Name: center of mass in discrete and continuous systems
Uploaded: 2026-01-12T12:24:32.000Z
Duration: 1 min 30 s
Channel: JoVE

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&#8217;s center of mass.For several discrete masses, the center of mass equals the sum of the moments &#8212;defined as mass times position&#8212; 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&#8217;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.

applications of integration to find centers of mass

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 xcm is given by<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:32.75%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/40591/40591_Equation_1_editor_40591.svg" alt="center of mass calculation; equation: x_cm = &#931;(m_i * x_i) / &#931;m_i; physics formula" data-altupdatedat="1776139283000" data-alt="40591_Equation_1.svg" /></figure>where mi is the mass of the ith object, and xi is its position measured from a chosen origin. The numerator represents the total moment of the system, defined as the sum of mass&#8211;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:<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:30.08%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/40591/40591_Equation_2_editor_40591.svg" alt="Static equilibrium equation; total moment integration formula; educational diagram." data-altupdatedat="1776139283000" data-alt="40591_Equation_2.svg" /></figure>The total mass of the rod is<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:25.73%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/40591/40591_Equation_3_editor_40591.svg" alt="Calculating total mass, equation, &#8747;&#955;(x)dx, mathematical integration, analysis, physics concept." data-altupdatedat="1776139283000" data-alt="40591_Equation_3.svg" /></figure>Taking the ratio of the total moment to the total mass yields the center of mass of the rod,<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:39.77%;"><img style="aspect-ratio:300/72;" src="https://cloudfront.jove.com/files/ftp_upload/40591/40591_Equation_4_editor_40591.svg" alt="Centre of mass calculation equation, integral formula, educational physics diagram." data-altupdatedat="1776139283000" data-alt="40591_Equation_4.svg" /></figure>This integral formulation generalizes the discrete case and provides a consistent definition of the center of mass for continuous systems.

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, ...

applications-of-integration-to-find-centers-of-mass

Education

JoVE Core

Calculus

Unit

Application of Techniques of Integration

KEY TERMS AND CONCEPTS

center of mass in discrete and continuous systems

Center of mass can be found from rotational equilibrium and from integration. A balanced plank on a pivot shows the idea clearly. When the net torque about the pivot is zero, the system is in balance. Torque is the force on an object times its perpendicular distance from the pivot. At that point, the pivot lies at the center of mass.

For a set of discrete point masses, the center of mass is the position where the masses balance by weight. Each mass contributes its mass times its position, measured from a chosen origin. The center of mass x cm equals the total moment divided by the total mass. Here, the moment is the sum of all mass-position products. This setup makes the net torque about the center of mass equal to zero.

The same idea also works for a continuous object such as a thin rod. The rod can be treated as a one-dimensional object with a linear mass density, ρ(x). Small pieces of the rod have mass dm = ρ(x)dx, where dx is a tiny length segment. The moment of each piece about one end is xdm. Adding all of these pieces with a Riemann sum leads to an integral as dx approaches 0.

The total mass of the rod is found by integrating the mass elements. The center of mass is then the ratio of the total moment to the total mass. This integral form extends the discrete formula to continuous systems. It gives a consistent way to locate the center of mass for a rod or any similar object.

Center of Mass in Discrete and Continuous Systems

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key terms and concepts

Learn how rotational equilibrium, torque, and integration are used to find the center of mass for point masses and a thin rod.

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Center of Mass in Discrete and Continuous Systems

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