Center of Mass: Definition, Equation, How to Find (w/ Examples)

Have you ever seen one of those toy birds that is able to balance on your fingertip by its beak without tipping over, as if by magic? It isn’t magic that allows the bird to balance at all, but the simple physics associated with center of mass.

Understanding the physics behind the center of mass allows you not only to understand conservation of momentum and other related physics, but can also inform stability and dynamics in the sports you play, as well as allow you to perform some creative balancing acts.

Definition of Center of Mass

An object’s center of mass, sometimes also called the center of gravity, can be thought of as the point where the total mass of an object or a system can be treated as a point mass. In certain situations, external forces can be treated as though they are acting on the center of mass of the object.

For the toy bird balancing on your fingertip, the center of mass is at its beak. This might seem wrong at first, which is why the act of balancing appears magical. Indeed, for a bird sitting on a branch, its center of mass is somewhere in its body. But the balancing bird toy often has weighted wings that span outward and forward, causing it to balance differently.

The center of mass can be determined for a single object – such as the balancing bird – or it can be computed for a system of several objects, as you will see in a later section.

Center of Mass for a Single Object

There will always be a single point on a rigid body that is the location of that body’s center of mass. The position of the center of mass of an object depends on the distribution of mass.

If an object is of uniform density, its center of mass is easier to determine. For example, in a circle of uniform density, the center of mass is the center of the circle. (This would not be the case, however, if the circle was denser on one side than the other).

In fact, the center of mass will always be at the geometric center of the object when density is uniform. (This geometric center is called the centroid.)

If the density is not uniform, there are other ways to determine the center of mass. Some of these methods involve the use of calculus, which is beyond the scope of this article. But one simple way to determine the center of mass of a rigid object is to simply try to balance it on your fingertip. The center of mass will be at the balancing point.

Another method, useful for planar objects, is as follows:

  • Suspend the shape from one edge point along with a plumb line.
  • Draw a line on the shape that lines up with the plumb line.
  • Suspend the shape from a different edge point along with a plumb line.
  • Draw a line on the shape that lines up with the new plumb line.
  • The two lines drawn should intersect at a single point.
  • This unique intersection point is the location of the center of mass.

For some objects, however, it is possible for the balance point to be outside the bounds of the object itself. Think of a ring, for example. The center of mass for a ring shape is in the center, where no part of the ring exists at all.

Center of Mass of a System of Particles

The position of the center of mass for a system of particles can be thought of as their average mass position.

The same idea can be used as for a rigid object if you imagine this system of particles are all connected by rigid, massless plane. The center of mass would then be the balance point of that system.

To determine the center of mass of a system of particles mathematically, the following simple formula can be used:

\vec{r} = \frac{1}{M}(m_1\vec{r_1} + m_2\vec{r_2} + ...

Where M is the total mass of the system, mi are the individual masses and ri are their position vectors.

In one dimension (for masses distributed along a straight line) you can replace r with x.

In two dimensions, you can find the x-coordinate and y-coordinate of the center of mass separately as:

x_{cm} = \frac{1}{M}(m_1x_1 + m_2x_2 + ... \\ \text{ }\\ y_{cm} = \frac{1}{M}(m_1y_1 + m_2y_2 + ...

Examples of Calculating the Center of Mass

Example 1: Find the coordinates of the center of mass of the following system of particles: particle of mass 0.1 kg located at (1, 2), particle of mass 0.05 kg located at (2, 4) and particle of mass 0.075 kg located at (2, 1).

Solution 1: Apply the formula for the x-coordinate of the center of mass as follows:

x_{cm} = \frac{1}{M}(m_1x_1 + m_2x_2 + m_3x_3) \\\text{ }\\= \frac{1}{0.1 + 0.05 + 0.075}(0.1(1) + 0.05(2) + 0.075(2))\\\text{ }\\=0.079

Then apply the formula for the y-coordinate of the center of mass as follows:

y_{cm} = \frac{1}{M}(m_1y_1 + m_2y_2 + m_3y_3) \\\text{ }\\= \frac{1}{0.1 + 0.05 + 0.075}(0.1(2) + 0.05(4) + 0.075(1))\\\text{ }\\=2.11

So the location of the center of mass is (0.079, 2.11).

Example 2: Find the location of the center of mass of a uniform density equilateral triangle whose vertices lie at points (0, 0), (1, 0) and (1/2, √3/2).

Solution 2: You need to find the geometric center of this equilateral triangle with side length 1. The x-coordinate of the geometric center is straightforward – it is simply 1/2.

The y-coordinate is a little trickier. It will occur at the location that a line from the top of the triangle to the point (0, 1/2) intersects with a line from any of the other vertices to the midpoint of one of the opposite side. If you sketch such an arrangement, you will find yourself with a 30-60-90 right triangle whose long leg is 0.5 and short leg is the y-coordinate. The relationship between these sides is √3y = 1/2, hence y = √3/6, and the coordinates of the center of mass are (1/2, √3/6).

Motion of the Center of Mass

The location of the center of mass of an object or system of objects can be used as a reference point in many physics calculations.

When working with a system of interacting particles, for example, finding the center of mass of the system allows for an understanding of linear momentum. When linear momentum is conserved, the center of mass of the system will move with a constant velocity even as the objects themselves bounce off one another.

For a falling rigid object, gravity can be treated as acting on that object’s center of mass, even if that object is rotating.

The same is true of projectiles. Imagine tossing a hammer, and as it flies through an arc in the air, it rotates end over end. This might seem like complex motion to model at first, but it turns out that the center of mass of the hammer moves in a nice smooth parabolic path.

A simple experiment can be performed which demonstrates this by taping a small piece of glow tape to the hammer’s center of mass, and then tossing the hammer as described in a dark room. The glow tape will appear to move in a smooth arc, like a tossed ball.

A Simple Experiment: Find the Center of Mass of a Broom

A fun center-of-mass experiment that you can perform at home involves using a simple technique for finding the center of mass of a broom. All you need for this experiment is one broom and two hands.

With your hands relatively far apart, hold up the broom on the end of two pointer fingers. Then, slowly bring your hands closer together, sliding them underneath the broom. As you move your hands closer together you may notice one hand wants to slide along the underside of the broom handle while the other one stays put for a while before sliding.

The entire time your hands move, the broom remains balanced. Eventually, when your two hands meet, they will meet at the location of the broom’s center of mass.

Center of Mass of the Human Body

The center of mass of the human body is located somewhere near the navel (belly button). In men the center of mass tends to be a little higher since they carry more body mass in their upper body, and in women, the center of mass is lower because they carry more mass in their hips.

If you stand on one foot, your center of mass will shift toward the side of the foot you are standing on. You may notice yourself leaning more toward that side. This is because in order to stay balanced, your center of mass needs to stay over the foot you are balancing on or else you will tip over.

If you stand with one leg and hip against a wall and try to lift your other leg, you will likely find it impossible because the wall prevents your weight from shifting over the balance leg.

Another thing to try is standing with your back to the wall and your heels touching the wall. Then try to bend forward and touch the floor without bending your legs. Women may be more successful at this task than men because their center of mass is lower in their body and may end up still being over their toes as they lean forward.

Center of Mass and Stability

The location of the center of mass relative to an object’s base determines its stability. Something is considered stably balanced if, when tipped slightly and then released, it then returns back to its original position instead of tipping further and falling over.

Consider a three-dimensional pyramid shape. If balanced on its base, it is stable. If you lift one end slightly and let it go, it falls back down. But if you try to balance the pyramid on its tip, then any deviations from perfect balance will cause it to fall over.

You can determine if an object will fall back to its original position or tip over by looking at the location of the center of mass relative to the base. Once the center of mass moves past the base, the object will tip over.

If you play sports, you might be familiar with the ready position where you stand with a wide stance and knees bent. This keeps your center of mass low, and the wide base makes you more stable. Consider how hard someone would have to push you to tip you over if you are in the ready position vs. when you are standing up straight with your feet together.

Some cars have problems with rolling over when they take sharp turns. This is because of the location of their center of mass. If the center of mass of a vehicle is too high and the base is not wide enough, then it doesn’t take much to cause it to tip over. It’s always best for the stability of a vehicle to have most of the weight as low as possible.

References

About the Author

Gayle Towell is a freelance writer and editor living in Oregon. She earned masters degrees in both mathematics and physics from the University of Oregon after completing a double major at Smith College, and has spent over a decade teaching these subjects to college students. Also a prolific writer of fiction, and founder of Microfiction Monday Magazine, you can learn more about Gayle at gtowell.com.