Drill a hole in earth

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Gramps

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Drill a hole in earth
« on: 08/12/2004 19:37:18 »
If i drill a hole through the center of the earth that cross the entire planet and then i drop a ball into the hole:

1. What is going to be the speed of the ball at the other end of the planet?

2. How much time is going to take the ball of going from one end to another?

Ok, I have:

Mass of Earth= 5.974 * 10 ^24 Kg
R of Earth = 6. 374 * 10 ^ 6 m
G = 6.67 * 10 ^ -11
diameter of Earth = 1.27 *10^7 m

I need to know the speed the ball use to go from one end of the hole to the other end. I thought the speed was 0 m/s but i think i'm wrong.

Should i use $$V= \sqrt {2gd}$$ ?

Now i have to look for the time it takes the ball to travel from one point to the other. So the ball is going to pass Earth's core.
One point i want to clear is that r = R.
The proffesor told me that we need to use earth's diameter to find time.

If there is something wrong with something i said, please correct me.

gsmollin

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Re: Drill a hole in earth
« Reply #1 on: 09/12/2004 14:32:30 »
The development of this problem is an important part of Newtonian gravitational theory. You need to consult a good physics textbook. I used Resnick and Halliday, "Physics, Part 1", Sec 16-7, "Gravitational Effect of a Spherical Distribution of Mass".

Here are the upshots:

Assumptions: Neglect all frictional effects and assume the earth has a uniform density.

The motion of the ball is simple harmonic. So the answer to question 1 is "0".

The answer to question 2 is time = 42.1 minutes for a trip from one surface to the opposite. Therefore period of oscillation is 84.2 minutes.

As I said, the development of this subject is a tour-de-force of Newtonian gravitational physics, needing three-dimensional calculus, analytical geometry, and insight of historical importance. I would never think of depriving you of the thrill of finding it yourself in your college library's physics book section. Enjoy your research.
"F = ma, E = mc^2, and you can't push a string."

schug

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Re: Drill a hole in earth
« Reply #2 on: 13/12/2004 16:23:35 »
If I don't make this assumptions, the ball would make an another motion.How could this motion look like?
I think it would stop after e few periods in the middle of the earth, because of the gravitional force and the friction.Or does it looks different?

P.S.:Sorry for my English, it's not the best[]

gsmollin

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Re: Drill a hole in earth
« Reply #3 on: 14/12/2004 18:25:58 »
If we add friction to the problem, then the ball stops in the middle of the earth. The motion of the ball depends on the kind and amount of this friction. In the simplest case, we assume a damped harmonic motion. The ball may oscillate back and forth a number of times before settling at the center of the earth. We call that under-damped. It may never oscillate at all, but just fall to the center. We call that over-damped.

In any case, with the addition of friction, the ball never reaches the opposite surface.
"F = ma, E = mc^2, and you can't push a string."