The 42nd Period | STEM Education Hub

Use this as a quick reference for the force–energy relationship, reading equilibrium from a U(x) graph, and the potential energy of common systems.

🧭 Plot Summary

Potential energy is energy stored by position rather than motion — and it comes with a strict membership requirement: a system only has potential energy if the objects inside it interact exclusively through conservative forces. There's no such thing as "the potential energy of friction," because friction's work depends on the path taken, not just the start and end points. Potential energy is a scalar, and exactly where you call U = 0 is entirely your own choice — a convenience for the problem at hand, since only ΔU ever shows up in any physical prediction. A conservative force and its potential energy are two sides of the same coin: ΔU = −∫F_cf·dr gets you from force to energy, and in one dimension, F(x) = −dU/dx gets you back — the force always points toward decreasing potential energy. That relationship turns a U(x) graph into a map of a system's equilibria: a local minimum is a stable equilibrium, where a small nudge creates a restoring force pulling the system back, while a local maximum is unstable, where a small nudge creates a force that pushes the system further away. Common systems have their own potential energy formulas worth knowing by heart: an ideal spring stores U_s = ½k(Δx)², gravity between two masses in general stores U_g = −Gm₁m₂/r, and near a planet's surface — where the gravitational field is essentially constant — that simplifies to the familiar ΔU_g = mgΔy. When a system has more than two objects, its total potential energy is simply the sum of the potential energy of every individual pair.

What you'll do in this lesson

Explain what potential energy is, and why it can only ever be defined for objects interacting through conservative forces.
Recognize potential energy as a scalar tied to position, where choosing U = 0 is always a free, convenient decision.
Move between a conservative force and its potential energy in both directions — finding ΔU from F, and finding F(x) from the slope of U(x).
Read a U(x) graph to find stable equilibria (local minima) and unstable equilibria (local maxima), and connect each to the direction of the restoring or repelling force.
Apply the potential energy of common systems: an ideal spring, and gravity both near a planet's surface and in its general, large-scale form.
Find the total potential energy of a system of more than two objects by summing the potential energy of every pair.

Why it matters

Potential energy is the missing piece that turns the work–energy theorem from Lesson 3.2 into full-blown conservation of energy in Lesson 3.4 — and the force–from–slope idea you'll build here, F(x) = −dU/dx, is a reasoning tool the AP exam reuses constantly, especially in Mathematical Routines questions that hand you an unfamiliar U(x) and ask what the system does next.

✅ Self-Check Before You Roll On

Check off each item as you get there. These aren't grades — they're your own signal.

I can explain that a system of two or more objects has potential energy only if those objects interact with each other exclusively through conservative forces.I know potential energy is a scalar quantity tied to the position of objects within a system, and that choosing where U = 0 is always a free decision the observer makes to simplify the analysis — only ΔU ever matters physically.I can relate a conservative force to the potential energy it's associated with: ΔU = −∫F_cf·dr.I can find a conservative force in one dimension from the slope of its potential energy function, F(x) = −dU/dx, and I know the force points toward decreasing potential energy.I can explain what makes an equilibrium stable (a small displacement creates a restoring force back toward equilibrium) versus unstable (a small displacement creates a force that pushes further away).I can identify stable equilibria as local minima and unstable equilibria as local maxima on a graph of U(x).I can apply the elastic potential energy of an ideal spring, U_s = ½k(Δx)².I can apply the general form of gravitational potential energy between two masses, U_g = −Gm₁m₂/r.I can apply — and explain the justification for — the near-surface approximation ΔU_g = mgΔy, and I know it isn't valid far from a planet's surface.I can find the total potential energy of a system with more than two objects by summing the potential energy of every individual pair of objects.

🔬 Deep Dive

Your textbook for this topic: conservative forces and the zero-of-PE choice, the force–energy relationship, equilibrium from U(x) graphs, and the potential energy of common systems — with interactive tools.

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🧭 Guided Practice

Step through worked examples for every essential knowledge statement in 3.3.A. Three examples each, revealed one step at a time.

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📝 Assignment

20 questions covering the whole lesson. Every answer gets feedback right away, and a wrong answer tells you the most likely reason why.

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📌 AP Exam Note

Don't mix up the two gravitational PE formulas: the general form, U_g = −Gm₁m₂/r, is always negative with its zero chosen at infinite separation, while the near-surface approximation, ΔU_g = mgΔy, is only ever a change with whatever zero you pick — they're different tools for different regimes, and the exam likes to test whether you know which one applies.