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Module 5 -- Center of Mass: definition - PER wiki

# Module 5 -- Center of Mass: definition

#### Learning Goals

After you finish this module you should be able to:

• Calculate the center of mass of a system of several point particles.
• Calculate the center of mass of a system of several extended object.
• Explain how to calculate the center of mass of a rigid object.

#### Introduction

When describing the translational motion of an extended object, we can represent the object as a point particle, as if all its mass was concentrated at one point. This point, located at the average location of the distributed mass, is called the center of mass. For example, a uniform rod has its center of mass right at its middle. In this module we will learn how to calculate the center of mass of an object or a group of objects.

Moreover, we will show that the overall translational motion of a system can be described in terms of the center of mass of the system. The system under consideration can be a group of particles, such as the molecules of a gas in a container, or a group of extended objects such as the boy and the girl playing on the icy lake from the previous unit.

The athlete shown in the movie below is an example of a non-rigid extended object. The motion of all the different parts of his body can be complicated, but the motion of the athlete's center of mass (red solid path) is the same as the motion of a point particle with a mass equal to the athlete's mass. The two dash lines represent the paths of other parts of the extended object (the head and right foot). While the athlete is in air, the external force acting on the athlete is gravity. As a result, his center of mass follows a projectile motion.

We will see that the location of the center of mass can be either fixed relative to an object, as in the example of the athlete, or it can be a point in space (e.g. a discrete distribution of masses as in examples 1 and 2 below).

### The center of mass of a system of 2 point particles

Consider a system formed by two point particles of masses m1 and m2 located at positions $\vec{r}_1$ and $\vec{r}_2$ measured with respect to point Q as shown in the figure below. The position of the center of mass with respect to the coordinate system with origin at point Q is obtained as:

The center of mass, cm in the figure, represents the mean location of the mass of the system. It is located along a line between the particles and closer to the more massive particle. In the example below m1 > m2.

### The center of mass of a system of N particles

The concept of center of mass can be extended to a system of N particles with masses m1, m2,...mN located at the positions $\vec{r}_1, \vec{r}_2...\vec{r}_N$:

$\vec{R}_{cm}=\frac{m_1\vec{r}_1+m_2\vec{r}_2+...}{m_! + m_2 +...} = \frac{\sum_{i=1}^Nm_i\vec{r}_1}{\sum_{i=1}^Nm_i}$

### The center of mass of a rigid object

A rigid object can be thought as a system containing a very large number of particles and the summation of the expression above becomes an integral over the continuous distribution of mass:

$\vec{R}_{cm}=\frac{\int\vec{r}dm}{\int dm}=\frac{\int\vec{r}dm}{M}$

where M is the total mass of the object.

To obtain the position of the center of mass of a rigid object it is necessary to know how is the mass distributed in volume and then solve three integrals, one for each component of the position vector:

$X_{cm}=\frac{\int x\; dm}{M} \mbox{, } Y_{cm}=\frac{\int y\; dm}{M} \mbox{, } Z_{cm}=\frac{\int z\; dm}{M}$

Center of mass of a symmetric object: If the object is symmetric and the mass is distributed uniformly throughout the object's volume, then the center of mass lies on an axis of symmetry. For example, the center of mass of a uniform sphere or cube lies at its geometric center.

Center of mass of an asymmetric object: If an object has an irregular shape the center of mass can be determined experimentally as shown in the video below by TSG @ MIT Physics

This video is provided by the Technical Services Group of the MIT Department of Physics.
Used with permission.

#### The center of mass and the center of gravity of a rigid object

The gravitational force acts on each element of mass within a rigid object. Adding the forces on each element, we obtain the total gravitational force $M\vec{g}$ on the object of mass M. This force is applied at a given point in the object that is called the center of gravity of the object. If the acceleration of gravity $\vec{g}$ is the same throughout the object, then the center of gravity coincides with the center of mass. When an object is close to the Earth surface this is a very good approximation. If an object is pivoted on its center of gravity it balances in any orientation (no rotation). This is not true when an object is far from the Earth surface, for example an artificial satellite.