Why is the potential energy equal to the negative integral of a force?

Question

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 1

Force is a vector quantity. In any problem involving vector quantities, a sign convention will either be specified, or must be defined.

Example: An object is raised to a height of 10 meters and has a weight (gravitational force) of 10 newtons. The potential energy is 100 N-m with respect to the ground, which is a typical reference point. Implicit in this is a sign convention that "up" is positive.

Sign conventions can be completely arbitrary, but--within a given problem--they have to be consistent.