Count each interacting pair once
You will be able to: Sum unique pair energies and avoid double counting interactions.
How do you add potential energy when there are more than two objects?
Three small masses have three gravitational pairs: AB, AC and BC. Adding “the energy of A” and then “the energy of B” carelessly can count the same AB interaction twice. A pair list keeps the bookkeeping clear.
A useful starting point: Gravity beyond the constant-field approximation →
Words and symbols before equations
- Pair energy U_ij
- Potential energy associated with one interaction between objects i and j.
- Unique pair
- An unordered pair: AB and BA describe the same interaction.
- Superposition of energies
- Add the scalar energies of independent pair interactions.
- Σ over i < j
- A compact instruction to count each pair once.
What this picture assumes
Three equal point masses at 0, 1 and c reference lengths on a line. Energy unit is Gm²/L. Points never coincide. External supports maintain the shown configuration; this is not a time simulation.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- U_AB = −1, U_AC = -0.3333, U_BC = -0.5 in units Gm²/L. Total U = -1.833 Gm²/L. AB stays fixed while C moves.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
For separated point masses, U_total = −G Σ over i<j (m_i m_j/r_ij), choosing all pair energies zero at infinite separation. Three objects give three terms, not six. Energy is a scalar sum; you do not add these terms as force vectors.
If one mass moves while two others stay fixed, only the pairs involving the moving mass change energy. The fixed pair still contributes to the total value but cancels in ΔU. External supports may be required to hold the other masses fixed.
Pairwise addition also applies when a system contains several ideal springs: add each spring’s ½kx² once. Do not assume a single “height” or “separation” can describe every internal interaction of a multi-object system.
A worked example, step by step
Three equal point masses m lie on a line at 0, L and 3L, with L > 0. Express total gravitational potential energy in units of Gm²/L.
- List unique pairs: AB separated by L, BC by 2L, and AC by 3L.
- Their energies are −Gm²/L, −Gm²/(2L), and −Gm²/(3L).
- Add once per pair: U = −(1 + ½ + ⅓)Gm²/L = −(11/6)Gm²/L.
- The result is negative with the chosen reference. Doubling every separation halves the magnitude of U.
Potential energy belongs to each interaction once. AB and BA are not two independent gravitational pairs.
If the AB distance stays fixed, can U_total still change?
Compare with an explanation
Yes. AC and BC may change even while the AB term stays constant.
Predict. Change one thing. Explain.
Move the third mass while keeping A at 0 and B at 1 reference length. Predict which two pair energies change and which stays fixed. Read the energy bars separately from the position diagram.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
U_AB = −1, U_AC = -0.3333, U_BC = -0.5 in units Gm²/L. Total U = -1.833 Gm²/L. AB stays fixed while C moves.
Three equal point masses at 0, 1 and c reference lengths on a line. Energy unit is Gm²/L. Points never coincide. External supports maintain the shown configuration; this is not a time simulation.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant work, system boundary, energy or calculus relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionThree equal masses sit at 0, L and 2L. List pair energies, calculate total U, then state the new total if every separation doubles.
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Compare with the answer and four-point rubric
- 1 point: AB contributes −Gm²/L.
- 1 point: BC contributes −Gm²/L and AC contributes −Gm²/(2L).
- 1 point: Total U = −(5/2)Gm²/L.
- 1 point: After all distances double, total U = −(5/4)Gm²/L, half its previous value.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1How many terms are needed for three interacting objects?
Three unique pair terms.
RECALL 2Why are energy terms added without directions?
Potential energy is scalar, even when its associated forces have different directions.
RECALL 3What terms cancel when only one mass moves?
Terms between masses whose relative configurations stay unchanged.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Count each interacting pair once
- U_total = Σ over unique pairs U_ij.
- Gravity: U_ij = −Gm_i m_j/r_ij.
- For n objects, there are n(n − 1)/2 distinct pairs.
Remember: Potential energy belongs to each interaction once. AB and BA are not two independent gravitational pairs.
Conditions: Three equal point masses at 0, 1 and c reference lengths on a line. Energy unit is Gm²/L. Points never coincide. External supports maintain the shown configuration; this is not a time simulation.
Refresh Kid · AP Physics C: Mechanics Unit 3 (official Unit 3) · Objectives 3.3.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 3.3, objectives 3.3.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026 alongside the Fall 2026 clarifications. This is Mechanics Unit 3: Work, Energy, and Power. The unit covers Topics 3.1–3.5. Calculus connects work to force integrals, force to potential-energy derivatives, and power to the rate of energy transfer. Models distinguish object-only and multi-object systems; translational models exclude rotational energy unless explicitly noted. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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