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tjackson3

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## Homework Statement

This class has given me a number of problems that seem like they'd be ludicrously easy until I start them.

A commuting scheme involves boring a straight tunnel through the crust of the Earth between New York City and San Francisco. If you were to drop a ball down into the tunnel in NYC then how long does it take for the ball to pop up in SF? Assume that there is no friction or air resistance in the tunnel. Assume that the Earth is round and has constant density, and that at the surface of the Earth g = 10m/s^2. Finally assume that the triangle connecting NYC, SF and the center of the Earth is an equilateral triangle (60 degrees for each angle). Assume that the radius of the Earth is 6,400 km. Hint: Ignore the Earth's rotation and centrifugal force aects in this problem!

## Homework Equations

[tex]\vec{F} = m\vec{a}; \vec{F_g} = -G\frac{mM}{r^2}\hat{r}; \vec{g} = \vec{F}/m [/tex]

## The Attempt at a Solution

As best as I can figure, a picture of this would look like so:

http://img180.imageshack.us/img180/648/physprob.png

In the picture, R is the radius of the earth, and [tex]\theta[/tex] is the angle formed by the intersection of the radius that touches NYC and the ray from the center of the Earth to the position of the ball. The way I've considered approaching it is to say that the acceleration due to gravity will depend exclusively on the angle [tex]\theta[/tex], which itself depends on time. So

[tex]\ddot{x} = g\cos\theta[/tex]

However, since [tex]\theta[/tex] depends on time in a way that I cannot seem to define, I can't figure out how to integrate this. Am I on the right track? Thanks for your help!

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