Gravitational Potential energy doubt

Question

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

Answer 1

Prashant,

 

I do not know if your physics class is with calculus or not.  I will give you an answer using calculus.

 

You are probably already familiar with potential energy U as a function of height h:

 

U = mgh

 

But this formula is valid only for a constant acceleration of gravity g.  The variation in height above the surface of the earth has to be extremely small so we can say that g ≅ constant.

 

In general, allowing for variability over a wide distance range, the force of gravitation between the earth M and a mass m is

 

F = -GMm/r²

 

where r is the distance between the centers of the earth and the mass m.

 

Since F = -∂U/∂r, and using ∞ as our reference point,

 

U(r) = -∫_∞^r F(r')dr' = -∫_∞^r-GMm/r'² dr' = GMm∫dr'/r'² U(r) = -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.