Video Transcript
In this video, our topic is the
center of mass of particles. We’ll be looking at how we can
calculate the center of mass of a collection of particles with different individual
masses and positions in 2D space. The concept of a center of mass is
generally useful when we’re thinking about the effects of gravity on objects,
specifically when we think about balancing them. The center of mass of an object or
a set of objects is a point in space which is effectively the average position of
all the mass in the system.
Being able to find this point is
useful as, for certain calculations, we can treat the entire system as if it were a
single mass located at the center of mass. In other words, we can effectively
assume that all of the mass in a system is concentrated at this point. We can think of the center of mass
of a system as its balance point. Consider this example where a tray
of objects is being supported with one hand. Effectively, it’s possible to treat
the tray and the set of objects as if they were a single mass, located at the center
of mass. Supporting the tray by a point
directly below the center of mass ensures that the tray remains balanced.
In this video, we’re focusing on
finding the center of mass of sets of particles. So let’s start with a simple
example. Let’s say we have two particles
named A and B with masses 𝑀 A and 𝑀 B, respectively. Note that when we say particle, we
simply mean an object with negligible size. Remember that we can think of the
center of mass as being the average position of all the mass in the system. So in this case, where our system
consists of two particles, finding the center of mass is fairly intuitive. It’s simply located at the midpoint
of the two particles.
If we had a coordinate axis to our
diagram, then we can find the coordinate of the position of the center of mass by
averaging the position coordinates of A and B. In this case, particle A has an
𝑥-coordinate of zero and particle B has an 𝑥-coordinate of five. So we can find the 𝑥-coordinate of
the center of mass, which we write COM sub 𝑥 by adding together the position
coordinates of A and B and dividing by the number of particles. This gives us five over two or 2.5
as the 𝑥-coordinate is the center of mass.
Note that in this example we said
that the mass of particle A is equal to the mass of particle B, so what happens if
their masses are not equal? If one of the particles has a
larger mass — let’s say particle B has a higher mass than particle A — then we would
expect the position of the center of mass to be closer to the particle B than
particle A. And in fact, large masses do have a
greater influence over the position of the center of mass than small masses.
So, to factor this into our
calculation of the position of the center of mass, we now need to calculate what’s
known as a weighted average of the particles’ positions. So, rather than just adding
together the position coordinates of each of the particles and dividing by the
number of particles, we multiply each particle’s position by its mass and divide by
the total mass. So, in this case, we take the
position of A, that’s zero, multiply it by the mass of A, 𝑀 A, and add it to the
position of B, which is five, multiplied by the mass of B, 𝑀 B. We then divide by the total mass in
the system, which, in this case, is the mass of A plus the mass of B. We can generalize this expression
by writing 𝑥 A for the position of A and 𝑥 B for the position of B.
Now this equation tells us the
𝑥-coordinate of the center of mass for any two particles A and B, where the
positions of the two particles are 𝑥 A and 𝑥 B and their masses are 𝑀 A and 𝑀
B. This equation can be further
generalized to deal with systems of more than two particles. So, for any number of particles,
the 𝑥-coordinate of the center of mass is equal to the sum of each particle’s mass
multiplied by its position divided by the sum of each particle’s mass, in other
words, the total mass in the system.
This idea can easily be extended to
more than one dimension. For example, if we have several
particles are distributed through two-dimensional space, then the 𝑥-coordinate of
their center of mass will be given by this expression and the 𝑦-coordinate of their
center of mass will be given by this expression, which is the same except it
references the 𝑦-coordinate of each particle rather than the 𝑥-coordinate.
So as a quick example, let’s make
up some masses for these particles and have a go at calculating the center of
mass. Let’s say the mass of particle A is
one kilogram, the mass of particle B is two kilograms, the mass of particle C is
three kilograms, and the mass of particle D is four kilograms. It’s worth noting that all of the
masses need to be expressed in terms of the same units. But as long as the units of mass
are the same as each other, the units used won’t actually affect the position of the
center of mass. Since all these masses are given in
kilograms, we might as well forget about units of mass for this example. So we could say that these
particles just have masses one, two, three, and four, respectively.
Okay, so we’re looking for the
coordinates of the center of mass of these four particles, which we could write like
this. First, let’s use this formula to
work out the 𝑥-coordinate of the center of mass. To do this, we first need to find
the sum of the product of each particle’s mass and 𝑥-coordinate. So particle A has a mass of one and
an 𝑥-coordinate of one. And the product of these is, of
course, one times one. Particle B has a mass of two and an
𝑥-coordinate of 2.5. So we add on the product of two and
2.5. Particle C has a mass of three and
an 𝑥-coordinate of six, giving us three times six. And particle D has a mass of four
and an 𝑥-coordinate of three, so we can add on four times three. We then divide all of this by the
sum of all of the masses, in this case, one plus two plus three plus four.
Evaluating the numerator of this
expression, one times one is one, two times 2.5 is five, three times six is 18, and
four times three is 12. The sum of all these values is
36. And looking at the denominator, one
plus two plus three plus four is 10. So, we found that the 𝑥-coordinate
of the center of mass is 36 over 10 or 3.6. Next, we can use this formula to
find the 𝑦-coordinate of the center of mass. To do this, we first need to sum up
the mass of each particle multiplied by its 𝑦-coordinate. The mass of particle A is one and
its 𝑦-coordinate is two, giving us one times two. The mass of particle B is two and
its 𝑦-coordinate is four, giving us two times four. The mass of particle C is three and
its 𝑦-coordinate is three, giving us three times three. And particle D has a mass of four
and a 𝑦-coordinate of one, giving us four times one.
Once again, we divide all of this
by the sum of the masses of the particles, which we already calculated to be 10. Evaluating the numerator again, we
have one times two, which is two, two times four, which is eight, three times three,
which is nine, and four times one, which is of course four. Adding all of these together gives
us 23. And so the 𝑦-coordinate of the
center of mass is given by 23 over 10 or 2.3. So, in conclusion, the center of
mass of these four particles is located more or less here, at the coordinates 3.6,
2.3. Okay, so now that we’ve talked a
bit about what a center of mass is and had a go at calculating the coordinates of
the center of mass of a system of particles using these two equations, let’s have a
look at some more example questions.
Three particles are placed on a
line. Particle A of mass four
kilograms is located at the origin, particle B of mass six kilograms at nine,
six, and particle C of mass 10 kilograms at six, four. Determine the coordinates of
the center of mass of the three particles.
A good place to start with this
question is to draw the positions of the three particles on pair of coordinate
axes. So here’s particle A at the
origin, here’s particle B at coordinate nine, six, and here’s particle C at
coordinate six, four. This question askes us to find
the center of mass of the three particles. We can recall that the center
of mass is effectively the average position of all of the mass in a system. We can calculate the exact
position of the center of mass of a system of particles by finding the average
position of those particles weighted according to their mass.
Specifically, we can calculate
the coordinates of the center of mass of a system consisting of particles in
two-dimensional space using these two equations. The equation on the left tells
us that the 𝑥-coordinate of the center of mass, written COM sub 𝑥, can be
found by adding together the product of each particle’s mass and 𝑥-coordinate
and dividing this quantity by the sum of all of the masses of the particles. The equation on the right shows
us that the 𝑦-coordinate of the center of mass can be calculated in a similar
way, only this time we sum the product of each particle’s mass and
𝑦-coordinate. Let’s first apply the equation
on the left to find the 𝑥-coordinate of the center of mass of the three
particles.
The numerator of this
expression tells us that we need to take the mass of each particle multiplied by
the 𝑥-coordinate of each particle and then sum these quantities together. In other words, we multiply the
mass of particle A, 𝑀 A, by the 𝑥-coordinate of particle A, 𝑥 A, then add the
mass of particle B multiplied by the 𝑥-coordinate of particle B and then add
the mass of particle C multiplied by the 𝑥-coordinate of particle C. The denominator of this
expression is the sum of all of the masses. In this case, that means we’re
dividing by the mass of particle A plus the mass of particle B plus the mass of
particle C.
Fortunately, all the
information we need is available in the question. The mass of particle A is four
kilograms, and it’s located at the origin, which means the 𝑥-coordinate is
zero. So the mass of A multiplied by
the 𝑥-coordinate of A is four times zero. Next, we know that the mass of
particle B is six kilograms, and its 𝑥-coordinate is nine. So we add six times nine. Finally, the mass of particle C
is 10 kilograms, and its 𝑥-coordinate is six. So, we add 10 times six.
We then divide all this by the
mass of A plus the mass of B plus the mass of C, which is four kilograms plus
six kilograms plus 10 kilograms. Evaluating the numerator, we
have four times zero, which is zero, six times nine, which is 54, and 10 times
six, which is 60. 54 plus 60 in the numerator is
114, and four plus six plus 10 in the denominator is 20, which expressed as a
decimal is 5.7. So the 𝑥-coordinate of the
center of mass of these three particles is 5.7.
Now we just need to find the
𝑦-coordinate of the center of mass. And we can do this using the
equation on the right. So first, we multiply each
particle’s mass by its 𝑦-coordinate and sum these together. Particle A has a mass of four
kilograms, and because it’s located at the origin, we know it has a
𝑦-coordinate of zero. Particle B has a mass of six
and a 𝑦-coordinate of six. And particle C has a mass of 10
and a 𝑦-coordinate of four. We then divide all of this by
the sum of the masses of all the particles, which we previously calculated to be
20. Now, evaluating the numerator,
we have four times zero, which is zero, six times six, which is 36, and 10 times
four, which is 40. Summing the values in the
numerator gives us 76, and that’s divided by 20, which expressed as decimal is
3.8.
So, if the 𝑥-coordinate is 5.7
and the 𝑦-coordinate is 3.8, then the coordinates of the center of mass are
5.7, 3.8. And this is the final answer to
our question.
Next, let’s look at an example
where we’re not told the exact coordinates of the particles in the system.
The figure shows a system of
point masses placed at the vertices of a square of side length six units. The mass placed at each point
is detailed in the table. Determine the coordinates of
the center of gravity of the system.
So, looking at the diagram, we
can see that we indeed have four point masses, which we could also call
particles, positioned at the corners or vertices of a square. The table shows us that the
point mass of position A has a mass of 75 kilograms, the mass at point B has a
mass of 29 kilograms, the one at point C has a mass of 71 kilograms, and the one
at point D has a mass of 85 kilograms. The question asks us to find
the center of gravity of the system, which is the same as asking for the center
of mass. We can recall that the center
of mass is effectively the average position of all the mass in the system and
that the 𝑥- and 𝑦-coordinates of the center of mass are given by this formula
and this formula, respectively.
So, let’s start by using the
formula on the left to find the 𝑥-coordinate of the center of mass. If we look at the numerator on
the right-hand side of this expression, we can see that we need to find the sum
of each mass multiplied by its 𝑥-coordinate, so we can rewrite the numerator as
the mass at position A multiplied by the 𝑥-coordinate of position A plus the
mass at position B multiplied by the 𝑥-coordinate at B and so on for each of
the four points. The denominator of this
expression is the sum of all of the masses in our system. So we could rewrite the
denominator as the mass at point A plus the mass at point B and so on. So all we need to do now is
substitute in the masses and positions of all of the point masses.
Now, we’ve been given the
masses of each of these point masses in the table. However, we haven’t explicitly
been told the 𝑥-coordinates of all of these points. In order to work out the
𝑥-coordinates, we need to use the fact that the point masses are positioned at
vertices of a square of side length six units. This means that all of these
lengths on the diagram are six units long. We can see that position A is
the origin. So this has coordinates zero,
zero. Since position B is six units
above the origin, we know that its coordinates must be zero, six. Position C is six units to the
right of position B, in other words, six units in the 𝑥-direction. So its coordinates must be six,
six. And finally, we can see that
position D is six units in the 𝑥-direction from the origin, so it must have
coordinates six, zero.
Now that we have the
coordinates of all of the points and their masses, we can substitute the
𝑥-coordinates and the masses into this expression in order to find the
𝑥-coordinate of the center of mass. So firstly, we can see that the
mass of A is 75 kilograms and it has an 𝑥-coordinate of zero. So this gives us 75 times
zero. Next, we have the mass of B
multiplied by the 𝑥-coordinate of B, which is 29 times zero, the mass at point
C multiplied by the 𝑥-coordinate of C is 71 times six, and the mass of D times
the 𝑥-coordinate of D is 85 times six.
We then divide all of this by
the sum of all of the masses. Evaluating the numerator, we
can see that this term and this term are both a number multiplied by zero. So both of these terms are just
equal to zero, leaving us with 71 times six plus 85 times six. And plugging this into our
calculator gives us 936. The sum in the denominator is
equal to 260. And this fraction simplifies to
18 over five. And this is the 𝑥-coordinate
of the center of gravity of the system. Now, we just need to use this
formula to find the 𝑦-coordinate of the center of mass. We basically do exactly the
same thing, only using the 𝑦-coordinate of each point rather than the
𝑥-coordinate.
So we have the mass of A
multiplied by the 𝑦-coordinate of A plus the mass of B multiplied by the
𝑦-coordinate of B plus the mass of C multiplied by the 𝑦-coordinate of C and
the mass of D multiplied by the 𝑦-coordinate of D. All of this is then divided by
the sum of all of the masses of the particles, which we previously calculated to
be 260. Once again, in the numerator,
two of the terms are just equal to zero. And evaluating the rest of the
expression gives us 600 over a 260. We can simplify this fraction
to 30 over 13. And this is the 𝑦-coordinate
of the center of mass. So, putting the 𝑥- and
𝑦-coordinates together, the coordinates of the center of gravity or center of
mass of the system are 18 over five, 30 over 13.
Okay, so now we’ve answered a
couple of practice problems, let’s review the key points that we’ve looked at in
this video. We’ve seen that the center of mass
of a system of particles is the average position of the particles weighted according
to their mass. We can calculate the spatial
coordinates of the center of mass independently from each other. For a two-dimensional system, this
would be the 𝑥- and 𝑦-coordinates. And this can be accomplished using
these two formulas. The 𝑥-coordinate of the center of
mass, denoted COM 𝑥, is given by the sum of the mass of each particle multiplied by
its 𝑥-coordinate divided by the sum of the masses of all the particles. Similarly, the 𝑦-coordinate of the
center of mass, denoted COM 𝑦, is given by the sum of the mass of each particle
multiplied by its 𝑦-coordinate divided by the sum of the masses of the
particles.
