To my understanding the work done on an object is defined mathematically as:
$$W = \vec{F}\cdot\vec{S}=|\vec{F}||\vec{S}|cos\theta$$
This, I understand. My problem is that I don't understand that if the angle $\theta$ is 90 degrees how can the work done by $\vec{F}$ on the object is zero. For example; say you have a particle and the direction of the displacement is directly to the right, and you also have a force vector acting on the particle that is straight up(like the normal force on a box that is standing on a flat surface). How is it possible that the force vector is not doing any work? Must the particle not take a different route because of the force vector acting upward on the particle, like if you add the vectors together?
There has to be something wrong with my reasoning, but what is it?
Best Answer
Because it's not any work, but the work done by a force that produces a displacement.
In the scenario you describe, somehow that force is not doing any work on the particle. This could be because the particle is restricted by another force to not go perpendicular and then the sum of forces in the perpendicular direction is zero.
In the second scenario, with the box and the normal force, it's the same. That force doesn't do any work since in the direction of that force there is zero movement. Which is analogous to say that the cosine of the angle between the displacement and such force is 90°.