Choose a system and locate its center of mass
You will be able to: Choose a system boundary and calculate a mass-weighted position.
Where is the representative position of several objects?
Two carts sit at the 0 m and 4 m marks. One has mass 1 kg and the other 3 kg. Their center of mass is at 3 m, closer to the heavier cart.
A useful starting point: Differentiate motion component by component →
Words and symbols before equations
- System
- The objects selected for an analysis; everything outside is the environment.
- Internal interaction
- An interaction between two objects inside the chosen boundary.
- Center of mass x_cm
- Mass-weighted average position, in metres.
- Σ, sigma
- Add the indicated quantity over all parts of the system.
What this picture assumes
Two point masses at x = 0 and 4 m. The marker is a weighted position; the connector is a location guide, not a massive rod.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- Masses 1 and 3 kg; total 4 kg. Center of mass at x = 3 m between 0 and 4 m.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
Choose what you need to explain before drawing the boundary. For one cart, the other cart’s push is external. For both carts together, their mutual pushes are internal. A system can exchange energy or mass with its surroundings; the constant-total-mass examples here do not describe a rocket losing fuel.
For point-like parts, x_cm = Σmᵢxᵢ/Σmᵢ. Multiply each mass by its position, add those products, then divide by total mass. Symmetry puts the center at the geometric midpoint only for an appropriate symmetric mass distribution.
Each part can move differently while the system has one center-of-mass position. The point model can represent translational motion, but it does not describe rotation, stretching or rearrangement inside the system. Internal structure matters whenever the question depends on it.
A worked example, step by step
Find x_cm for 2 kg at x = 1 m and 4 kg at x = 7 m.
- Use the same origin for both positions.
- Total mass M = 2 + 4 = 6 kg.
- Weighted sum = 2(1) + 4(7) = 30 kg·m; x_cm = 30/6 = 5 m.
- The result lies between 1 and 7 m and closer to the 4 kg mass, a useful reasonableness check.
The center of mass is not necessarily a material point, and it is not generally the midpoint.
Can a ring’s center of mass lie in empty space?
Compare with an explanation
Yes. Symmetry places it at the center even if no material is there.
Predict. Change one thing. Explain.
Hold the carts at 0 and 4 m. Increase only the right-hand mass. Predict which way the center moves, then explain why it remains between the carts.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Masses 1 and 3 kg; total 4 kg. Center of mass at x = 3 m between 0 and 4 m.
Two point masses at x = 0 and 4 m. The marker is a weighted position; the connector is a location guide, not a massive rod.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant force, system boundary, acceleration 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 questionTwo masses, 1 kg at 0 m and 3 kg at 8 m, form a system. Find total mass, center of mass, and explain how the interaction classification changes if only the first object is selected.
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Compare with the answer and four-point rubric
- 1 point: Total mass is 4 kg.
- 1 point: Weighted sum is 24 kg·m.
- 1 point: x_cm = 6 m, closer to the heavier object.
- 1 point: Their mutual interaction is internal for both together but external to the first object alone.
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 1Why does the heavier object pull x_cm toward itself?
Its position receives a larger mass weight in the average.
RECALL 2Does a system boundary have to be a physical wall?
No. It is a choice made for the analysis.
RECALL 3When is a point model insufficient?
When internal shape, rotation or deformation is essential to the question.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Choose a system and locate its center of mass
- x_cm = Σmᵢxᵢ/M; M = Σmᵢ.
- Apply the same weighted average separately to y and z.
- Whether a force is internal or external depends on the chosen system.
Remember: The center of mass is not necessarily a material point, and it is not generally the midpoint.
Conditions: Two point masses at x = 0 and 4 m. The marker is a weighted position; the connector is a location guide, not a massive rod.
Refresh Kid · AP Physics C: Mechanics Unit 2 (official Unit 2) · Objectives 2.1.A; 2.1.B · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 2.1, objectives 2.1.A; 2.1.B. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026 alongside the Fall 2026 clarifications. This is Mechanics Unit 2: Force and Translational Dynamics. The unit covers Topics 2.1–2.10. Calculus is introduced where it is needed for continuous mass and velocity-dependent forces. Shell theorem is applied without requiring a proof; spring combinations are purely series or purely parallel. 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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