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I have some confusion about the potential energy created by a gravitational field.

http://ocw.mit.edu/courses/physics/8-01t-physics-i-fall-2004/assignments/ic_sol_w07d3_1.pdf

This link says that the energy of an object falling through the centre of the earth is $\frac{1}{2}mgR_e$. So shouldn't the energy required to lift an object to a height $h$ above the surface be $$ mMG(\frac{1}{2R_e}+\frac{1}{h})~~?$$ This is not consistent with the commonly used $$\frac{mMG}{h+R_e}.$$ Why does the potential energy decrease as you go higher?!

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In the link, it takes the center of the Earth as reference point of potential energy. While the commonly used formulas takes infinity as the reference point.

If you want to use the center of the earth as the reference point, then because the forces inside and outside have different forms, the potential energy outside the earth should be

$$\frac{1}{2}mgR_e-\int_{R_e}^r \frac{GMm}{r^2}dr$$ $$=\frac{1}{2}mgR_e+GMm\left(\frac{1}{R_e}-\frac{1}{r}\right)$$ $$=\frac{1}{2}mgR_e+mgR_e-GMm\frac{1}{r}$$ $$=\frac{3}{2}mgR_e-\frac{mgR_e^2}{r}$$

Note that the potential energy still increases with height outside the Earth.

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