Work Done by a Constant Force

When a constant force $\vec F$ acts on a body while the body undergoes a displacement $\vec d$, the work done by that force is defined as the scalar (dot) product of the two vectors:

where $F = |\vec F|$, $d = |\vec d|$, and $\theta$ is the angle between the direction of the force and the direction of the displacement. Because work is a dot product of two vectors, it is itself a pure number (a scalar) -- it has a magnitude but no direction of its own, even though both of the quantities that produced it, force and displacement, are vectors. The SI unit of work is the joule ($\text{J}$), defined as $1\ \text{J} = 1\ \text{N}\cdot\text{m}$: the work done when a force of $1\ \text{newton}$ moves its point of application through $1\ \text{metre}$ in the direction of the force.

The factor $\cos\theta$ makes precise an idea that is intuitively obvious once stated: only the component of the force along the direction of motion contributes to the work done; any component of the force perpendicular to the displacement contributes nothing at all, however large that perpendicular component may be. Three special cases are worth fixing clearly:

• When $\theta = 0^\circ$ (force and displacement in the same direction), $\cos\theta = 1$ and $W = Fd$, its largest possible positive value for the given $F$ and $d$ -- for example, gravity doing work on a freely falling body.
• When $\theta = 90^\circ$ (force perpendicular to displacement), $\cos\theta = 0$ and $W = 0$ -- for example, the normal reaction of a horizontal floor on a body sliding along it does zero work, however large the normal force, because the floor's push is always perpendicular to the body's horizontal motion; similarly, the centripetal force in uniform circular motion, always perpendicular to the instantaneous velocity, does zero work at every instant.
• When $\theta = 180^\circ$ (force directly opposing the displacement), $\cos\theta = -1$ and $W = -Fd$, a negative quantity -- for example, kinetic friction opposing a sliding body's motion, or an applied braking force opposing a moving vehicle, both of which remove kinetic energy from the body rather than adding to it. …