Many of us as kids would have tried digging through our sand boxes to get to the center of the Earth, which obviously is pretty much impossible. An interesting purpose to do this would be to reach the country on the exact opposite side of you. The straight route is always the fastest, isn’t it? You can figure out which country you will end up in, but if you decide to take that path, you may want to understand gravity and center of the Earth a little better.
We can safely consider a ball instead of just throwing a poor human down in that scary tunnel. Knowing that digging a tunnel of that size through the center of the planet is far from realistically possible, we will make a few assumptions to help understand the scenario better. A body falling through the center will experience a variety of forces including centrifugal force due to Earth’s spin, friction, Coriolis forces, etc. Assuming all of these effects to be negligible (zero) and also ignoring the fact that earth’s density is not uniform, the simple answer is that the body will move in simple harmonic motion (SHM), shuttling back and forth all the way.
While you are above the surface of the Earth, force of gravity is quantified by the relation F=G(m1*m2)/r^2. But once you move below the surface, the force reduces linearly as you move towards the center according to the relation F=G(4*pi*r*rho*m)/3, which equates to F=kr, if you take all the constants as ‘k.’
The force of attraction will linearly decrease as you move towards center and the speed reaches its maximum of about 7900 m/s when you reach the center. It will take about 84.5 minutes to get to the other side with speed decreasing as you move away from the center. When you do get to the other end, you will need some serious support to hold on to, or you are coming back all the way for another ride of 84 minutes, and this will happen over and over and over again.
In simpler words, Earth,’s gravity would pull any object dropped to its center. As the object reaches the center, the speed will continue to increase. The object will reach peak speed at the center, and as it begins to move in opposite direction, it will decelerate, reach the end, and fall back into the hole again.
The video below by Matt Anderson explains the same theory in a tad more detail.