Work Done by Constant Forces

Previously, we introduced work as $W = Fd$ when force and displacement are in the same direction. However, forces often act at angles to the displacement. In such cases, only the component of force parallel to the displacement does work and transfers energy.

The dot product of force and displacement vectors captures this precisely:

$$W = \vec{F} \cdot \vec{d}$$

Recall from our study of dot products that for vectors $\vec{F}$ and $\vec{d}$ separated by angle $\theta$:

$$W = \vec{F} \cdot \vec{d} = F d \cos\theta$$

This formula reveals that only the component parallel to displacement, $F_{\parallel} = F \cos\theta$, contributes to work. The perpendicular component $F_{\perp} = F \sin\theta$ does no work because it acts at a right angle to the motion. It may change the direction of motion or press the object against a surface, but it does not transfer energy along the displacement.

Consider a person pushing a lawn mower with force $F$ at angle $\theta$ below the horizontal. The mower moves horizontally through displacement $d$. We decompose the applied force:

$$F_x = F \cos\theta$$ $$F_y = F \sin\theta$$