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What is a system, and how does treating an object or group of objects as a single point at its center of mass simplify the analysis of motion?

Topic 2.1 Systems and Center of Mass: define a system and its center of mass, and explain how the center of mass of a system moves in response to external forces.

A focused answer to AP Physics 1 Topic 2.1, covering what a system is, internal versus external forces, the center of mass and how to locate it, and how the center of mass responds only to external forces, with full worked examples.

Generated by Claude Opus 4.89 min answerUpdated 2026-06-04

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this topic is asking
Systems, internal and external forces
The center of mass
How the center of mass moves
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What this topic is asking

The College Board (Topic 2.1) wants you to define a system, distinguish internal from external forces, and use the center of mass as the single point whose motion represents the system as a whole. The central result is that the center of mass responds only to external forces, which is why we can treat an extended object, or even a group of objects, as a single particle in dynamics problems.

Systems, internal and external forces

Choosing the system is a decision you make, and it determines which forces count as external. If you treat two colliding carts as one system, the forces they exert on each other are internal; if you analyze just one cart, that same contact force becomes external. Internal forces always come in Newton's-third-law pairs within the system, so they cancel when you consider the system as a whole.

The center of mass

For two objects, the center of mass is always on the line between them, nearer the more massive one. For a uniform rod it is the midpoint; for a uniform disk it is the center. Locating the center of mass lets you replace a complicated extended body with a single point carrying all its mass.

How the center of mass moves

The reason the center of mass is so useful is the rule governing its motion:

F_{net,\,external} = M_{total}\, a_{cm}

The center of mass accelerates exactly as a single particle of the total mass would under the net external force. Internal forces never appear, because each is cancelled by its third-law partner inside the system. So a wrench-throwing astronaut, a bursting firework, or two blocks shoved apart by a spring all share one feature: while only internal forces act, the center of mass keeps doing whatever it was doing (staying at rest or moving at constant velocity), even as the pieces scatter.

This is why dynamics problems can treat a car, a person, or a planet as a point. Whatever internal happenings occur (the engine's pistons, the person's muscles, the planet's churning interior), the overall translational motion is governed solely by the external forces acting on the center of mass. When a high-jumper arches over a bar, parts of the body follow complicated paths, but the center of mass traces a simple parabola set by gravity alone, because gravity is the only external force. Recognizing which forces are internal to your chosen system, and therefore irrelevant to the center-of-mass motion, is a powerful simplification that recurs throughout Unit 2 and again in momentum problems.

Locating a center of mass and predicting its motion

A $3.0$ kg cart sits at $x = 1.0$ m and a $1.0$ kg cart at $x = 5.0$ m on a frictionless track, initially at rest. A spring between them is released. Find the center of mass and state where it is after the spring fires.

step 1 Apply the center-of-mass formula

$x_{cm} = \dfrac{m_1 x_1 + m_2 x_2}{m_1 + m_2} = \dfrac{(3.0)(1.0) + (1.0)(5.0)}{3.0 + 1.0} = \dfrac{3.0 + 5.0}{4.0} = 2.0$ m.

step 2 Identify the forces

The spring force acts between the two carts, so it is internal to the two-cart system. The track is frictionless and horizontal, so there is no net external horizontal force.

step 3 Apply the center-of-mass rule

With zero net external force, $a_{cm} = 0$ . The center of mass does not accelerate, and it started at rest, so it stays put.

step 4 Conclusion

After the spring fires, the carts move apart (the lighter one faster), but the center of mass remains at $x = 2.0$ m. It sits closer to the heavier $3.0$ kg cart, as expected.

Try this

Q1. A $4.0$ kg mass is at $x = 0$ and a $4.0$ kg mass at $x = 6.0$ m. Calculate the center of mass. [2 points]

Cue. Equal masses: $x_{cm} = \dfrac{(4.0)(0) + (4.0)(6.0)}{8.0} = 3.0$ m, the midpoint.

Q2. State whether a force between two parts of a chosen system can change the system's center-of-mass velocity. [1 point]

Cue. No; internal forces cannot accelerate the center of mass.

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2024 (style)3 marksSection II (short FRQ, quantitative). Two blocks lie on a frictionless surface along the

x

-axis: a

2.0

kg block at

x = 0

and a

6.0

kg block at

x = 4.0

m. (a) Calculate the position of the center of mass. (b) The two blocks are connected by a compressed spring that is released, pushing them apart. Explain what happens to the center of mass and why.

Show worked answer →

A 3-point FRQ on locating the center of mass and reasoning about internal forces.

(a) Center of mass (2 points): $x_{cm} = \dfrac{m_1 x_1 + m_2 x_2}{m_1 + m_2} = \dfrac{(2.0)(0) + (6.0)(4.0)}{2.0 + 6.0} = \dfrac{24}{8.0} = 3.0$ m. (1 point) correct formula, (1 point) correct value, closer to the heavier block.
(b) Explain (1 point): the spring force is internal to the two-block system, so it cannot move the center of mass. With no external horizontal force, the center of mass stays at rest at $x = 3.0$ m even as the blocks fly apart.

Markers reward the mass-weighted average and the recognition that internal forces do not move the center of mass.

AP 2022 (style)1 marksSection I (multiple choice). An astronaut floating in deep space throws a wrench. While the wrench is in flight, what happens to the center of mass of the astronaut-plus-wrench system? (A) it accelerates toward the wrench (B) it remains at rest or moves at constant velocity (C) it accelerates toward the astronaut (D) it speeds up. Justify your reasoning.

Show worked answer →

A 1-point conceptual MCQ. The answer is (B).

The force the astronaut exerts on the wrench (and the reaction on the astronaut) is internal to the chosen system, and there are no external forces in deep space. With zero net external force, the center of mass does not accelerate: it stays at rest if it started at rest, or moves at constant velocity. The astronaut and wrench move apart, but their center of mass is unaffected. The trap is thinking the throwing force moves the whole system; internal forces cannot.

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Sources & how we know this

AP Physics 1: Algebra-Based Course and Exam Description — College Board (2024)