AP® Physics C: Mechanics · Unit 3 · Topic 3.2
Conservative and Nonconservative Forces
A conservative force does work that depends only on start and end points, so it gives back all the energy you put in. A nonconservative force like friction depends on the path, and it bleeds energy away as heat that never returns.
Lift a box to a shelf and gravity does negative work on the way up, then positive work of the same size on the way down. The net work over that round trip is zero. That's the signature of a conservative force: the work done depends only on where you start and where you end, not on the route. Because of that path-independence, you can build a potential energy function for it, gravitational PE = mgh, spring PE = (1/2)kx2, that stores the energy and hands it back exactly.
Friction fails this test completely. Push a box across a floor and back to where it started, and friction has done negative work the entire time, on both legs of the trip. Walk it in a loop and the work is never zero; it just keeps draining more. That's the formal way to check: compute the work done by a force around a closed path. Zero means conservative, gravity, springs, electric forces from fixed charges. Nonzero means nonconservative, friction, air resistance, any force that converts mechanical energy into heat or sound. Since the work depends on how far you dragged the box, not just where it ended up, there's no PE function to write down for it.
This distinction is what makes energy conservation useful as a shortcut. When every force in a problem is conservative, total mechanical energy, kinetic plus potential, stays constant, and you can skip straight from start to finish without tracking the motion in between. The moment friction or drag shows up, you need the full energy accounting: initial KE plus PE equals final KE plus PE plus whatever got lost as heat. That lost term is exactly the work done against the nonconservative force, and it's always positive, a one-way withdrawal from the mechanical energy account.
Key terms
- conservative force
- A force whose work on an object depends only on the start and end positions, not the path taken.
- nonconservative force
- A force, like friction or drag, whose work depends on the path length and removes usable mechanical energy as heat.
- closed-loop test
- Checking whether the net work done by a force around any closed path is zero; zero means the force is conservative.
- potential energy function
- A stored-energy expression, like mgh, that exists only for conservative forces because their work is path-independent.
- energy dissipated
- Mechanical energy converted to heat by a nonconservative force, equal to the friction force times distance traveled.
Watch it explained
- How it worksHow does walking a closed loop test whether a force is conservative?
- Worked exampleFind the energy lost to friction when a 5 kg box slides 6 m with a kinetic coefficient of 0.30.
- Common mistakeWhy isn't there a potential energy function that goes with friction?
- Going furtherWhat changes about the work when the path from A to B gets longer but the endpoints stay put?
Check yourself
End of topic.