radius of the surface.

Assuming a uniform mass density inside, the gravitational acceleration inside is proportional to the radius and, thus, goes to zero at the center.