Reply #29: Assumptions are important

[caraher]

The original problem is really about a non-rotating sphere and a tunnel in which there is no friction or air resistance. Given those assumptions energy will be conserved and an object dropped into the tunnel would indeed oscillate with the stated period - and do so with undiminished "amplitude" (that is, you would make it all the way through to the opposite side and back; repeat forever).

Realistically... well, technologically physically constructing such a tunnel is a non-starter. But lets suppose that - somehow - there exists such a tunnel. Is there air in the tunnel? That causes drag; the sum of kinetic and gravitational energy decreases steadily, and you don't make it through.

The case considering rotation is rather interesting... if the tunnel were truly frictionless, rotation would not matter! As someone mentioned, you would hit a wall because of this, but if you assume a frictionless tunnel that doesn't matter. As you fell, without friction the sum of kinetic and gravitational energies would be constant. The rotational speed of Earth would have to increase as you approached the center, but not by a noticeable amount. (It's like considering how jumping moves the location of the center of mass of Earth...)

But this business about "gravitational drag" in the previous post is incorrect. It is correct that the net gravitational attraction decreases as you approach the center of Earth, where it is zero, and after that it increases as you move out. But absent any "losses" the point at which you reach the extreme of your travel is when you are the same distance from the center as the distance from which you started (assuming you fall from rest). It's simple conservation of energy.

The argument outlined about "terminal velocity" is only valid in the context of air drag. Terminal velocity is by definition the speed at which the force due to air resistance is equal in magnitude and opposite the direction of weight. Weight does decrease as you move inward, but if you are considering an air-filled tunnel not making it to the other side can be explained simply by noting that that drag is a "loss" like friction.

I'm also not sure if the terminal velocity would change in quite the way you think, because the density and pressure of air probably wouldn't be constant as a function of height, even if you somehow kept the tunnel at the same temperature. (Though off the top of my head I'm not quite sure exactly how it would change; I'd expect pressure to increase all the way down, but at a diminishing rate thanks to the diminishing value of "g" as you approached the center.)

The ball analogy is not applicable. A real ball deforms when it hits the ground. This deformation stores energy, some of which is released as the ball rebounds but some of which converts to thermal energy. The random thermal energy is not available for conversion into either kinetic or gravitational energy, so it effect it is "lost." That's the difference between dropping a rubber ball and a lump of clay - the clay deforms irreversibly, while the ball's deformation is largely reversible. The surface from which the ball bounces also experiences this deformation and (usually imperceptible) heating.

In addition, the ball experiences air drag, whether or not it is perfectly elastic. These two "loss" mechanisms are why real balls eventually stop bouncing.

The one thing I may be missing would be some kind "drag" that might come up through general relativity. I don't know whether there's something there for me to miss; this is simply a small disclaimer!