Physics C: Mechanics · CED Unit 3: Work, Energy, Power (14-17% of exam score) · 18 min read · Updated 2026-05-11
Work, Energy, Power (Calculus-Based) — AP Physics C: Mechanics
AP Physics C: Mechanics · CED Unit 3: Work, Energy, Power (14-17% of exam score) · 18 min read
1. Work as a Line Integral★★☆☆☆⏱ 4 min
Work quantifies energy transferred to a system by a force acting over a displacement. For AP Physics C, we use a calculus definition that works for variable forces and curved paths, unlike the algebra-based $W=Fd\cos\theta$ for constant forces.
W = \int_{r_i}^{r_f} \vec{F} \cdot d\vec{r} = \int_{r_i}^{r_f} F dr \cos\theta
Exam tip: Always remember to take the dot product — perpendicular force components contribute zero work
2. The Work-Energy Theorem★★☆☆☆⏱ 4 min
The work-energy theorem is one of the most powerful tools in mechanics, because it relates net work done on an object directly to its change in kinetic energy, no integration of acceleration required. It holds for all forces, constant or variable, on any path.
3. Conservative Forces and Potential Energy★★★☆☆⏱ 4 min
For conservative forces, work done between two points is independent of the path taken, and work done over a closed path is zero. This property allows us to define a potential energy function that stores energy associated with the conservative force. Non-conservative forces like friction have path-dependent work, so no potential energy can be defined for them.
Power is the rate of energy transfer or work done. Unlike average power, which is total work divided by total time, instantaneous power measures the rate of work done at a specific moment in time, which is what is most commonly asked on AP Physics C exams.
5. Equilibrium Analysis from Potential Energy★★★☆☆⏱ 3 min
Equilibrium occurs when net force on a system is zero. Since force is the negative derivative of potential energy for conservative systems, we can find and classify equilibrium points directly from the potential energy function:
**Stable equilibrium**: $U(x)$ is at a local minimum, $\frac{d^2U}{dx^2} > 0$. A small displacement produces a restoring force back to equilibrium.
**Unstable equilibrium**: $U(x)$ is at a local maximum, $\frac{d^2U}{dx^2} < 0$. A small displacement produces a force that pushes the system further from equilibrium.
**Neutral equilibrium**: $U(x)$ is constant over a region, $\frac{d^2U}{dx^2} = 0$. A small displacement leaves the system at rest in the new position.
Common Pitfalls
Why: Habits from algebra-based AP Physics 1 lead students to forget variable forces require integration
Why: Students mix up work done on the system vs work done by the system
Why: Students confuse $P_{avg}=W/\Delta t$ with $P_{inst}=\vec{F}\cdot\vec{v}$
Why: Students do not separate conservative and non-conservative work correctly
Why: Students forget a zero first derivative only confirms an equilibrium point, not its type