Center of Mass

Definition

The center of mass $(\bar{x}, \bar{y})$ is the point at which the entire mass of a system can be considered concentrated. It is the balance point of the system. $\bar{x} = \frac{M_y}{M}, \quad \bar{y} = \frac{M_x}{M}$

• How to read: “The x-coordinate of the center of mass, x-bar, is equal to the moment about the y-axis divided by the total mass M; and the y-coordinate, y-bar, is equal to the moment about the x-axis divided by the total mass M.”
• Meaning: Each coordinate of the center of mass is the first moment about the perpendicular axis divided by total mass.

Why It Matters

Locating the center of mass is critical for the stability of any physical structure, from skyscrapers to spacecraft. If ignored, uneven weight distribution leads to structural failure, uncontrolled rotation, or inefficient energy use. It is the mathematical key to balance in a physical world.

Core Concepts

• Mass ($M$): The total amount of matter in the system (integral of density).

• Moment ($M_x, M_y$): A measure of the tendency of a mass to cause rotation about an axis. $M_y = \int x \, dm$ and $M_x = \int y \, dm$.

  • How to read: “The moment about the y-axis is the integral of x with respect to mass; and the moment about the x-axis is the integral of y with respect to mass.”
  • Meaning: First moments weight each mass element by its distance from the respective axis.

• Centroid: If density is constant, the center of mass depends only on geometry and is called the centroid.

Connected Concepts

• Moments of Inertia
• Graphs of Functions: Properties
• Library of Functions, Piecewise Functions