Circular orbits: gravity, energy and angular momentum
You will be able to: Connect gravitational force, circular speed, potential energy and angular momentum.
Why does a satellite in a higher circular orbit move more slowly?
Imagine two small satellites circling the same massive planet. The outer satellite has a larger circular path, but its required orbital speed is lower. Gravity is weaker there, and the balance for circular motion sets a smaller speed.
A useful starting point: When a wheel slips: reason about the contact point →
Words and symbols before equations
- M and m
- Masses of the central body and satellite, with M much larger than m.
- r
- Distance between the centers, not altitude above the surface.
- G
- Universal gravitational constant; GM has units m³/s².
- U and E
- Gravitational potential energy and total mechanical energy K+U; U=0 at infinite separation.
What this picture assumes
Scaled teaching system, not Earth: GM=100 m³/s², satellite m=2 kg, central body radius below 2 m and much larger mass. Gravity only. Each setting is a different circular orbit, not an orbit transfer.
Connect the picture to the physics
Gravity provides the inward force: GMm/r²=mv²/r. Cancel m and one r to get v²=GM/r. This is a circular-orbit result. It assumes gravity alone acts and the central body’s motion is negligible compared with the light satellite’s motion.
Use U=−GMm/r rather than the near-surface approximation mgh. The negative sign comes from choosing zero potential at infinite separation. K=½mv²=GMm/(2r); E=K+U=−GMm/(2r). Within one ideal circular orbit, r, speed, K, U, E and angular-momentum magnitude are constant.
Velocity is perpendicular to the radius, so |L|=mrv=m√(GMr). Across different circular orbits of the same satellite, larger r gives lower speed but larger L and a less-negative E. Moving between these orbits requires an interaction such as thrust; they are not different points on one gravity-only circular path.
A worked example, step by step
Use a simplified gravity model with GM=100 m³/s². A 2 kg satellite follows a circular orbit at r=4 m outside a compact central body. Find v, K, U, E and |L|.
- v=√(100/4)=5 m/s.
- K=½(2)(25)=25 J; U=−100(2)/4=−50 J.
- E=25−50=−25 J.
- |L|=mrv=(2)(4)(5)=40 kg·m²/s. The scaled numbers illustrate the relationships, not an Earth orbit.
Orbital r is center-to-center distance. Increasing a negative energy toward zero is an increase, not a decrease.
Does constant speed in a circular orbit mean zero acceleration?
Compare with an explanation
No. Velocity changes direction, and gravity supplies inward acceleration v²/r.
Predict. Change one thing. Explain.
Compare separate circular orbits at different radii in a scaled gravity model. Predict the direction of change of speed, total energy and angular momentum. The slider does not simulate a satellite migrating between orbits.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
r=4 m; v=5 m/s; K=25 J; U=-50 J; E=-25 J; |L|=40 kg·m²/s. All remain constant within this chosen circular orbit.
Scaled teaching system, not Earth: GM=100 m³/s², satellite m=2 kg, central body radius below 2 m and much larger mass. Gravity only. Each setting is a different circular orbit, not an orbit transfer.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant physical relationship or contact condition 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 questionIn a scaled model, GM=144 m³/s², m=1 kg and circular radius r=4 m. (a) Find speed. (b) Find K and U. (c) Find E and |L|. (d) Predict speed at a separate circular orbit of radius 16 m.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: v=√(144/4)=6 m/s.
- 1 point: K=18 J; U=−36 J.
- 1 point: E=−18 J and |L|=1×4×6=24 kg·m²/s.
- 1 point: 3 m/s, half the speed because radius quadrupled.
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 1Which distance appears in orbital equations?
Center-to-center separation r.
RECALL 2Where is U defined as zero?
At infinite separation.
RECALL 3For a circular orbit, how do K and U relate?
K=−U/2, so E=U/2=−K.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Circular orbits: gravity, energy and angular momentum
- Circular orbit: v=√(GM/r).
- U=−GMm/r; K=GMm/(2r); E=−GMm/(2r).
- Circular |L|=mrv. Gravity-only ideal circular motion keeps all of these constant.
Remember: Orbital r is center-to-center distance. Increasing a negative energy toward zero is an increase, not a decrease.
Conditions: Scaled teaching system, not Earth: GM=100 m³/s², satellite m=2 kg, central body radius below 2 m and much larger mass. Gravity only. Each setting is a different circular orbit, not an orbit transfer.
Refresh Kid · Unit 6 · Objectives 6.6.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 6.6, objectives 6.6.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. Fall-2026 corrections also checked. 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.
Want to work through this with a tutor?
Bring your question about Circular orbits: gravity, energy and angular momentum. Your explanation and answers remain free to access.