In the rough approximation that the density of the Earth is uniform throughout its interior, the gravitational field strength (force per unit mass) inside the Earth at a distance r from the center is gr/R, where R is the radius of the Earth. (In actual fact, the outer layers of rock have lower density than the inner core of molten iron.)

Question

1. Using the uniform-density approximation, find an expression for the amount of energy required to move a mass m from the center of the Earth to the surface.

I wanted to do the following:
We know from the given information that F (from the gravitational field strength) = mg(r/R)
Then using W=Fd, you could get (mg(r/R))(R), where R cancels out and W=mgr. But Force isn't a constant. So how would you solve this problem then?

2. Calculate the ratio of the energy you found, to the energy required to move the mass from Earth's surface to a very large distance away.

Answer 1

1. For the work, you need to calculate the integral of F*dr from r=0 to R

F = M*g*r/R
Integral of F*dr = M*g r^2/(2R) @ r=R
= M*g*R/2

2. To remove the object from the earth's graviational field, the work required is

integral of F*dr = M*g*R^2/r^2 dr from r=R to infinity
= M*g*R2/R - 0
= M*g*R

Answer 2

I've tried both of those answers before but it simply asks me "How is g related to G, M, and R?"

I'm not really sure what else to do.

Answer 3

g = G Me/R^2, the value of the acceleration of gravity at the surface of the earth. Me is the mass of the earth. The M is my equations is the mass of the moved object.

Answer 4

I didn't understand what you did for part 2?

Answer 5

I don't understand what you did in part 2 either. where did the r^2 come from?