Work

Work is a transfer, not something an object carries around. The AP Physics 1 CED defines it as the amount of energy transferred into or out of a system by a force exerted on that system over a distance, and the equation sheet prints $W = F_{\parallel} d = F d \cos\theta$.

The cosine carries the sign, with $\theta$ measured between the force and the displacement.

• At $\theta = 0^\circ$ the transfer is fully positive, $W = Fd$.
• At $\theta = 90^\circ$ it is zero. A perpendicular force can change the direction of the motion without changing the kinetic energy, which is why the normal force on a level floor, the string tension on a conical pendulum and the inward force in circular motion all do no work whatsoever.
• At $\theta = 180^\circ$ it is negative, $W = -Fd$. Kinetic friction on a sliding block is the standard case, and the minus sign means energy is leaving.

Work is a scalar quantity that may be positive, negative or zero, measured in joules, where one joule is one newton meter. Negative work is not less work than none, it is energy travelling the other way.

For a force that varies, work is the area under a graph of the parallel component against displacement.

Adding up the signed works and setting the total equal to $\Delta K$ is the work energy theorem, and that procedure is the job of the work energy theorem guide.