Center of Mass Reference Cheat Sheet

Key Facts

• For particles on a line, the center of mass is $x_{cm}=\frac{\sum m_i x_i}{\sum m_i}$.
• In two dimensions, the center of mass coordinates are $x_{cm}=\frac{\sum m_i x_i}{\sum m_i}$ and $y_{cm}=\frac{\sum m_i y_i}{\sum m_i}$.
• For a continuous object, the center of mass is found with $x_{cm}=\frac{1}{M}\int x\,dm$, $y_{cm}=\frac{1}{M}\int y\,dm$, and $z_{cm}=\frac{1}{M}\int z\,dm$.
• The total mass of a system is $M=\sum m_i$ for particles and $M=\int dm$ for continuous objects.
• If an object has uniform density and symmetry, its center of mass lies on every line, plane, or axis of symmetry.
• The velocity of the center of mass is $\vec{v}_{cm}=\frac{\sum m_i \vec{v}_i}{M}$.
• The acceleration of the center of mass obeys $\sum \vec{F}_{ext}=M\vec{a}_{cm}$.
• For an isolated system with $\sum \vec{F}_{ext}=0$, the center of mass moves with constant velocity, so $\vec{v}_{cm}$ is constant.

Vocabulary

Center of Mass

The point where the mass of a system can be treated as concentrated for analyzing translational motion.

Particle System

A collection of separate masses whose center of mass is found by taking a mass-weighted average of their positions.

Continuous Body

An object with mass spread continuously through space, often requiring integrals to find its center of mass.

Uniform Density

A condition where mass is evenly distributed, so density is constant throughout the object.

External Force

A force exerted on a system by something outside the chosen system boundary.

Symmetry Axis

A line or axis that divides a uniform object into matching mass distributions, causing the center of mass to lie on it.