How does an object gain potential energy?

[adjurovich]

Yes. It's the notion that mechanical work can be generalized to a complete understanding of energy transfers. It cannot. The concept of internal energy must first be introduced. Later it can be followed by heat and the 1st Law of Thermodynamics.

This was one of the great intellectual accomplishments of the 19th century and led to what we now call the conservation of energy.

And that's almost correct. The energy a system has due to the relative position of its constituents. There are other textbooks that do this correctly, but they are in the minority.

It is incorrect because you are using mechanical work to draw conclusions about the more general concept of energy.

It goes into increasing the potential energy of the system. It comes from the person doing the lifting.

Development of energy concepts in introductory physics courses
Arnold B. Arons
Citation: Am. J. Phys. 67, 1063 (1999); doi: 10.1119/1.19182
View online: http://dx.doi.org/10.1119/1.19182
View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v67/i12
Published by the American Association of Physics Teachers

I think I almost understood it. The last question I will ask is: if the energy coming from external work is stored in the system as potential energy. What happens to the energy from work of gravity?

 

[Mister T]

The work is not external. It's internal to the Earth-object system.

 

[adjurovich]

The work is not external. It's internal to the Earth-object system.

Let’s be a little more precise: work done by gravity or work done by some force $F$?

 

[Mister T]

Let’s be a little more precise: work done by gravity or work done by some force $F$?

I answered the question you asked:

What happens to the energy from work of gravity?

As I told you before, the energy expended by the agent separating the object from Earth is stored as potential energy in the object-Earth system.

 

[adjurovich]

I answered the question you asked:

Thanks a lot!! I get it now.

 

[A.T.]

if the energy coming from external work is stored in the system as potential energy. What happens to the energy from work of gravity?

Work is transfer of mechanical energy. If an external force does positive work on one of the bodies then it transfers energy to that body. Some of that energy might go into KE of that body, and some into the PE of the system. The later transfer is represented by the negative work by gravity on that body.

It might help to replace the gravitational field with a spring. Here it's more clear where the PE is stored, while the whole energy transfer scheme is the same.

 

[PeroK]

I think there is a general problem here of starting with the equations and then trying to interpret them. Whereas, the meaning of an equation is inherent in the definitions and assumptions that led to the equation in the first place.

In the case of $F =ma$ it is often forgotten that $F$ is the total external force and that this equation is not valid independent of the definition of the three quantities $F, m$ and $a$.

The same applies to the work energy theorem. That equation holds under a precise set of hypotheses. And if it appears to fail, then one or more of the hypotheses must not apply.