A force on its own does nothing measurable. Make it move something, and it starts spending — and what it spends is work. $W = F d \cos\theta$, read aloud W equals F d cosine theta. $W$ is the work in joules; $F$ is the constant force in newtons; $d$ is the distance the thing actually moved, in metres; and $\theta$ — theta — is the angle between the direction of the force and the direction of the motion, in degrees. One joule is one newton pushed through one metre, which is why a joule and a newton-metre are the same algebra.

The cosine is the whole subtlety, and it is best learned at its limits. Push straight along the motion and $\theta = 0$, $\cos 0 = 1$, and every newton counts. Push at an angle and only the along-the-motion share is paid in; the rest presses sideways and buys you nothing. Push square across the motion, $\theta = 90^\circ$, $\cos 90^\circ = 0$, and the work is exactly zero. Carry a toolbox across a level floor at a steady walk and you do no work on it whatever — your arm holds it up, the motion is horizontal, and the two are at right angles. Your muscles disagree loudly, but physics is not taking questions on that.

Two habits. First, $d$ is the distance MOVED, not the distance you walked around the shop — push a stuck crate for an hour and if it never budges, the work is zero. Second, watch the sign: a force opposing the motion has $\theta = 180^\circ$, $\cos 180^\circ = -1$, and it does NEGATIVE work — it takes energy out. Friction lives there permanently, and the rest of this chapter is about what it takes.