Where will a ball thrown into a hole dug from North Pole to South Pole end up?

[Neha Sanghvi]

Hey, could someone please help me out? This is a silly problem, but I am unable to figure it out.
The problem is that a 1 m diameter hole is dug from North Pole to South Pole. Then, a ball is thrown in the hole. Where will the ball go? Will it remain at the North Pole, or go to the South Pole or get stuck in between ( at the centre )?

According to me, it should be the last one, the ball goes to the centre bcoz the gravitational force should be the highest there. But, then it is a hole. So, someone please help me out!

Thanks.

 

[Doc Al]

What makes you think that the gravitational force is highest at the center of the Earth? (It's actually zero at the center.) You should know that if you toss a ball into a hole, it will fall, picking up speed as it falls further. (Ignore air resistance and other complications.) So what do you think will happen? When the ball reaches the center (if it does) how will it be moving? (Fast? Slow?) Will it keep going past the center? Why or why not? Think it through.

 

[Severian596]

I'll add to Doc Al's advice by suggesting that you apply a limit to this process; ask yourself what happens at the boundaries of your experiement if you assume one way or the other. If you assert that the ball will continue falling downward indefinitely, away from the north pole, what would an observer standing at the "south pole hole" see?

What makes you think that the gravitational force is highest at the center of the Earth? (It's actually zero at the center.) You should know that if you toss a ball into a hole, it will fall, picking up speed as it falls further. (Ignore air resistance and other complications.) So what do you think will happen? When the ball reaches the center (if it does) how will it be moving? (Fast? Slow?) Will it keep going past the center? Why or why not? Think it through.

 

[andrevdh]

Neha, In this case the ball is subjected to a conservative force, gravity. This means that the total mechanical energy of the system will be conserved kinetic, T, and potential energy, U. As the ball falls down the hole kinetic energy will therefore increases at the expence of potential energy. Secondly, the gravitational force that the ball experiences - Fg - is the total attraction of all of the little mass particles that is in the sphere "below" it (as it falls down the hole the mass particles in the shell outside of the sphere "above" it do not contribute to this attraction that it experiences (their attractions cancel out) - the total attraction therefore decrease as it falls inwards towards the centre of the Earth - see the attachment) - that is via Newton's universal gravitational law. Thirdly, the resultant attraction of all of these little mass particles gives the same result as if all of the mass were concentrated at the centre of the sphere! Fg therefore decreases to zero as the ball approaches the centre of the Earth and then increases again on the other side, but now Fg points in the opposite direction. Do you know of any other physical system that behaves similarly?

 

[HallsofIvy]

On the other hand, if we do allow for air resistance, the total energy of the ball decreases, eventually decreasing to 0. The ball will oscillate back and forth, rising a little less each time. Wherever its speed becomes 0, there will still be a net force back toward the center of the earth. The ball will eventually come to rest at the center of the earth.