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2.1 Systems and Center of Mass

Syllabus
2024
Topic
2.1
Level
—

Learning objectives

2.1A—Describe the properties and interactions of a systemDescribe the properties and interactions of a system.• System properties are determined by the interactions between objects within the system.• If the properties or interactions of the constituent objects within a system are not important in modeling the behavior of the macroscopic system, the system can itself be treated as a single object.• Systems may allow interactions between constituent parts of the system and the environment, which may result in the transfer of energy or mass.• Individual objects within a chosen system may behave differently from each other as well as from the system as a whole.• The internal structure of a system affects the analysis of that system.• As variables external to a system are changed, the system’s substructure may change.2.1B—Describe the location of a system’s center of mass with respect to the system’s constituent partsDescribe the location of a system’s center of mass with respect to the system’s constituent parts.• For systems with symmetrical mass distributions, the center of mass is located on lines of symmetry.• The location of a system’s center of mass along a given axis can be calculated using the equation• A system can be modeled as a singular object that is located at the system’s center of mass. BOUNDARY STATEMENT AP Physics 1 only expects students to calculate the center of mass for systems of five or fewer particles arranged in a two-dimensional configuration or for systems that are highly symmetrical. AP Physics 1: Algebra-Based Course and Exam Description Force and Translational Dynamics UNIT 2

A system begins with a chosen boundary

Choose the system

A system is the object or collection of objects chosen for analysis. Draw an imaginary boundary around it: objects inside are the system's constituent parts; everything outside is the environment. The system's properties depend on how its parts interact.

Read the boundary

Ask about the model Inside the boundary Across the boundary
What is tracked? Parts and their internal interactions Transfers between system and environment
What may change? Parts may move or behave differently from one another Energy or mass may enter or leave
Can details be ignored? Yes, if internal structure does not affect the macroscopic behavior No, if an external change alters the system's substructure or behavior

Decide how much detail matters

Example: choose a cart plus the blocks fixed on it as one system. If the blocks remain fixed and their arrangement does not matter for the motion being studied, model the whole system as one object. If the blocks slide relative to the cart, the internal structure matters and the parts must be analyzed.

Do not assume isolation

A system is not automatically isolated, rigid, or uniform. The boundary is a modeling choice. State what is inside it, identify relevant interactions, and reconsider the single-object model when external conditions change the system's substructure.

Center of mass is a mass-weighted position

Interpret the model

The center of mass is the position at which a system can be represented as one object for translational analysis. It is a mass-weighted location: larger masses pull the center of mass closer to themselves. In the formula below, mim_i is each particle's mass, x⃗i\vec{x}_i is its signed position from the chosen origin, and ∑imi\sum_i m_i is the total mass.

Weight each position by mass

x⃗cm=∑imix⃗i∑imi\vec{x}_{cm}=\frac{\sum_i m_i\vec{x}_i}{\sum_i m_i}

Calculate and interpret

Hypothetical example: a 2.0 kg2.0\,\mathrm{kg} particle is at x=0 mx=0\,\mathrm{m} and a 3.0 kg3.0\,\mathrm{kg} particle is at x=4.0 mx=4.0\,\mathrm{m}.

xcm=(2.0 kg)(0 m)+(3.0 kg)(4.0 m)2.0 kg+3.0 kg=2.4 m.x_{cm}=\dfrac{(2.0\,\mathrm{kg})(0\,\mathrm{m})+(3.0\,\mathrm{kg})(4.0\,\mathrm{m})}{2.0\,\mathrm{kg}+3.0\,\mathrm{kg}}=2.4\,\mathrm{m}.

The result lies between the particles and closer to the heavier 3.0 kg3.0\,\mathrm{kg} particle, as expected.

Use symmetry and respect scope

For a symmetrical mass distribution, the center of mass lies on every applicable line of symmetry; it need not coincide with material. It is not generally the geometric midpoint. AP Physics 1 calculations are limited to five or fewer particles in a two-dimensional arrangement or to highly symmetrical systems.