Physics C: Mechanics · Work, Energy, and Power (Unit 3) · 14 min read · Updated 2026-05-11
Forces and Potential Energy — AP Physics C: Mechanics
AP Physics C: Mechanics · Work, Energy, and Power (Unit 3) · 14 min read
1. 1D Relation Between Force and Potential Energy★★☆☆☆⏱ 4 min
The relationship between conservative force and potential energy comes directly from the definition of potential energy change: the change in potential energy of a system equals the negative work done by the conservative force.
To find the instantaneous force at position $x$, take the derivative of both sides using the Fundamental Theorem of Calculus, which gives the core 1D formula.
F_x(x) = - \frac{dU}{dx}
The negative sign has a clear physical meaning: a conservative force always points in the direction of decreasing potential energy. If $dU/dx > 0$, force points toward decreasing $x$, which matches this rule.
Exam tip: AP MCQs almost always include a distractor option with the correct magnitude but opposite sign. Always confirm your sign matches the rule that force points toward lower potential energy before selecting your answer.
2. Equilibrium and Stability Classification★★★☆☆⏱ 4 min
Equilibrium occurs when the net force on a particle is zero. Using the $F_x = -dU/dx$ relation, this means the equilibrium condition is $\frac{dU}{dx} = 0$: the slope of the $U(x)$ graph is zero at any equilibrium point. We classify equilibrium into three types based on the curvature of $U(x)$ at the equilibrium point:
**Stable equilibrium**: $U(x)$ is at a local minimum, so $\frac{d^2U}{dx^2} > 0$. Any displacement creates a restoring force pointing back to equilibrium. Intuitively, this is a ball at the bottom of a valley.
**Unstable equilibrium**: $U(x)$ is at a local maximum, so $\frac{d^2U}{dx^2} < 0$. Any displacement creates a force that pushes the particle further away from equilibrium, like a ball at the top of a hill.
**Neutral equilibrium**: $U(x)$ is flat around the point, so there is no restoring or repelling force for any displacement, like a ball on flat ground.
Exam tip: For MCQ questions that give you a graph of $U(x)$ (not an algebraic function), use the 'ball-on-a-hill' rule to classify stability instantly, no calculation required.
3. Conservative Forces in Multiple Dimensions★★★★☆⏱ 3 min
For motion in 2 or 3 dimensions, the force-potential energy relation extends using the gradient operator. The force vector is the negative gradient of the potential energy function.
Each component of the force is the negative partial derivative of $U$ with respect to that coordinate. When taking a partial derivative with respect to one coordinate, treat all other coordinates as constants. Physically, the force vector always points in the direction of maximum decrease of potential energy, extending the 1D intuition to multiple dimensions. AP Physics C almost exclusively tests 2D cases for this topic.
Exam tip: Don’t forget to differentiate linear terms (like $3x$ in this example) when calculating partial derivatives. Students often omit these simple terms, leading to incorrect component values.
4. AP-Style Practice Worked Examples★★★★☆⏱ 3 min
Common Pitfalls
Why: Students memorize the relation as 'force is the derivative of potential energy' and omit the sign derived from the work-energy relation for conservative forces.
Why: Students confuse the condition for equilibrium (first derivative zero) with the condition for stability (second derivative sign).
Why: Students forget that $U$ depends on multiple variables, so the derivative with respect to $x$ only accounts for variation in $x$, holding $y$ constant.
Why: Students confuse the value of $U$ at equilibrium with its curvature, since potential energy can have any zero reference point.
Why: Students forget that potential energy is only defined for conservative forces.