### Video Transcript

In this video, we’re talking about
the units of measured quantities. We’re going to learn what units
are, why they’re important, and also how to work with them practically.

As we get started, it’s helpful to
admit that, sometimes, even the idea of using units can seem a little strange. This is often the case when we’re
taking our very first class in physics. That’s because, up until this
point, we’ve probably gotten used to working with numbers just by themselves, as
pure numbers. For example, when we’re in math
class and we’re working towards a solution, often that answer, whatever it is,
involves a pure number by itself, without units. But then we get to physics
class. And suddenly, from our perspective,
all of these units start attaching themselves to numbers. It’s not just 33 anymore. It’s 33 meters. It’s no longer eight, but eight
seconds. It’s not 1.7536, but 1.7536 amperes
of electrical current.

So, along with understanding what
units are and why they’re important, part of our challenge is just to remember them
in the first place. To remember that when we have a
measured quantity, that quantity must have some unit attached to it. There’s actually a very good reason
why this is so, why measured quantities need units.

Imagine that we had plans for
building a tree fort in a tree in our backyard. Naturally, our fort would be
designed so that it could fit the dimensions of the tree. And let’s say that the dimensions
of our tree house on the floor were 10 feet wide by 10 feet long.

Now, if we were to go to the
hardware store to purchase the wood we need to build a tree house, to get the right
amount of wood, we would need to know the units, that is, the length of wood we need
to buy. If we wanted to buy four large
pieces to help frame out the base of our tree fort, each of these pieces would need
to be 10 feet long. If we didn’t specify the units, if
we just said each board needs to be 10 long, the person listening might ask 10
what? 10 inches, 10 feet, 10 meters.

In order to communicate how much
wood we actually want to purchase, we’ll need to know the units involved. And it’s the same for building
anything, whether it’s a tree fort or an automobile or a rocket ship. To know how much of something there
is, we need to include units in that measurement. So units then tell us just what
quantity a given number refers to.

All by themselves, a given number
could refer to anything. But when we combine a number with a
unit, we know what we’re talking about. So while a number all by itself is
an abstract quantity, when we include a unit with that number, now it corresponds to
some physical value. This is why units are so important
when it comes to making measurements.

So if using units is a fairly new
thing, don’t worry. There’s a very good reason for
learning to do it. And as we’ll see, including units
when we do a calculation or make a measurement can actually help us check whether
we’ve done things correctly. In other words, using units can
help increase our confidence that we’ve arrived at the right answer.

Now, just as an aside, there are
different systems of units out there. One system measures distance, say,
in units of feet or in miles. And another measures distance in
units of meters or kilometers. The system of units that we’ll use
and the one we can consider standard is called the SI system. In this system, distances are
measured in units called meters, time is measured in seconds, mass is measured in
kilograms, and so on.

To get an idea of how these units
work, let’s imagine now that we’re not drawing up plans for a tree house, but rather
drawing up plans for a real full-sized house. Say that we measure 10 meters out
in one direction and then perpendicular to that 10 meters out in another.

Now, if we make this into a square
this way, then let’s say that the area of that square is the footprint or the
foundation for this house. And if we wanted to know the total
area of this footprint, we could figure that out using the two measured lengths we
have here, 10 meters and 10 meters. Here’s how this works.

To solve for the area in this
square, we’ll call it 𝐴. We know that we’ll need to multiply
one side of the square by another side. Or since the sides are the same
length, we could say a side by itself. We can see that this is going to
involve multiplying 10 by 10. That’s because each of these
distances is 10 meters.

But remember, each one of these
distances isn’t just 10 something, but they’re specifically 10 meters. What we’re going to do then is not
just multiply one number by another number, 10 by 10. But we’re also going to multiply a
unit by a unit, meters by meters. What we’re finding out is that when
we multiply one measurement by another measurement, not just the numbers but also
the units are involved. And it’s for that reason that if we
wrote down that this is equal to 100 meters, then that answer would be
incorrect. It’s true that 10 times 10 is
100. But we also need to multiply the
units, meters, together. If we think about it just in terms
of the units, then what we’re doing here is multiplying a meter by a meter. And the question is, what is
that?

One way to think of this is to
replace our unit, in this case meters, with a variable. 𝑥 is a common representation of a
variable. And from math class, we’re probably
used to seeing that if we multiply 𝑥 by 𝑥, the answer is 𝑥 squared. And since 𝑥 is a variable, that
means we can put any particular quantity in for 𝑥. And this relationship will be
true.

So if we substitute in meters for
our variable 𝑥, that tells us that a meter times a meter is a meter squared. So when we multiply 10 meters by 10
meters, the correct answer is 100 meters squared. We’ve combined both the numbers as
well as the units.

Now let’s say that this was a
question in homework or on an exam. The question was, what is the area
𝐴 of the footprint of this house? Earlier on, we said that using
units in calculations can actually increase our chances of getting the right
result. And we see that here in this
example of calculating the area 𝐴. Here’s why that is.

When we’re asked to solve for an
area 𝐴 and we know we’re working in the SI system of units, then right away, we
know that our answer should have units of meters squared. Unless the problem statement says
otherwise, these are the units we’d expect. Knowing this, when we calculate our
answer for 𝐴, we can look and see if the units match up with the units we’d
expect. If we do, then we have increased
confidence in our answer. We’ve probably solved for it in the
right way. So rather than tripping us up or
becoming something we might forget, using units can be helpful.

Now, so far, we’ve seen that units
of distance, meters and meters, can combine together. But actually, this is true even if
the units we’re considering are very unlike one another. As an example of that, let’s
consider an acceleration.

Now acceleration, we know, has
units of meters per second per second, or meters per second squared. Now since we’re focusing on units,
the number of meters per second squared is not so important here. So let’s just call that 𝑥. Knowing this acceleration, let’s
say that we wanted to multiply it by a time. And because we’re in the SI system,
that time has units of seconds. Once more, the particular number of
this unit isn’t so important. Let’s just call it 𝑦.

What we’re trying to do then is
multiply 𝑥 meters per second squared by 𝑦 seconds. We can see these quantities are
totally different. This one is an acceleration,
whereas this one is a time. But nonetheless, we’re still able
to multiply these units by one another. The product of these two values
would be 𝑥 times 𝑦. That’s the number. And then, as far as the units, they
would be meters per second squared multiplied by seconds.

Once again, it can be helpful to
think of our particular units as though they were variables. In this case, we can imagine that
our unit s for seconds is a variable. Thinking of it that way, we want to
figure out what will s divided by s squared be. Looking at this fraction, we can
see that one factor of s will cancel out. It’s s divided by s, which is equal
to one. So the fraction overall is equal to
one divided by s.

Overall then, when we multiply an
acceleration, meters per second squared, by a time in seconds, our final unit will
be meters per second, which is a speed. Even when units are very unlike one
another, like we saw here, we can still combine them through multiplication to come
up with a final equivalent unit. Now that we’ve seen how units can
combine together, let’s get a bit of practice through an example exercise.

Which of the following is an
appropriate symbol for the unit of a quantity found by dividing a temperature by a
distance?

Now, as we look over the five
answer options, we see these different symbols representing different physical
quantities. And we need to be careful not to
confuse these quantities with one another.

For example, the m in option A is
different from the m in option B. Those don’t represent the same
thing. Notice also that the k in option D
is lowercase, whereas the Ks in other options are uppercase. This difference is intentional, and
these symbols mean different things.

Now our question asks us, which
symbol correctly represents a temperature divided by a distance? As a starting point, we can
establish that we’re working with a particular system of units. That’s called the SI or
international system. The reason it’s important to know
what system we’re working with is because temperatures and distances are represented
differently in different unit systems.

Within the SI system, temperature
is represented using a unit called a Kelvin. So the temperature of an object
would be reported in these units. We would say it’s 10 Kelvin or 87
Kelvin or something like that. This unit is abbreviated using a
capital K. And looking back at our answer
choices, we see this capital K in four of our five answer options.

Now moving on to distance, in the
international system, the SI system for short, the unit of distance is the
meter. And we abbreviate this unit using a
lowercase m. Along with measuring distances in
units of meters, we can also measure distances in units of kilometers, abbreviated
km. And specifically, this is a
lowercase k that comes in front of the m for meters.

Now that we know this, that solves
the mystery of why this k in answer option D is different from all the other Ks. This lowercase k refers to the kilo
in kilometers, whereas these uppercase Ks in the other options refer to a
temperature in Kelvin. This means, by the way, that answer
option D does have a distance in it. But it has no temperature. Therefore, it won’t be our answer
to this question.

Now that we know that temperature
in Kelvin is abbreviated capital K and distance in meters is abbreviated lowercase
m, we can see what the symbol will look like when we divide a temperature by a
distance. Our answer then should show us
capital K divided by lowercase m. That’s a temperature divided by a
distance in the SI system.

Looking at option A, we might think
that this is a distance in meters multiplied by a temperature in Kelvin. But actually, this m is a
prefix. It stands for “milli” or one
thousandth. The reason we know this m is not a
distance in meters is because there’s no multiplication sign between these two
letters. If there were, this would be a
distance in meters multiplied by a temperature in Kelvin. But because there’s not, we know
that that m stands for “milli.” So we’re talking about thousandths
of a Kelvin. In other words, this answer option
has a temperature, but it has no distance involved. Therefore, it also won’t be our
answer.

Moving on to option B, this shows
us a temperature in Kelvin divided by a distance in meters. This matches the symbol that we
were looking for. So option B looks like it might be
our answer. Before we make it our final choice
though, let’s look at options C and E.

Option C has a temperature in
Kelvin multiplied by a distance in meters. Because there’s no division
involved here, we won’t choose that. And then, in option E, we have a
temperature in Kelvin divided not by a distance in meters, but by a quantity in
meters times meters.

Now if we think about it, if we
have a distance of one meter and then we multiply that by a distance of one meter,
then the result is not a distance along a line, but rather an area. And the area is one meter
squared. It’s this unit, square meters, that
we see in the denominator of option E. So E is not showing us a
temperature divided by a distance. But it’s showing us a temperature
divided by an area. Therefore, it also isn’t the symbol
we’re looking for.

So then option B really is our
final answer. The symbol for the quantity found
by dividing a temperature by a distance in the SI system is capital K for Kelvin,
divided by lowercase m for meters.

Let’s take a moment now to
summarize what we’ve learned about units of measured quantities. Starting out, we saw that units are
so useful because they tell us what particular quantity a number refers to. If we were looking just at a pure
number all by itself, we wouldn’t know what that number referred to. But when we attach a unit to the
number, say in this case meters per second, then we know that this particular
number, six, refers to a speed, and specifically a speed in meters per second.

Because units help us identify what
a certain number refers to, units are included in all quantities that we
measure. Another thing we learned was that
units, just like numbers, can be combined through multiplication or division. We could take a time in seconds,
for example, and multiply that by a temperature in Kelvin. Or we could take a mass in
kilograms, say, and divide that by a distance in meters.

As a special case of unit
multiplication, we also saw that a unit can be multiplied by itself. Say we have a distance in meters
and we multiply that by another distance in meters. Then, in this case, the final unit
would be meters times meters, or meters squared. And lastly, we saw that when we use
units, they help us check our calculations for accuracy.