I'm confused about how the energy is conserved and the signs of the works are also confusing for me. I have an example on my mind I would like to ask. Imagine a book on the ground. We want to lift this book and the velocity is zero initially. To accelerate the book, I apply force on the book a little more than the gravitational field force and then for the rest of the path I apply force equal to the magnitude of the gravitational field force to keep it's velocity constant. So the net work done on the book by me and the gravitational field equals to zero. Now my question is that even though there is zero work done on the book(net work is zero after all) ,we say that the potential energy is increased for the book.How can we give energy to something when there is no net work done on it?I feel I'm making some logical mistake...

There is a difference between work and energy

Work done on an object is nothing but the increase in kinetic energy of the body due to the applied force

Energy on the other hand, is the capacity to do work. A body at rest possesses energy as well(unless it is at the center of the Earth. A body at rest at any height above the surface of the Earth possesses gravitational potential energy, for example.

From another perspective, any body possessing heat (i.e. unless it is at absolute zero) has heat energy as well, another form of energy. It does not depend whether it is at rest or if work was done on it for some time. It still possesses energy.

I hence hope the difference between energy and work is clear !

• No, objects don't possess potential energy. Systems do. The gravitational potential energy of the "body" and Earth which you described could have any value of potential energy, depending on your reference point. Also, objects do not possess heat. They have internal (or thermal) energy. Heat is energy transferred by differences in temperature. Finally, work can also decrease the kinetic energy of a body. – Bill N Mar 25 '17 at 18:32

Your confusion arises because you have not defined the system that you are dealing with.

System: book alone.
External forces: force on book due to gravitational attraction of the Earth (magnitude $mg$) and you push the book up (magnitude $mg$).

The book starts with a certain upward velocity, rises a vertical height $h$ with the velocity staying constant.

Work done by you $+mgh$ as the force moves in the same direction as the force.
Work done by force on book due to attraction of Earth is $-mgh$ as the force and the displacement are in opposite directions.
Net work done on the book $-mgh + mgh = 0$.
Kinetic energy gained by the book $= 0$.

The book alone cannot have any gravitational potential energy.
It is the system consisting of the book and the Earth which has the gravitational potential energy.

System: book and Earth External forces: force that you provide to life the book up.
The gravitational attraction forces, book attracting Earth and Earth attracting book are internal forces.

The book starts with a certain upward velocity, rises a vertical height $h$ with the velocity staying constant.

Work done by you lift the book is $+mgh$. Change in kinetic energy of system $= 0$.
Increase in the gravitational potential energy of the system $= mgh$.

Update as a result of a comment
There is often a query about the motion of the motion of the book.
The OP does work accelerating the book which then moves at constant velocity and finally the book does work on the OP equal to the amount of work did at the start of the motion of the book and the book stops.
So the net amount of work done by the OP in starting the book moving and stopping the book moving is zero.

• This is correct, but it does ignore part of the OP. In the OP, the book starts at rest so the change in KE is not zero. You might want to include a discussion of that. – garyp Mar 25 '17 at 20:36
• @garyp Thank you for the comment. I have added an update to explain that the net work done by the OP in starting and then stopping the book is zero. – Farcher Mar 25 '17 at 22:34
• Similarly,we can think of a book falling down from some height. – Osman kaçık Mar 26 '17 at 9:43
• System:Book alone Internal forces:- External forces: Gravitational force by the Earth on the book Result:Book gains kinetic energy Now we define book+Earth as our system. System:Earth and book External forces:- Internal forces:Gravitational forces between Earth and book Result:Kinetic energies of the Earth and book cancel each other and there is no net change of energy for the system. Now the question is,where is the potential energy change for the system? I know that it's equal to the kinetic energy gained by the book but how ? – Osman kaçık Mar 26 '17 at 10:11
• If you let the book go both the book and the earth will gain kinetic energy and lose an equal amount of gravitational potential energy. There is no cancelling out of kinetic energy what is equal and opposite for the book and the Earth is the momentum of each. – Farcher Mar 26 '17 at 10:22