Applications of Integration to Find Centers of Mass

Rotational equilibrium occurs when net torque about a pivot is zero. The pivot aligns with the system's center of mass at this point. Torque equals force times distance from the pivot. When all torques balance, the system is in equilibrium, and the pivot location defines where the center of mass is positioned.

The center of mass equals the total moment divided by total mass. The moment is the sum of each mass multiplied by its position. Mathematically, xcm = (m₁x₁ + m₂x₂ + ... + mₙxₙ) / (m₁ + m₂ + ... + mₙ). This formula ensures net torque about the center of mass is zero.

A continuous rod is treated as composed of infinite infinitesimal mass elements. Each element has mass dm = ρ(x)dx, where ρ is linear mass density and dx is infinitesimal length. The moment of each element is x·dm. Summing all elemental moments using a Riemann sum yields an integral that determines the rod's center of mass.

As the size of small mass elements approaches zero, the discrete Riemann sum becomes continuous. The limit of summing infinitesimal contributions defines an integral. This mathematical transition allows exact calculation of total moment for continuous objects like rods, replacing approximation with precise integration.

Linear mass density ρ(x) describes mass per unit length along a rod. Each infinitesimal element's mass is calculated as dm = ρ(x)dx. This density function allows integration over the rod's length to find total mass and total moment, enabling center of mass calculation for non-uniform continuous systems.

For discrete masses, moment is mass times position summed for all objects. For continuous systems, each infinitesimal element contributes moment x·dm. Integration of these elemental moments replaces discrete summation, providing a unified framework. This extension shows how applications of integration to probability density functions and other continuous distributions follow similar principles.

Center of mass is defined as the ratio of total moment to total mass. Total moment represents the weighted sum of all mass-position products, while total mass is the sum of all individual masses. This ratio identifies the single point where the system's mass is effectively concentrated for rotational purposes.