Why is the potential energy equals to the negative integral of a force? I am really confused with this negative sign. For example, why there is a negative sign in the gravitational potential energy and what does it mean?
I read that the negative sign means you do the same force but in the opposite direction. Doesn't that mean the object shouldn't move?
Answer
When you do conservative work on an object, the work you do is equal to the negative change in potential energy $W_c = - \Delta U$. As an example, if you lift an object against Earth's gravity, the work will be $-mgh$. Gravity is doing work on the object by pulling it towards the Earth, but since you are pushing it in the other direction, the work you do on the box (and therefore the force) is negative. The field does negative work when you increase a particle's potential energy.
Mathematically, it is just that $F=\frac{dW}{dx}$, which means that if the work is conservative, then $F=\frac{-dU}{dx}$, since $W_c = - \Delta U$. Then $-dU = Fdx$, so $U = - \int F dx$.
We can also say that work is negative when the force and displacement are in opposite directions, since $W = \vec F \cdot d\vec x = Fdxcos\phi$. When $\phi=\pi$, then $\cos\phi = -1$. An example of this conceptually is friction. An object sliding down a plane has kinetic friction acting on it. The friction is in the direction (up the ramp) opposite to the object's motion/displacement. So we say that the friction force is doing negative work.
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