# Is this thought regarding work and potential energy correct?

I am really confused about this. The potential energy acquired by an object is equal to $mgh$, if we are on any planet, where $g$ is the planet's gravitational acceleration, which directly comes from the definition of work, that is $W = F.s$. But, suppose I am holding a book, and I lift it with a force greater than its weight $mg$, will it acquire more potential energy than before, just because $F$hand $>F$bookweight ?

In addition to this, I want to ask this. Suppose I am standing in space, near a planet, where there in no air resistance, no external disturbance, just me and the planet. An suppose I am standing on the planet, and there is a book on the ground, and there is a gravitational field between us, which is attracting me and the planet. Now, if I pick up the book and take it to a certain height more than its weight on the planet, will it have acquired more potential energy? (Remember there is no drag, no resistance nothing)

If you apply a greater force than $mg$ you will do more work and the book will accelerate, so as well as gaining gravitational potential energy it will also gain kinetic energy.
The work that you do will equal the sum of the gain in gravitational potential energy and the gain in kinetic energy.

Applying a force greater than the book's weight will cause it to accelerate upwards. The books potential energy near the surface of the planet depends, as you said, on the mass of the book, the acceleration due to gravity and the height above the surface.

The formula for gravitational potential energy is actually $U = -G \frac{Mm}{r}$, but near the surface of the planet, we don't worry about changes in in $g$, so it is approximately constant, which is where the formula $U = mgh$ comes from.

So your greater force will cause an acceleration, but it is not the acceleration that increases the book's gravitational potential energy.

Lifting a book from the surface of the planet requires you to do work against the gravitational field of the planet, and the book's potential energy will increase.

Start with $U = mgh$. This indeed comes directly from the work performed by the gravitational force: $$U_{end}-U_{begin} = mg\Delta h = W_{grav} = \vec{F}\vec{s}$$

If the motion and the gravitational force are in the same direction, the work done is positive, which is the same as stating that the potential energy lowers. If motion and the gravitational force are opposite, the work is negative and the potential energy increases.

Now, if you perform a upwards force on the book that is bigger than the gravitational force on the book, not only do you perform positive work on the book, but the gravitational force also performs negative work on the book (the book goes up, so the direction of gravitation and movement is opposite), so the gravitational potential energy will increase.

In short: I think you forgot that not only you performed work, the gravitational force did as well. All together:

$$W_{you}+W_{grav} = \Delta E_{kinetic}$$ or $$W_{you} = \Delta E_{kinetic} + \Delta U$$

From wikipedia:

Potential energy is equal (in magnitude, but negative) to the work done by the gravitational field moving a body to its given position in space from infinity.

So, it isn't matter how much work is done by other forces acting on a body. What is matter is work done by the gravitational field.