Probing Local Realism

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You are partially correct, however the main force that acts upon the ball would simply be its inertia.

Gravity also plays a role, and the influence of gravity would get stronger as the ball travels to the more dense core of the earth, but the former force is stronger thus causing it to oscillate. We can illustrate the motion of the ball as a sinusoidal function with a decreasing amplitude over time.

The question I wanted to raise is how strong does a planet's gravity have to be to overtake the inertia of the ball? As a person with a degree in physics and math, I though the best way to proceed would be to simply to take the gravitational formula and make it equal to the harmonic oscillation formula (in an above post), and solve for the unknown. Not sure it is correct however.

If you jumped down this hypothetical hole (at the north pole, and ending at the south pole so the Earth's spin wont matter) you would accelerate as you fell at the same rate you would accelerate when falling off a high place on the earth's surface.

You would keep accelerating until you passed the center of the Earth. Then you would steadily decelerate. And keep on decelerating. Until you would finally stop just as you reach the opening at the other end of the tunnel on the surface of the other side of the Earth. So you would then have to grab the edge of that opening on the other side of the Earth as fast as you could so you don't fall back down the hole!

This is because you would have the whole earth pulling you towards the center of the earth during both halves of the journey. So it would cancel out.

But that's assuming that the tunnel is also a vacuum, and you are wearing a space suit to breath.

If there was normal atmospheric pressure air filling this tunnel its whole length then the drag with the air would slow you down. You would probably stop maybe ten percent below the surface of the planet on the opposite side, then fall back down, pass the center of the earth again, stop 20 percent below the surface of the first side, and fall back down again, and so on..like a swinging pendulum slowly stopping, until you came to rest at the center of the earth (except you would burn up from the friction with the air long before that).

Okay: the earth (for whatever reason) is not spinning (or that isnt a factor). Ocean water is not flowing down the tunnel, and there is no air in the tunnel (or magically air friction is suspended for the purpose of the experiment).

So its all about Newtonian motion. All of the messy stuff is not there to complicate it.

So you drop your ball down this mine shaft that goes all of the way down through the center of the earth, and continues straight up to the point on the surface on the opposite side of the earth (which odds are would be on the bottom of the ocean if your end was on dry land, but for the sake of the story we will say that both ends are on dry land).

So what happens to the ball?

The mass of the whole planet is pulling gravitationally on the ball as it fall down the hole so it accelerates at the same rate of that a falling object falls at the surface of the earth (32 feet per second squared).

As it falls miles down into the earth more and more of the earth's mass grows above it pulling it gravitationally upward. And gradually less and less of the earth is below it. So its rate of acceleration slows somewhat. But its is still moving fast and is still accelerating as it rushes past the center of the earth. There is no air in the tunnel so the ball never reaches terminal velocity. But the moment it passes the center of the earth it starts to decelerate because more the of the earth's mass is excerting gravity behind it than in front of it acting as a break. But it still keeps moving forward away from the center of the earth. It finnally grinds to a halt, and it halts more or less at the same moment that it reaches the opening of the tunnel at the opposite side of the earth. So some person on the other side needs to grab it before it falls back down.

Ohhhh...I get it.

Its basically a trick question.

you're gonna tell us that the ball already has momentum when you drop it down the hole. Momentum imparted to it by some motion of the earth. We've already agreed to take the Earth's rotation on its axis off the table (the tunnel goes between the poles, or something magical happens to stop the earth rotating, or something).

So that leaves the other motion: the earth revolving around the sun. So for some reason the ball free falling through the earth causes centripetal force to make the ball slam into the side of the tunnel before it gets very far not because of the Earth's rotation, but because of it revolution around the Sun.

Is something like THAT where you're going with this?