Velocity for an object at the center of the earth?

ukballer1012

Let's assume there is a hole that goes through the center of the earth with ideal conditions (earth is spherical and gravity is the only force). If you dropped an object with a mass of, say 20kg, what is its velocity at the center of the earth? I know the force is much like that of a spring and thus is linear but have no clue how to calculate this. Please show work.

Nugatory

You'll need to make another ideal-conditions assumption as well: that the density of the earth is uniform.

If you make this assumption, you'll be able to the calculate the gravitational force on your mass when it's at a distance r<R_E (where R_E is the radius of the earth) from the center of the earth. Because the gravitational field of a hollow spherical shell is always zero in the interior of the shell, you can ignore everything outside of r when calculating the force at r.

Once you have the force as a function of r, you can get the work done by dropping the object into the hole from the surface of the earth by integrating between R_E and 0. That will give you the kinetic energy of the dropped object at r=0, and E=(mv²)/2 will get the velocity from there.