Work

🧭 Plot Summary

Work is the bridge connecting force — everything you built in Unit 2 — to energy — everything the rest of Unit 3 is about. Work is the amount of energy a force transfers into or out of a system as that system moves through some distance. Conservative forces (like gravity and springs) do work that only depends on where you start and end, and that work is zero over any closed loop — which is exactly why potential energy can only ever be defined for a conservative force. Nonconservative forces (most commonly friction and air resistance) do work that depends on the whole path taken. Only the force component parallel to an object's displacement ever changes its energy; a perpendicular component can redirect an object's motion without touching its kinetic energy at all. For a force that changes along the way, finding work means reaching for the work integral, built from the dot product between force and displacement — and for systems that can change shape (a walking person, a car's spinning tires), the point where a force is actually applied can move a different distance than the system's center of mass, which is exactly when the full "system" model is required instead of the simpler "object" model. Once you know the total work done on an object, the work–energy theorem hands you its change in kinetic energy directly.

What you'll do in this lesson

• Define work, and distinguish conservative forces from nonconservative forces, including where potential energy does and doesn't apply.
• Recognize work as a scalar that can be positive, negative, or zero.
• Compute work from a variable force using the work integral and the dot product.
• Explain why only the parallel component of a force does work, and how a perpendicular component can redirect motion without changing kinetic energy.
• Apply the work–energy theorem to relate net work to a change in kinetic energy — and recognize when a deformable system, not just a single object, must be modeled.
• Find the energy dissipated by friction.
• Find work as the area under a force–vs–displacement graph.

Why it matters

The AP exam's very first free-response question every year is the Mathematical Routines question — and work is one of its favorite subjects. This lesson's integral, dot product, work–energy theorem, and object-vs-system distinction are directly, explicitly testable as multi-step derivations, and they set up every remaining lesson in Unit 3.

✅ Self-Check Before You Roll On

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