The 42nd Period | STEM Education Hub

Use this as a quick reference for which systems have potential energy, choosing U = 0, spring and gravitational potential energy, and adding up the pairs in a larger system.

🧭 Plot Summary

Potential energy is energy stored in the arrangement of a system. It needs at least two objects interacting through a conservative force, such as gravity or an ideal spring, so friction and air resistance never have one. It is a scalar that depends only on where the objects are, not on the route they took to get there, and it belongs to the system rather than to any single object. Where you call the potential energy zero is your choice, and it changes the value of U but never the change ΔU. Three equations cover the common cases: an ideal spring stores U_s = ½kx²; near a planet's surface a height change gives ΔU_g = mgΔh; and for two spherical masses at any separation, U_g = −Gm₁m₂/r, which is negative because zero is defined at infinite separation. In a system with more than two objects, you add the potential energy of every pair.

What you'll do in this lesson

Identify which systems have potential energy, and why only conservative interactions qualify.
Treat potential energy as a scalar that belongs to a system and depends on the positions of its objects.
Choose the zero of potential energy for convenience, and show that ΔU is independent of that choice.
Calculate elastic, near-surface gravitational, and general gravitational potential energy.
Add up the potential energy of every pair of objects in a system with three or more objects.

Why it matters

Kinetic energy (Lesson 3.1) and potential energy are the two kinds of mechanical energy, and work (Lesson 3.2) is what moves energy between them and the surroundings. In Lesson 3.4 you'll put them together in conservation of energy, where the trick of choosing a convenient U = 0 and tracking only ΔU makes hard problems easy. Spring and gravity energy also come back in oscillations and orbits later in the course.

✅ 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 potential energy belongs to a system of two or more objects that interact through conservative forces, and why friction and air resistance have no potential energy.I can explain that potential energy is a scalar that depends on the positions of the objects in the system, not on the path taken.I can choose where U = 0 for a problem, and I know that U changes with that choice but ΔU does not.I can calculate the elastic potential energy of an ideal spring with U_s = ½kx², and explain why doubling the stretch quadruples the energy.I can use ΔU_g = mgΔh near a planet's surface, and I know why it only works when g is nearly constant.I can use U_g = −Gm₁m₂/r for two spherical masses, and explain why the values are negative.I can find the total potential energy of a system with more than two objects by adding the potential energy of each pair.

🔬 Deep Dive

Your textbook for this topic: which systems have potential energy, choosing U = 0, springs, gravity near and far from a planet, and multi-object systems.

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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

Potential energy questions are almost always about changes. Pick a convenient zero, find U at the start and the end, and subtract. Don't use mgΔh when the height change is comparable to the planet's radius: use −Gm₁m₂/r instead. For springs, remember x is squared, so doubling the stretch gives four times the energy, and stretching or compressing by the same amount stores the same energy. With three or more objects, count every pair.