Applications of Integration to Find Centers of Mass

Rotational equilibrium provides a natural framework for defining the center of mass of a system. For a plank balanced on a pivot with two unequal masses, equilibrium is achieved when the net torque about the pivot is zero. Torque is defined as the product of a force and its perpendicular distance from the pivot. When the torques due to all forces cancel, the pivot coincides with the center of mass of the system.

For a system composed of several discrete point masses, the center of mass lies at the position where the weighted contributions of all masses balance. Mathematically, the center of mass x_cm is given by

where m_i is the mass of the i^th object, and x_i is its position measured from a chosen origin. The numerator represents the total moment of the system, defined as the sum of mass–position products, while the denominator is the total mass. This formulation ensures that the net torque about the center of mass is zero.

The same concept extends to continuous systems, such as a thin rod that can be modeled as a one-dimensional object with a linear mass density (x). In this case, the rod is treated as being composed of an infinite number of infinitesimal mass elements. Each small element has a mass dm = (x)dx, where dx is an infinitesimal length of the rod.

The moment of each element about one end of the rod is xdm. The total moment is obtained by summing all such elemental moments using a Riemann sum, which becomes an integral in the limit as dx tends to 0:

The total mass of the rod is

Taking the ratio of the total moment to the total mass yields the center of mass of the rod,

This integral formulation generalizes the discrete case and provides a consistent definition of the center of mass for continuous systems.