I am now teaching Calculus of several variables this semester. In apllications of integrals, the problem of finding the work done in moving an object under a force $F$ is one of the most common problems.
Let's assume that an object is moving from a point A to a point B (on a horizontal line) under a constant force $F$ pointing an angle $\alpha$ (w.r.t. the horizontal line). Then in most textbooks (Calculus), the work done in this case is defined by $W=|F||AB|\cos(\alpha)$. By accepting this definition, then one can meaningfully find a way to evaluate the work done in moving an object along a curve (in 2D or 3D) and under a non-constant force (this leads to line integrals).
When teaching this, the most uncomfortable fact is "why do we have such a definition of the work" in the first place. Since the word "work" should say something, it might give confusion.
My question: how do we give intuition to the definition of work above? For example, how do we give an explanation for the fact that the work is negative if $\alpha$ is greater than 90 degrees?
Thanks so much for any hints.