Bob Miller wrote:
My sense is that you are wrong. That as you drop your 24,000 mile perpindicular speed will fall to match the speed of the earth around you but I have no argument.
Conservation of energy and momentum. Here's my version of what happens.The effect should be related, but different from, a comet zooming by a planet. In that scenario, the trajectory of the comet is bent as the gravitational pull attracts the comet, and then it zooms away as it gets further from the planet (or sun). The trajectory is affected more if the comet comes in close than if it's distant, because the centripetal force is proportional to the square of the distance between the two bodies.
The difference here is that while the comet sees the massive planet or sun as an external body of constant mass, in this case, you are inside the planet.
Your argument has a problem to. As you approach the center of the Earth with a tangential 24,000 mph you will be spinning around it at a faster and faster speed till at the center you would be going at an infinite speed in circles at a point.
So in this case, no matter where on earth you begin, you start plunging down because the force of gravity pulling you in is far greater than the centrifugal force caused by the earth's rotation, with that horizontal west to east velocity component. But the further down you get, the less the gravitational pull toward the core. Remember that at the core, all the forces on you cancel out, so there's nothing pulling you in when you're very close to the core.
My current thinking is: 1. Falling in from either pole creates a straight shot through the globe.2. Falling in right at the equator sends you in a curved path to the other side, where the path never gets very close to the core. (Simply because the further you fall in, the less the core attracts you. And the effect is opposite as you get away from the core.)
3. Falling in from somewhere between a pole and the equator creates a helical path, where you rotate around the axis all the way down to the other side.
In both 2 and 3, the momentum created by the horizontal velocity has to be conserved, but the gravitational pull on you as you approach the core is reduced. That's why there's no way you would reach any sort of infinite circular speed. There's nothing to pull you all the way to the core or to the axis. You're balacing the momentum created by the rotation on the surface with an ever decreasing gravitational pull as you approach the axis or the core.
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