Applications of Integration to Find Centers of Mass

Consider a plank balanced on a pivot with two unequal masses. The system achieves rotational equilibrium when the net torque is zero.

Torque is the force times the distance from the pivot. In rotational equilibrium, the pivot is aligned with the system’s center of mass.

For several discrete masses, the center of mass equals the sum of the moments —defined as mass times position— due to each mass divided by the total mass.

This concept also applies to a continuous system, such as a thin rod treated as a one-dimensional object with a linear mass density. The center of mass is given by the ratio of the rod’s total moment to its total mass.

Here, the rod is treated as composed of an infinite number of small mass elements. Each small element has a mass equal to the linear density multiplied by a tiny length.

The moment of each element equals its distance from one of the ends of the rod multiplied by its small mass.

To calculate the total moment of the rod, these small elemental moments are summed using a Riemann sum.

As the size of these small elements approaches zero, the Riemann sum transitions into an integral.

Taking the ratio of the total moment to the total mass gives the center of mass of the rod.