Physics C: Mechanics · Unit 3: Work, Energy, and Power · 14 min read · Updated 2026-05-11
Conservation of Energy — AP Physics C: Mechanics
AP Physics C: Mechanics · Unit 3: Work, Energy, and Power · 14 min read
1. Conservative vs Non-Conservative Forces★★☆☆☆⏱ 3 min
A conservative force is defined by path independence: the work it does moving an object between two points depends only on the start and end positions, not the path taken. Equivalently, work done by a conservative force around any closed path is zero. This property allows us to define a unique potential energy function $U$ for any conservative force.
2. The General Conservation of Energy Equation★★☆☆☆⏱ 4 min
The conservation of energy equation is derived directly from the work-energy theorem, which states that total work done on a system equals the change in kinetic energy: $W_{total} = \Delta K$. We split total work into work done by conservative forces and non-conservative forces: $W_c + W_{nc} = \Delta K$. Substituting $W_c = -\Delta U$ (from the definition of potential energy) gives the general equation that works for any system:
W_{nc} = \Delta K + \Delta U = (K_f + U_f) - (K_i + U_i) = \Delta E_{mech}
This is the most useful form for AP problems: the net work done by all non-conservative forces equals the change in total mechanical energy ($E_{mech} = K + U$) of the system. For an isolated (closed) system with no non-conservative work, $W_{nc} = 0$, so mechanical energy is conserved:
K_i + U_i = K_f + U_f
When dissipative forces like friction are present, $W_{nc}$ is negative, so $\Delta E_{mech}$ is negative. Mechanical energy is converted to internal (heat) energy, but total energy (including internal) remains conserved: $K + U + E_{int} = \text{constant}$.
3. Energy Diagrams and Equilibrium Classification★★★☆☆⏱ 4 min
For an object moving in one dimension with a known potential energy function $U(x)$ and constant total energy (isolated system), an energy diagram plots $U(x)$ vs position $x$, with a horizontal line for constant total energy $E_{tot}$. This lets you analyze motion without solving differential equations. The relation between force and potential energy gives:
F(x) = -\frac{dU}{dx}
Force on the object is the negative slope of the $U(x)$ graph. Equilibrium occurs when $F(x) = 0$, so the slope of $U(x)$ is zero ($\frac{dU}{dx} = 0$). Equilibrium is classified by the curvature (second derivative) of $U(x)$:
**Stable equilibrium**: Local minimum of $U(x)$, $\frac{d^2U}{dx^2} > 0$. Displacement creates a restoring force back to equilibrium.
**Unstable equilibrium**: Local maximum of $U(x)$, $\frac{d^2U}{dx^2} < 0$. Displacement creates a force that pushes the object further away.
**Neutral equilibrium**: Flat $U(x)$, $\frac{d^2U}{dx^2} = 0$. No force acts on a displaced object.
Turning points of motion occur where $E_{tot} = U(x)$, because kinetic energy $K = E_{tot} - U(x) = 0$ at these points, so the object stops and reverses direction. Any region where $U(x) > E_{tot}$ is forbidden, since kinetic energy cannot be negative.
4. AP-Style Worked Practice Examples★★★☆☆⏱ 3 min
Common Pitfalls
Why: Students confuse two approaches to energy problems; if gravity is treated as an external force doing work, it should not be included in potential energy, and vice versa.
Why: Students incorrectly extend gravity's path-independence property to non-conservative forces like friction.
Why: Students mix up slope (for equilibrium condition) and curvature (for stability classification) when analyzing energy diagrams.
Why: Students overapply the simpler near-Earth formula to problems where gravity changes significantly with height.
Why: Students confuse velocity direction (which can be negative) with speed (which is always a positive magnitude).
Why: Students often omit the negative sign when calculating $W_{nc}$ for kinetic friction.