Video Transcript
In this video, our topic is center
of mass. The concept of center of mass is
useful whenever we’re thinking about balancing some object or some system.
We can start out by thinking of
single objects like this one. This object because it has mass
also has a center of that mass. We can think of that this way. If this object were compressed down
to a single point and all of its mass was concentrated at that point, then this
would be the center of mass of this object. We can say that this is the point
where all of the object’s mass is effectively concentrated. And that means that if this object
is in a uniform gravitational field, then this is the point from which the weight
force on the object acts. We can consider this to be the
definition of an object center of mass.
And as we consider this definition,
it’s worth reminding ourselves that this is true when the object is in a uniform
gravitational field. If it’s not, then the weight force
acts from a different location called the center of weight, then the center of mass
of the object. But for our purposes, we’ll assume
that center of mass and center of weight are the same.
Now, given an object, it’s often
possible to estimate visually where its center of mass is located. For example, say that we have a
uniform plank of wood. Uniform means that the density of
the wood is the same all throughout its volume. In that case, this plank’s center
of mass is located at its geometric center. If we were to look halfway along
its length and also halfway along its height, then the location at which lines from
those points intersect is this plank’s center of mass.
And we can make this estimation
with other shapes too. Say, for example, we have a
spherical object or an equilateral triangle, or even a system of two identical
masses connected by a thin rod. Just like with our plank, we can
visually estimate where all the mass of these objects or systems of objects is
effectively concentrated. Consider our uniform sphere. We can find the geometric center of
this object by drawing any two diameters through it. And the point at which these two
diameters intersect will be the sphere’s center of mass.
Likewise, for our uniform-density
equilateral triangle, we can find the geometric center of this object by marking out
the midpoints of each side of the triangle. And then, if we draw lines normal
to those sides through those midway points, then, again, the point of intersection
of those lines is this triangle center of mass. Just like with our uniform sphere,
this point on our triangle is the point from which the weight force acts.
And now, let’s look at this object,
which is technically a compound object or a system of objects. It includes this block here and
this identical block over here as well as a thin rod in line with the midpoints of
these two blocks. Now, we can see that if we were to
consider these two blocks individually, then their centers of mass would be at their
geometric centers, just like we saw with our other shapes. But thinking of the system of
masses as a whole, we can say that the system center of mass will be the average
position of these two centers of mass. That is, it will be midway between
them. If we were to draw a line
connecting these two points, then the midpoint of that line would be the center of
mass of this system of masses.
Now, we mentioned a moment ago that
this object here is called a compound object because it’s made up of more than one
shape. It’s not unusual to find a compound
object and want to visually find its center of mass. So for the sake of getting some
practice, let’s look at another example of this.
Consider this object right
here. Now, at first glance, it doesn’t
seem like a compound shape. And indeed, we’ll say it’s one
object with a uniform density. But in order to visually locate its
center of mass, we are going to divide this object up into two different pieces. We’re going to make it a compound
shape. The reason we’re going to do this
is because right now it’s hard to tell visually where the center of mass of this
L-shaped object is. But by dividing it up into smaller
pieces, we can locate that center of mass more accurately.
Remember, from the example of our
wooden plank that if we have a uniform rectangular object, then we were able to
visually locate its center of mass fairly well. Over here with our L-shaped object,
we’re going to use a similar approach by dividing this into two rectangular
bits. There are a couple of different
ways we could choose to do this. One way is to make an imaginary
dividing line right here so that now our object is divided into two rectangles, one
up here and then one longer one down here. Now that our object is divided up
this way, we can apply the same approach we did for our wooden plank to find the
centers of mass of these two individual rectangles.
When we sketch in lines that are
perpendicular to the midpoints of the sides of these rectangles, we see that where
the lines intersect are the centers of mass of each one. With that done, remember how with
our shape over here we drew a line that joined the two individual centers of mass of
the smaller blocks. We’re going to do the same thing
over here. Here, we’ve sketched in a dashed
line connecting these two points. And we know that the overall center
of mass for this L-shaped object lies somewhere along this line. Just where the center of mass is
along this line, we don’t exactly know. But with this line drawn in, we can
do a better job than we would have otherwise estimating where this point is.
At this point, though, a question
may come up. When we divided up this object, we
chose to make one rectangle here and the other one was right here. But what if we had divided it
differently? After all, another natural way to
divide this larger object into two rectangles would be to make an imaginary line
here so that now our two rectangles would look like this. If we had done this, the question
may come up: would we have found a different center of mass for our overall
object? And this brings up the very
important point that the overall center of mass of our object does not, and
therefore should not, depend on how we divide our object up into a compound
shape.
We can recognize that whether we
draw our imaginary dividing line here or here like we did before, the overall shape
is still the same and therefore it has the same center of mass no matter what. So, however we choose to divide up
this object into two rectangles, that choice does not change the location of the
object center of mass. And therefore we ought to get the
same location or thereabouts if we’re estimating it visually either way. To see how this works, let’s now
find the centers of mass of these two newer rectangles.
Once again, we draw in lines
perpendicular to the midpoints of the sides of the rectangles. And we mark where these lines
intersect; these are the centers of mass of our two new rectangles. Then what we do is draw a line
connecting these two centers of mass. Just like we said with the pink
line, the center of mass of our overall shape lies somewhere along this orange
one. And in fact, we’re now seeing an
advantage of dividing up our shape into two different rectangles in two different
ways.
Notice that these two lines, the
pink one and the orange one, intersect. That point of intersection is right
about here. And it’s this location marked by
our green dot that indicates the center of mass of our overall L-shaped object. Notice that this point lies along
both the pink and the orange lines. So, we could’ve estimated its
location based on just having one of those lines. But since we have two, we now have
the advantage of finding where they intersect and marking that point.
So, whenever we have a more
complicated-looking object or even a system of objects, it’s possible to divide it
up into simpler shapes, find the centers of mass of those individual shapes, and
then figure out the overall center of mass based on those individual centers of
mass. We saw here that when we do this
division into smaller, simpler shapes, no matter how we do it, the object’s center
of mass is not affected. Therefore, we should get the same
answer no matter how the object is divided up.
Now, if we look at the location
we’ve marked for this L-shaped object center of mass, we see it’s right on the edge
of the object, and this brings up an important point about center of mass. As it turns out, an object’s center
of mass need not be located within the object; that is, it could actually be at a
point in empty space. For example, say that this arm of
our L-shaped object was just a little bit longer. Say that it had an overall shape
like this.
This change in our object shape
would have an effect on its center of mass. Specifically, it would move that
point a little bit up and a little bit off to the right. Say that it resulted in our new
center of mass with the object being here, for example. This second dot we’ve drawn is
clearly outside of our L-shaped object. It’s an empty space between the
arms of the object, and yet it is this object’s center of mass.
Or consider another example of an
object’s center of mass being in empty space. Let’s think back to our sphere,
which we said is a uniform sphere, implying that it has uniform density. Let’s say that instead, this is a
hollow sphere with an empty center. Even though all the space in here
is unoccupied by any amount of mass, we can see that, still, based on the shape of
our empty sphere, its center of mass is still located at its geometric center. This point is an empty space, but
that’s no problem. It’s still the object’s center of
mass. So if we encounter cases like this,
it’s nothing to worry about. It’s entirely possible that an
object’s center of mass is not located within the mass of the object.
Now, earlier, we talked about how
this idea of an object’s center of mass is related to the object’s balance. Here’s what we mean by that. Say that our plank here is resting
on a flat surface, and say that that surface looks like this. We can say that right now, our
wooden plank is in a stable position. We can say that because its center
of mass located right here is still over top of the flat surface the plank is
resting on. And so long as a vertical line that
goes through the plank center of mass would also intersect with this flat surface,
that would remain true. Our plank would still be
stable.
So even if we were to push the
plank to the right a bit, because the center of mass of the plank remains over top
of the surface, the plank continues to be stable. But if we were to nudge the plank
just a bit farther, then we would reach a point where the center of mass is no
longer supported by the surface. In this case, the normal reaction
force of the surface against the weight of the plank — that is, the upward acting
force that the surface exerts on the plank — would not pass through the plank’s
center of mass. Therefore, our plank is unstable,
and it would tend to rotate and start to fall off our surface.
So when we think about object’s
stability, it’s important to have the center of mass of that object or system of
objects in mind. If that point is supported, say, by
some underlying surface, then the object is stable and won’t tend to start
rotating. On the other hand, a normal
reaction force that does not pass to the center of mass of an object is the sign of
an unstable object that will tend to be put in motion.
Knowing all this about center of
mass, let’s get some practice by way of an example.
The object shown in the diagram has
a uniform density. Which of the points A, B, C, and D
is closest to the center of mass of the object?
Looking over at our diagram, we see
our object and these four points A, B, C, and D marked out on it. The question is “which of these
four points is closest to the overall object’s center of mass?” We can recall that an object’s
center of mass is the location at which all of its mass can effectively be
concentrated. Or, put another way, an object’s
center of mass is the point from which its weight force acts. Now, since the object in our
example has a uniform density, that means that if we’re able to find the geometric
center of this overall object, then that point will coincide with the center of
mass.
We can see that this particular
object is roughly symmetrical. It’s roughly as wide as it is tall
and, except for this missing tile right here and this missing half-tile right here,
is comprised of mass with a uniform density. Now, it would be easier to find the
center of mass of this object if it was symmetrical both horizontally and
vertically. And in fact, we can already see
that there’s a vertical line that evenly divides our object into two halves. So, we know that the object’s
center of mass will be somewhere along this vertical line. That’s not much help for our
answer, though, because all of our answer options A, B, C, and D lie along this line
anyway.
So now, let’s think of object’s
symmetry in the vertical direction. We can see that our object is not
symmetrical this way. But what if it was? That is, what if we added on a
half-tile right here that is the same density as the rest of our object? If that half-tile were here, then
we could draw a horizontal line like this and say that our shape above the line is
the same as our shape below it. It’s symmetric. And that would mean that our
object’s center of mass would lie somewhere along this horizontal line. And in fact, we know where it would
lie. It would lie right here, in line
with the four points A, B, C, and D.
Now, we see that this point we’ve
marked in as our hypothetical center of mass is about equidistant from points A and
B. But we also know it’s not the accurate location of this object’s center of mass,
because this half-tile is not really there. Instead, that space is just
empty. And that means that the actual
location of our object’s center of mass will be a little bit farther up than we’ve
drawn it. We see that a bit farther up from
this point, but not very much farther, is point B. And so, that will be our answer
for which of these four points is closest to the center of mass of the object. Point B is just slightly above the
object’s vertical midpoint.
Let’s summarize now what we’ve
learned about center of mass. First, we saw that in a uniform
gravity field, an object’s center of mass is the point from which its weight force
acts. We saw further that for an object
of uniform density, its center of mass is its geometric center. Compound objects have an overall
center of mass, which can be located by dividing the object into simpler shapes. Along with this, the location of an
object’s center of mass can be estimated visually. And lastly, we learned that an
object is considered stable if the normal reaction force on that object passes
through its center of mass.
