Video: Center of Mass

In this lesson, we will learn how to find an object’s center of mass and to analyze the effect of the position of the center of mass on an object’s stability.

14:16

Video Transcript

In this video, we’re talking about the concept of center of mass. Every object that has mass has a center to that mass. And as we’ll see, knowing an object’s center of mass is important if we want to balance it or orient it a certain way.

As a quick example of this, imagine a person standing on both legs, then say that without shifting their weight they lift up one of their legs. We know from experience what will happen. That person, unfortunately, will start to fall over. This is because perched like this on one leg, they’re no longer supporting their center of mass. We know the solution to this issue is to shift weight. When a person does this, they bring their center of mass directly in line with their one point of support.

We’ll get back to this connection between balancing an object and knowing its center of mass in a moment. But for now, let’s define this term that we’ve been talking about, an object’s center of mass. Say that we have some object. Here’s our object at rest on the ground. We know that our object is composed of matter, stuff that takes up space. Some of that matter is here. Some of it is over here. Some is over here, some over here and so on and so forth, all the little chunks of matter that make up our overall object.

Now, if we were to consider a pair of these chunks, say this chunk here and this chunk here. Now, if we were to try to find the center of mass for just these two chunks of matter, we know that center would lie somewhere along the line between these two chunks. Specifically, if the chunks were the same size and had the same density, that is the same mass overall, then the center of mass of these two pieces would lie exactly halfway between them. Another way to say this is to say that the effective mass of these two little chunks of matter right here is located at a point exactly between them, right here, this dot we’ve drawn.

And then, going further, we could do the same thing for another pair of chunks of matter, say this chunk and this junk together. If we just consider these two chunks all by themselves, then the center of mass of this system, would be somewhere along the line between the two. And again, if these two chunks had the same exact mass, then that center would be directly between them. Once more, we now have the effective location of these two chunks of matter. That location is right here, this second dot.

When we say effective location, here’s what we mean by that. Using one of these magenta dots as a starting point, say we use this one, then we can say that on the line along one side of that dot the mass and the distance from the dot of that mass is the same as the mass and the distance of the mass along the other side of that line. That is, if we consider the amount of mass on either side of our dot along this line, and we also include the distance of that mass from the dot. Then that combination of mass and distance is the same on either side of the dot. That’s what we mean when we say that this dot represents the center of mass of these two small chunks of matter.

And in general, we can say this. An object’s center of mass is the point from which the distribution of mass is equal in all directions. From this definition, we can see that as we consider these chunks of masses within our overall object. When we treat them in pairs, this pair here and then this pair here like we did. This definition of center of mass shows us that these two dots we’ve identified are the centers of mass of these two pairs of masses. They are the points from which the distribution of masses, in this case the two pairs of chunks of mass, are equal in all directions.

Now, instead of just considering a few of the little chunks of matter that make up our object, let’s consider all of them. Now, we’re starting to think about what is the overall center of mass for all of the small chunks of matter in our object. Assuming that each one of these little chunks represents the same amount of mass, then if we find the geometric center of all of those little chunks, that will be the point from which the distribution of mass is equal in all directions.

For example, say we draw a line through our center of mass. We can see that the distribution of mass within the object shape in this direction from the center of mass is the same as the distribution of mass along this opposite direction. And then, if we draw another line through our center of mass, we see the same thing. The distribution of mass in our object in this direction is the same as the distribution in this direction. And in fact, no matter what line we draw through, our center of mass we’ll find the same thing. This is what it means for the distribution of mass, which refers to how much mass we have and how far away it is from a point of interest, to be equal in all directions.

Now, let’s think about this practically. Let’s say that we have this system which consists of two masses, one of mass value 𝑚 and the other twice as massive. What we want to do is balance this system of masses on this triangle, that is balance it at a certain point. The question is, where along the distance between these two masses will we put the point of our triangle? This is the same as asking, where’s the center of mass of this system?

Going back to our definition of center of mass will help us figure this out. For the distribution of mass in this system to be equal in all directions, then the balance point that we’ll choose will be such that the distance from that balance point to our mass of two 𝑚 is half as big as the distance from that point to our mass of one 𝑚. By picking this location to put our balance point, we pick the one spot from which the distribution of masses in our system is equal in all directions. This spot is the system’s center of mass.

Now, at this point, it’s worth saying something about a term which is similar to the term center of mass. This other term is called the center of gravity. And just like all objects have a center of mass, so all objects have a center of gravity. Though these two terms are similar and are sometimes used interchangeably. Strictly speaking, they’re not completely identical.

To see why, let’s again consider our object here on the ground. We saw earlier that the center of mass of our object is right about there. That’s because it’s from this point, and this point alone, that the distribution of mass in our object is equal in all directions. The center of gravity, though, isn’t about the distribution of mass so much as it’s about the distribution of weight in the object. This brings up then the difference between mass and weight.

We know that if we have some object. Say that it has a mass 𝑚, that that mass is the same regardless of where the object is. This mass could be on the surface of Earth or in the upper atmosphere or way off in outer space. In all those cases, its mass is the exact same value. But what is not the same in those cases is this object’s weight. That’s because the object’s weight depends on the strength of the gravitational field it’s in.

If our mass is sitting on Earth’s surface, then we know the strength of that field. We call it lower case 𝑔, the acceleration due to gravity. But then, if we relocated our mass and put it way off in outer space, the gravitational force on our mass would be much weaker and, therefore, its weight would be much less. So, while the object’s mass never changes, its weight does.

And this brings us to the definition of the center of gravity. This is the point from which the distribution of an object’s weight is equal in all directions. So, we see that the difference between these two terms, center of gravity and center of mass, comes down to the difference between mass and weight.

Knowing this, let’s say that along with our mass over here, we introduce a second object. We know that gravitationally, these two masses will attract one another. And what’s more, if we bring the two masses closer together, then the strength of that attraction will increase. That’s because the gravitational field from one mass to the other is relatively stronger.

Now, coming back to our object here, which is resting on the ground on Earth. Let’s say we were to pick out a chunk of this object’s mass up here at the top and then one also down at the bottom. Let’s say that from a mass perspective, these two chunks are identical. They have the same mass. But do they have the same weight? In other words, does gravity act on them equally?

Well, precisely speaking, it doesn’t. Because this chunk right here is a little bit closer to the center of the Earth, the large mass attracting this object, than this chunk up here, the gravitational force of attraction on the lower chunk is stronger than the force of attraction on the upper chunk. Overall, this means that the bits of matter on the bottom half of our object are being pulled on, that is, have a greater weight, than the bits of matter on the top side of our object.

And this means that when it comes to the center of gravity for this object, that center is actually a little bit closer to the bottom of the object than to the top. It’s not in the exact same location as the center of mass. The difference in location between these two points, center of mass and center of gravity, is incredibly small for most objects we encounter. Finding that these two points are in different locations requires that we treat the gravitational field as variable, not the same, over the scale of our object.

In this case, we’ve been required to say that the gravitational field is stronger here than it is up here. Technically, true, but if our object is reasonably sized, say, a few meters in height, then, practically speaking, we do treat the gravitational field as constant over that object. Specifically, we do this by simply using 𝑔 to represent the acceleration due to gravity near the Earth.

This is the secret for understanding the differences between center of gravity and center of mass. If we treat the gravitational field acting on an object as constant at all points in that object, then center of mass and center of gravity have the exact same location. Based on this, we can now update our definition of center of mass. We can say that assuming gravity is constant, an object’s center of mass is the point from which the distribution of weight is equal in all directions.

Under this assumption then, we see that center of mass and center of gravity will be located at the same point. And because we typically do assume gravity is constant through using the gravitational constant 𝑔 whenever we’re near Earth’s surface, it’s usually safe to assume that these two points, center of mass and center of gravity, coincide. Now, before we get to an example to try out some of these ideas, there’s one last thing we should mention.

So far, we’ve assumed that all the objects we’ve worked with have a constant density. That is, if we were to pick a certain size chunk from one part of our object, the mass of that chunk would be the same as a similarly sized chunk somewhere else. When that’s true, when our object does have a constant density all throughout, it makes it a bit simpler to find its center of mass. But even if its density is not constant, the definition we have here still holds.

For example, imagine we had a circular object where one-half of that object had density two 𝜌 and the other half had density 𝜌. If we wanted to balance this object, say, on the end of a pencil, then we can see that that balance point would not be at the exact center of the object because of this density difference. Instead, its center of mass would be over closer to the side with greater density. This follows our definition of the center of mass being the point at which the distribution of weight is equal in all directions. Knowing all this, let’s now take a look at an example exercise.

The diagram shows a drinking glass under three different conditions. The glass is shown empty, and its center of mass is shown at its geometric center. Water is then added to the glass, and the center of mass of the partially filled glass is shown; this is the center of the combined mass of the glass and the water. Finally, an ice cube is placed in the water, and a pencil is used to push the ice cube into the water so it is at rest, with its topmost face just at the height of the water level. Which of the points A, B, C, and D most correctly shows the position of the center of the combined mass of the glass, water, and ice.

All right, let’s look at our diagram which shows this glass under three conditions. First, the glass is completely empty. And we see the location the glass’s center of mass marked out. Then, the glass is partially filled with water. And this water having mass affects the overall center of mass of this system. The centre of mass of the glass plus the water added in is right here. And then, in our last snapshot, we see that an ice cube has been added to this water and then held underwater by our pencil. And we want to know at which of the four points marked out, A, B, C, and D is the new center of mass of this ice-cube-water-and-glass system?

As we get started on this question, let’s clear a bit of space on screen and then recall what the definition of center of mass is. In a constant gravitational field, an object’s center of mass is the point from which the distribution of weight is equal in all directions. So, considering the center of mass of our empty glass, this is the point from which the weight distribution is equal in all directions.

Then, when we add water to the glass, we know that that center of mass will drop downward because now there’s more weight towards the bottom of the glass. And indeed, we see this to be true. The center of mass of this combined glass-and-water system is lower than the center of mass for the glass by itself.

Then, we do something interesting. We add an ice cube to our glass and we hold it underwater. We can see the effect of this from our drawing. The overall level of water in the glass will go up when the ice cube is submerged in it. That makes sense since the ice cube takes up space. If we look at the center of mass of our glass-and-water system before the ice cube was added in, we see the height of that center of mass aligns with point B in our diagram.

And now, with the ice cube added, since water rises in our glass, we know the overall center of mass will also go up. That’s because we now have more mass towards the top side of the glass than we did before the Ice cube was introduced. The only one of our four points with a higher center of mass than point B is point A. And therefore, that’s our answer. This is the new center of mass of our combined glass-and-water-and-ice-cube system. Since we’re raising, or increasing the elevation, of mass in our system, so we also raise the center of mass.

Let’s summarize now what we’ve learned about center of mass. Starting off, we saw that an object’s center of mass, abbreviated C.O.M., is the point from which the distribution of mass is equal in all directions. For a one-dimensional or two-dimensional object, we could visualize the center of mass as being the point at which the object will balance.

Then, we learned about this similar term, center of gravity. And we saw that center of gravity concerns object weight rather than mass. This means that, strictly speaking, an object’s center of mass and center of gravity are not located at the same exact point. Lastly, we saw that if we assume a constant gravitational field, as we often do near Earth’s surface, then an object’s center of mass and its center of gravity do overlap at the same exact location.

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