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.
