Prashant K.

Gravitational Potential energy doubt

Why gravitational potential gradient is negative when a body is taken away from earth? Please resolve my doubt.

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Arturo O.

There is a missing "carriage return" in the last equation of the answer, and I left out the limits in one of the integrals.  It is very difficult to edit the answer, but here is the correct final equation for U(r):

U(r) = GMm∫r dr'/r'2 = -GMm(1/r - 1/∞) = -GMm/r

So you can see that in the more general case, gravitational potential energy is negative, increases as you move away from the earth, and is zero at infinity.

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01/02/17

Francisco P.

I would like to add to Arturo's fine explanation to say that the gravitational force (negative gradient of the gravitational potential energy) in both cases is directed to the center of the gravitating body (Earth) and is opposite to the direction of the increase in distance between the center of the two bodies (r).

You can think of the direction of the force as the negative of the slope of the Ug vs. r graph at any position for both the exact potential (U = -GMm/r) and the linearized potential energy near Earth's surface (U = mgz) where z = r - R with R ≡ radius of Earth.  In both cases, U increases as r increases for r > 0.  The slope is positive, so the force is negative.

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01/02/17

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