Connect local height energy to inverse-square gravity

1. Orient: Connect local height energy to inverse-square gravity

Potential energy belongs to a system of interacting objects and its configuration, not to an isolated object without its interaction partner. For two nonoverlapping spherical masses, U=−GMm/r when zero is chosen at infinite separation. Finite-separation values are negative because positive work is needed to separate the attractive pair to infinity quasistatically.

2. Explain and derive

Only energy differences enter ordinary conservation calculations. Moving from ri to rf gives ΔU=GMm(1/ri−1/rf); work by gravity is−ΔU. An outward move has rf>ri, so U increases toward zero and gravity does negative work. Choosing a different additive zero changes U values but not ΔU or predicted motion.

Near a planet’s surface over a small height change, the gravitational field is nearly constant and ΔU≈mgΔh. It is an approximation to the inverse-square relation, not a formula to apply unchanged at any altitude. Using the initial field for a large outward move overestimates ΔU because the field weakens as the object moves away.

For more than two gravitating objects, sum the potential energy of each distinct pair once. For three equal masses at the corners of an equilateral triangle of side r, there are three pairs, giving total U=−3 Gm²/r. Counting each pair from both members would incorrectly double the energy. The simple widget varies one test object relative to a fixed spherical body; the pair-sum extension is a separate written construction.

3. Work an example

The hypothetical body has R=10 m and GM=1000 m³/s²; the test mass is 2 kg. Moving from 10 to 11 m changes U from−200 to−181.818 J, an increase 18.182 J. Initial g=10 m/s² gives local m g Δr=20 J, about 10% high. These rounded parameters describe an invented teaching body, not Earth.

4. Predict and investigate

Predict first: When an attractive pair moves farther apart, does negative potential energy become more negative or less negative?

At initial radius ratio 1, compare outward increments 0.1,0.5 and 1 body radius. Compute exact and local-constant-field energy changes and their discrepancy. Repeat from initial radius ratio 3 and explain why the same absolute increment is a smaller relative change there. Use the lesson controls and retain the settings with your evidence.

5. Transfer and explain

Try a new practice case

For a hypothetical pair with GMm=120 J m, separation changes from 2 m to 3 m. Find the potential-energy increase.

Your numerical result (J) Check number

Reveal worked check (practice, not mastery)

This extra practice is not saved and does not award mastery. Record your first answer, explanation and whether you revealed help in the lesson notebook. Use the lesson’s saved question separately.

6. Reflect and retain evidence

How would a third interacting body change the potential-energy ledger? List the distinct pairs before writing their terms.

Retain a first attempt, signed ledger, units, system boundary, assumptions and a changed-case prediction. Public worked checks are practice, not independent mastery evidence. Synthetic outputs are not physical measurements.

Framework and source notes

Original teaching prose and numerical cases. College Board AP Physics 1 CED, Unit 3 topic mapping: 3.3. Checked 30 October 2026; this is a content cross-reference, not AP authorization.