Center of mass

Definition

The center of mass is the point at which the total mass of a system can be considered to be concentrated for the purpose of analyzing translational motion. It is found by taking the weighted average of the positions of all mass elements in a body or system.

5 Must Know Facts For Your Next Test

1. The formula for the center of mass in one dimension is $\bar{x} = \frac{\sum m_i x_i}{\sum m_i}$, where $m_i$ are masses and $x_i$ are their positions.
2. In two dimensions, the center of mass coordinates $(\bar{x}, \bar{y})$ are calculated using $\bar{x} = \frac{\iint x \delta(x,y) \, dA}{M}$ and $\bar{y} = \frac{\iint y \delta(x,y) \, dA}{M}$, where $M$ is the total mass.
3. For continuous bodies, integrals replace summations in the center of mass calculations.
4. The concept of moments is crucial for determining centers of mass; moments measure how far a point is from an axis and how much force it exerts at that distance.
5. Symmetry can simplify finding the center of mass; if a body has uniform density and symmetrical geometry, its center of mass lies on its axis or plane of symmetry.

Review Questions

• How do you calculate the center of mass for a system with discrete masses along a line?
• What role does symmetry play in determining the center of mass?
• Explain how integration is used to find the center of mass for continuous distributions.

Related terms

Moment:

A measure of the tendency to cause motion about an axis; mathematically defined as force times distance from an axis.

Centroid:

The geometric center of a plane figure or solid body, often coinciding with its center of mass if density is uniform.

Moment Arm: \(d = r \sin(\theta)\), where \(r\) is distance from pivot point to force's line, and \(θ\) is angle between them; measures moment's effectiveness.