Travel Through Earth: Tunnel Time

[dcppc]

I'm trying to find out that if you dig a tunnel through the center of the earth, then how long will it take to get from one end to the other end?
I can't use x=1/2at^2+vt because gravity is constantly changing when I approach the center of the earth.
And if I find out the gravity, how will I incoporate time into the equation?

 

[Doc Al]

I assume you want to find the time it would take for an object dropped into the tunnel at one side of the Earth to reach the other side? (Ignoring air resistance; modeling the Earth as a uniform sphere.) Start by finding the gravitational force on the object as a function of distance from the center of the earth. Then you can write the differential equation that would represent the motion, using Newton's 2nd law.

 

[BobG]

This is a tough problem. The force due to gravity is defined by Newton's Universal Law of Gravitation:

F=-\frac{GMm}{r^2}

The tough part is that the big M, the mass of Earth, is constantly changing from the object's frame of reference. As you fall through the Earth, some of the Earth's mass is in front of you, causing your fall to speed up, and some is back of you, causing your fall to slow down.

Fortunately, all of the mass further away from the center of the Earth than the falling object can be disregarded. There's a couple of ways to explain that, but a geometric explanation is probably the easiest.

All of the mass closer to the Earth than the falling object can be plugged into Newton's Universal Law of Gravitation as normal. What you have to do is figure out the rate of change for the mass in addition to the rate of change in the distance.

The result winds up giving you a very simple relationship. See this brain teaser thread for a discussion we had about it a few months ago.

 

[HallsofIvy]

But inside the earth, the gravitational force depends only on mass lower and so decreases as r. Set up a(x)= Cr and use the fact that when r= radius of earth, a= -9.81 to find C.