# Mechanics: Chapter 9 – Introduction to Work

Work is the dot product of force and displacement. Force and displacement are vectors. Since the dot product returns a scalar, Work is a scalar quantity.

• $W = F \cdot s = Fscos(\theta)$

Where $W$ is the Work done, $F$ is the Force, $s$ is the displacement and $\theta$ is the angle between the direction of the force and the direction of the displacement. You can also calculate the dot product by expressing the the Force and displacement in terms of perpendicular unit vectors $\hat{i} \hat{j} \hat{k}$ etc. In which case, the dot project is just

• $(A_{x}\hat{i} + A_{y}\hat{j}) \cdot (B_{x}\hat{i} + B_{y}\hat{j}) = A_{x}B_{x} + A_{y}B_{y}$

When the force varies with displacement s, the work done is the integral of Force with respect to ds.

• $W = \int F \cdot ds$