Work

Work ($W$) is Energy transferred by a force acting through a displacement. For a constant force,

$$W=\vec{F}\cdot \Delta \vec{r}=F\Delta r\cos \theta$$

The dot product matters: only the component of the force along the displacement does work.

Sign of work

• Positive work: force component points with the motion; kinetic energy tends to increase.
• Negative work: force component points against the motion; kinetic energy tends to decrease.
• Zero work: force is perpendicular to displacement, or there is no displacement.

Example: when a ball falls, gravity does positive work because force and displacement both point downward. When the ball is thrown upward, gravity does negative work because force is downward while displacement is upward.

Work-energy theorem

Net work changes kinetic energy:

$$W_{net}=\Delta K$$

Work is not “the opposite of Potential energy”. Instead, potential energy is a way of storing energy so that a conservative force can later do work.

Unit: Joule.

Problem habit

When using work, identify the displacement first, then ask which force components lie along that displacement. A normal force on a sliding block often does zero work because it is perpendicular to motion. Friction usually does negative work because it points opposite the relative motion.