Work

In everyday speech, "work" describes almost any physical or mental effort -- pushing against an immovable wall until you are exhausted certainly feels like work. Physics defines work far more narrowly and precisely: work is done by a force only when that force produces a displacement of the point on which it acts, and only the part of the force that lies along the direction of that displacement counts.

If a force $\vec{F}$ acts on a body and displaces it by $d\vec{r}$, the work done is defined as the scalar (dot) product of the two vectors:

where $\theta$ is the angle between $\vec F$ and $d\vec r$. Because it is a dot product of two vectors, work itself is a scalar -- it has magnitude but no direction, and can be positive, negative, or zero. Its SI unit is the newton-metre, also called the joule (J), and its dimensional formula is $[ML^2T^{-2}]$.

This precise definition immediately produces three cases in which the physics definition of work departs sharply from everyday intuition -- work done is zero:

1. When the force is zero ($F = 0$): a body coasting on a frictionless horizontal surface needs no force to keep moving, so no work is done to sustain that motion.
2. When the displacement is zero ($dr = 0$): pushing as hard as you like against a rigid wall that does not move means zero work is done on the wall, no matter how tired you become.
3. When the force and the displacement are mutually perpendicular ($\theta = 90^\circ$): gravity does no work on a body moving purely horizontally, and the centripetal force in circular motion, always directed toward the centre, does no work on the orbiting body because it is always perpendicular to the (tangential) velocity. …