AP Physics C: Mechanics · Unit 3: Work, Energy, and Power · Lesson 3.3

Deep Dive: Potential Energy

🔬 Deep Dive
This is your textbook for this topic. Take your time. Read it more than once.
Back to Lesson 3.3
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What Is Potential Energy?

A system of two or more objects has potential energy only when those objects interact exclusively through conservative forces — directly building on Lesson 3.2's distinction. Like kinetic energy, potential energy is a scalar, but instead of depending on motion, it depends on position — specifically, the configuration of the objects within the system.

🔑There's no universal "correct" place to set potential energy equal to zero — that choice belongs entirely to whoever is analyzing the situation, made purely to simplify the math. Move that zero point anywhere you like, and every physically meaningful quantity (forces, changes in energy, final speeds) comes out identical.
Choice A: floor = U = 0U = mghChoice B: tabletop = U = 0U = mg(h − h(table))Same physical situation — either zero-point choice gives correct, consistent physics.

Formally, potential energy connects to conservative force through an integral — the mirror image of the work integral from Lesson 3.2:

ΔU = −∫(a to b) F⃗c(r⃗) · dr⃗
💡The minus sign matters: positive work done by a conservative force corresponds to a decrease in potential energy. That's exactly what you'd expect — a ball falling has gravity doing positive work on it while its gravitational potential energy drops.
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Force from the Slope of U(x)

Run the integral relationship backward, and something elegant falls out: in one dimension, the conservative force on a system is simply the negative slope of its potential energy function.

F(x) = −dU/dx
🔑This force always points toward decreasing potential energy — exactly like a ball rolling downhill on an actual U(x) landscape. Steep slopes mean strong forces; flat regions (zero slope) mean zero force.

Explore this directly below with a potential energy curve shaped like two valleys separated by a hill.

A potential energy curve with two valleys and a hill between them. Drag through x — the tangent line shows the local slope, and F = −dU/dx follows directly from it.

position x3.00 m
Ux →
dU/dx ≈ 1.80 →  F(x) = −dU/dx ≈ -1.80 N

The force points toward decreasing potential energy — downhill on this graph, exactly like a ball rolling on a real hill.

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Equilibrium from Potential Energy Graphs

The force-from-slope relationship turns any U(x) graph into a complete map of a system's equilibrium points.

Stable equilibrium

A small displacement produces a force pushing back toward equilibrium. Occurs at a local minimum of U(x).

Unstable equilibrium

A small displacement produces a force pushing further away from equilibrium. Occurs at a local maximum of U(x).

🔑A marble in a bowl (stable) versus a marble balanced on top of a dome (unstable) is the classic physical picture — but the math behind both is identical: check whether you're sitting at a valley or a peak on the U(x) graph.

Revisit the tool above — as you scan across the curve, watch how the equilibrium classification switches between the two valleys (stable) and the hill between them (unstable).

ExampleGuided Example — Classifying Equilibria

A system's potential energy is given by U(x) = x⁴ − 8x² (in joules, x in meters). Find all equilibrium positions and classify each as stable or unstable.

Step 1 — Find force as a function of position
F(x) = −dU/dx = −(4x³ − 16x) = −4x³ + 16x
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Potential Energy of Common Systems

Elastic potential energy

For an ideal spring, stretched or compressed a distance Δx from its equilibrium length:

Us = ½k(Δx)²

Gravitational potential energy — general form

For two roughly spherical masses (planets, moons, stars):

Ug = −Gm₁m₂/r
⚠️This is negative — and that's correct, not a mistake. As r increases toward infinity, Ug rises toward zero, its maximum possible value. The deeper two masses sit in each other's gravitational well, the more negative Ug becomes.

Gravitational potential energy — near-surface approximation

Near a planet's surface, where the gravitational field is nearly constant, the general form simplifies to the familiar linear approximation:

ΔUg = mgΔy

Compare the two gravitational forms directly below — and see exactly how far you can climb before the approximation starts to break down.

A 70 kg object rising above Earth's surface. Compare the exact gravitational potential energy change to the near-surface linear approximation — and see exactly where they start to diverge.

height (m)1,000 m
ΔUheight →
General (blue): 687.2 kJApproximation (gold, mgΔy): 686.0 kJDifference: 0.18%

Near the surface, the two curves sit almost exactly on top of each other — the approximation is excellent. Push the height out toward hundreds of kilometers and the curves visibly begin to separate, since g itself isn't really constant that far up.

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Potential Energy of Multi-Object Systems

For a system with more than two objects, total potential energy is just the sum of the potential energy of every individual pair of objects within that system.

ExampleWorked Example — Three-Mass Gravitational System

Three point masses (2 kg, 3 kg, 4 kg) sit at the corners of an equilateral triangle with 2 m sides. Find the total gravitational potential energy of the system.

← Back to Lesson 3.3Ready for 3.4? Conservation of Energy is where kinetic and potential energy finally trade places, formally.