[4th repost] Hole through the center of Earth?

Question

Cylindrical hole is drilled form North pole through
the center of Earth to South pole.

Two small balls separated horizontally by distance d = 1cm
dropped into the hole. What is horizonatal separation between
the balls when they reemerge after one full period of oscillation?


Althought this problem is non-relativistic, the geometric nature of gravitation clearly mainifests itself in the answer - please leave it as closed form expression.

Assume that Earth has uniform density.

Answer 1

A lot of time is being spent on this question, huh. All right, we know that the gravitational acceleration inside the Earth is of the form:

a(r') = G(4/3)πpr'

where p is the density of Earth's mass and r' is distance to the center. The effective repulsive gravitational acceleration inside an infinite cylinder is of the form:

a(r) = G(2)πpr

where r is the distance from the axis of the cylindrical hole. I had already posted an answer earlier, hoping that (4/3) = (2), in which case the balls would have dropped straight down, but alas, real life is messier. The net repulsive gravitational acceleration is

a(r) = r''(t) = (g/2R)r(t)

where g is 9.81 m/s² and R = 6372 km. Solving the differential equation and getting hyperbolic results, it comes out that for 2 balls separated by 1 cm at the beginning will, after a period lasting 5068 seconds for their return, be separated by 42.67 cm.

Answer 2

The earth's gravity is irrelevant. The balls are being accelerated toward the center of the earth (vertical dimension) equally. The horizontal acceleration of the balls (assuming they are of equal mass Mb) is given by Ab = MbG/r² where r is the distance between the centers of the balls and has nothing to do with their surface separation.

Since the round trip takes about 84 min in the absence of friction, the distance they will draw colser to each other is
∆X = ½Ab*(84*60)² *2

The '2' is because each ball will move this distance.

Answer 3

Hypothetically talking, confident. yet merely after a super form of rocking around the centre. you're able to shoot previous the centre due on your momentum from the autumn, and attain just about a similar height you have been at earlier falling in, yet then gravity could take over and additionally you're able to fall backtrack back. After each and every fall, you're able to lose some momentum, until you're caught on the centre of the planet.

Answer 4

It's hard to say because they will meet at the center because they are falling on different radii which must converge at the center. Once they bounce, all bets are off.