Calculating Center of Mass with Summation A coordinate system can be chosen arbitrarily to determine the center of mass by computing the weighted average of positions for discrete masses. For a square with equal masses at its corners, symmetry causes the individual contributions to cancel, placing the center of mass at the geometric center. The formula x₍cm₎ = (Σ Mᵢxᵢ)/(Σ Mᵢ) and its y counterpart precisely capture this concept when the coordinates are chosen appropriately.
Extending to Continuous Distributions via Integration For objects without distinct particles, the discrete summation becomes an integral over the mass distribution. Each infinitesimal mass element contributes according to its position, making the center of mass the ratio of the integral of position times dM to the total mass. This formulation adapts to irregular shapes and extends naturally into three dimensions by integrating over all coordinate components.
Finding Equilibrium Through Balancing The center of mass of an irregular object can be identified by finding its balance point on a supporting surface. Balancing the object ensures that the contact point with the table lies directly beneath the center of mass, and rotating the item produces intersecting vertical lines that pinpoint its location. Any slight tilt generates gravitational torque, emphasizing that proper alignment of the center of mass is crucial for maintaining stability.