KAHIBARO
Table of Contents
What the center of mass means for particles
When a system contains several particles, it is often useful to describe the whole system by one special point called the center of mass. This point is the mass weighted average position of all the particles. It tells us where the system is balanced in a mathematical sense.
For a collection of particles with masses $m_1, m_2, m_3, \dots$ located at positions $\vec r_1, \vec r_2, \vec r_3, \dots$, the center of mass position vector is
$$
\vec R_{\mathrm{cm}} = \frac{\sum_i m_i \vec r_i}{\sum_i m_i}
$$
The denominator is the total mass of the system,
$$
M = \sum_i m_i
$$
so the formula is often written as
$$
\vec R_{\mathrm{cm}} = \frac{1}{M}\sum_i m_i \vec r_i
$$
This definition applies to any set of particles, whether they are at rest or moving.
The center of mass of particles is found by taking the weighted average of position, using mass as the weight:
$$
\vec R_{\mathrm{cm}} = \frac{\sum_i m_i \vec r_i}{\sum_i m_i}
$$
Heavier particles influence the center of mass more strongly than lighter particles.
Coordinates of the center of mass
In Cartesian coordinates, the vector formula becomes separate formulas for each coordinate. If the particles have positions $(x_i, y_i, z_i)$, then the center of mass has coordinates
$$
x_{\mathrm{cm}} = \frac{\sum_i m_i x_i}{\sum_i m_i}
$$
$$
y_{\mathrm{cm}} = \frac{\sum_i m_i y_i}{\sum_i m_i}
$$
$$
z_{\mathrm{cm}} = \frac{\sum_i m_i z_i}{\sum_i m_i}
$$
This means you can calculate each coordinate independently.
If all particles lie on a line, only one coordinate is needed:
$$
x_{\mathrm{cm}} = \frac{\sum_i m_i x_i}{\sum_i m_i}
$$
Two particle system
The simplest case is two particles. Suppose masses $m_1$ and $m_2$ are located at positions $\vec r_1$ and $\vec r_2$. Then
$$
\vec R_{\mathrm{cm}} = \frac{m_1 \vec r_1 + m_2 \vec r_2}{m_1 + m_2}
$$
If the particles lie on the $x$ axis, then
$$
x_{\mathrm{cm}} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}
$$
If the masses are equal, the center of mass lies halfway between them. If one mass is larger, the center of mass lies closer to the heavier one.
Example on a line
Consider three particles on the $x$ axis:
| Particle | Mass | Position |
|---|---|---|
| 1 | $2\,\mathrm{kg}$ | $1\,\mathrm{m}$ |
| 2 | $3\,\mathrm{kg}$ | $4\,\mathrm{m}$ |
| 3 | $5\,\mathrm{kg}$ | $7\,\mathrm{m}$ |
The total mass is
$$
M = 2 + 3 + 5 = 10\,\mathrm{kg}
$$
The center of mass is
$$
x_{\mathrm{cm}} = \frac{(2)(1) + (3)(4) + (5)(7)}{10}
$$
$$
x_{\mathrm{cm}} = \frac{2 + 12 + 35}{10} = \frac{49}{10} = 4.9\,\mathrm{m}
$$
The result is closer to the particle of mass $5\,\mathrm{kg}$ because that mass has the greatest weight in the average.
Example in two dimensions
Now consider two particles in a plane:
| Particle | Mass | Position |
|---|---|---|
| 1 | $1\,\mathrm{kg}$ | $(2,1)\,\mathrm{m}$ |
| 2 | $3\,\mathrm{kg}$ | $(6,5)\,\mathrm{m}$ |
The total mass is
$$
M = 1 + 3 = 4\,\mathrm{kg}
$$
The $x$ coordinate of the center of mass is
$$
x_{\mathrm{cm}} = \frac{(1)(2) + (3)(6)}{4} = \frac{2 + 18}{4} = 5\,\mathrm{m}
$$
The $y$ coordinate is
$$
y_{\mathrm{cm}} = \frac{(1)(1) + (3)(5)}{4} = \frac{1 + 15}{4} = 4\,\mathrm{m}
$$
So the center of mass is
$$
(x_{\mathrm{cm}}, y_{\mathrm{cm}}) = (5,4)\,\mathrm{m}
$$
Physical interpretation
The center of mass is the point where the mass distribution of the particles can be treated as concentrated for many purposes. It is not required to coincide with an actual particle. It can lie between particles, and in some systems it may even lie in empty space.
For particle systems, the center of mass is especially useful because it gives one position that represents the whole set of particles. Instead of tracking only where each particle is located, we can also track where the system as a whole is located.
The center of mass is a geometric point defined by mass distribution. It does not have to be the location of any actual particle.
Special cases
If all particles have the same mass, say each mass is $m$, then the factor $m$ cancels:
$$
\vec R_{\mathrm{cm}} = \frac{m\sum_i \vec r_i}{Nm} = \frac{1}{N}\sum_i \vec r_i
$$
where $N$ is the number of particles. In this case, the center of mass is just the ordinary average of the position vectors.
For two equal masses on a line,
$$
x_{\mathrm{cm}} = \frac{x_1 + x_2}{2}
$$
which is the midpoint.
Useful calculation steps
To find the center of mass of particles, it helps to proceed in a clear order. First choose a coordinate system. Then write the position of each particle in that system. Multiply each coordinate by the corresponding mass. Add those products. Finally divide by the total mass.
| Step | Action |
|---|---|
| 1 | Choose coordinates |
| 2 | List each mass and position |
| 3 | Compute $m_i x_i$, and also $m_i y_i$, $m_i z_i$ if needed |
| 4 | Add the mass weighted coordinates |
| 5 | Divide by total mass |
Common mistakes
A common mistake is to divide by the number of particles instead of the total mass. That works only when all masses are equal. Another common mistake is to ignore the sign of coordinates. Positions to the left of the origin or below the axis must be entered as negative values when appropriate.
Do not use
$$
\frac{x_1 + x_2 + \cdots + x_n}{n}
$$
unless all particle masses are equal.
In general, the correct formula is
$$
x_{\mathrm{cm}} = \frac{\sum_i m_i x_i}{\sum_i m_i}
$$
Relation to symmetry
Symmetry can often help locate the center of mass quickly. If identical particles are arranged symmetrically about a point or axis, the center of mass must lie on that point or axis. For example, two equal masses placed equally far on opposite sides of the origin have
$$
x_{\mathrm{cm}} = 0
$$
because their contributions cancel.
This symmetry idea is a shortcut, but it comes from the same mass weighted average formula.
Final idea
For a system of particles, the center of mass is the single point that represents how the masses are distributed in space. Mathematically, it is found by averaging positions with mass weights. In one dimension, two dimensions, or three dimensions, the same principle applies, and it becomes a powerful tool for describing the motion and structure of many particle systems.