Physics
posted by Mehreen .
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.)
1. Using the uniformdensity 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.

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 
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. 
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.

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