Introduction

Hopefully the lesson on vectors taught you much. Now this is going to be a relatively short lesson that explains a few concepts that are going to be important for the rest of physics. Assuming you have been taught something similar or taken a science class that has utilized this, you may simply skip over this lesson. However, if you haven't learned this before, this is going to be interesting for you.

Before we get into what dimensional analysis is, we must understand what we're dealing with here. If you look at the two words separately, you get "Dimensional" and "Analysis". The first important thing about dimensional analysis is the dimensions part.

Dimensions

A dimension, is much like maybe what you see as a unit. If you've ever measured something, you know that you need to specify what units it is in. For example, if you measure length, you might use centimeters or inches, and if you're measuring volume, you might use liters, or gallons.

Something important to note is that there are so many different units you can use to measure something like length, or time. In the world of science, we have standardized a set of three basic units that we use to express everything. This is called the International System of Units, or just the SI unit system for short. These three units include meters for length, kilograms for mass, and seconds for time. Keep these in mind because we will be using them from now on!

These are all examples of dimensions, and they give meaning to a number. What do we mean by this? Take the number $6$, for example. While it might be useful in your math class or your everyday arithmetic, in the world of science, it is utterly useless! (well, not utterly, but you get the idea). The reason why I say this is because it could mean $6$ meters, $6$ seconds, $6$ meters/second, $6$ Newtons, $6$ Newton meters, and so on. See, without a unit, a simple "number" becomes practically meaningless, so that's why we stress the importance of units. Hopefully the lesson on vectors taught you much. Now this is going to be a relatively short lesson that explains a few concepts that are going to be important for the rest of physics. Assuming you have been taught something similar or taken a science class that has utilized this, you may simply skip over this lesson. However, if you haven't learned this before, this is going to be interesting for you.

Before we get into what dimensional analysis is, we must understand what we're dealing with here. If you look at the two words separately, you get "Dimensional" and "Analysis". The first important thing about dimensional analysis is the dimensions part.

Dimensions

A dimension, is much like maybe what you see as a unit. If you've ever measured something, you know that you need to specify what units it is in. For example, if you measure length, you might use centimeters or inches, and if you're measuring volume, you might use liters, or gallons.

Something important to note is that there are so many different units you can use to measure something like length, or time. In the world of science, we have standardized a set of three basic units that we use to express everything. This is called the International System of Units, or just the SI unit system for short. These three units include meters for length, kilograms for mass, and seconds for time. Keep these in mind because we will be using them from now on!

These are all examples of dimensions, and they give meaning to a number. What do we mean by this? Take the number $6$, for example. While it might be useful in your math class or your everyday arithmetic, in the world of science, it is utterly useless! (well, not utterly, but you get the idea). The reason why I say this is because it could mean $6$ meters, $6$ seconds, $6$ meters/second, $6$ Newtons, $6$ Newton meters, and so on. See, without a unit, a simple "number" becomes practically meaningless, so that's why we stress the importance of units.

Derived Units

Another helpful thing to know about this is the concept of derived units. Derived units are units that are usually made through a combination of multiple units. For example, the unit for velocity is meters per second, or $\frac{\textrm{m}}{\textrm{s}}$. What does this mean? This tells you that something is traveling $1$ meter per $1$ second (when I say $1$, even though there isn't any place that explicitly puts a "$1$" in front of the $\textrm{m}$ and the $\textrm{s}$, it is implied, because $1$ multiplied by anything is just itself). Another derived unit is the Newton, or N, and its derived form is this: $\frac{\textrm{kg} \cdot \textrm{m}}{\textrm{s}^2}$. Whew! That's a lot of units, right? If you break it down, though, it becomes more apparent. You'll learn this later on, but since Newtons is the unit to measure force, it tells you that $1$ Newton is the force needed to give a mass of $1 ~ \textrm{kg}$ with an acceleration of $1 \frac{\textrm{m}}{\textrm{s}^2}$! Isn't that cool?

Derived Units

Another helpful thing to know about this is the concept of derived units. Derived units are units that are usually made through a combination of multiple units. For example, the unit for velocity is meters per second, or $\frac{\textrm{m}}{\textrm{s}}$. What does this mean?

This tells you that something is traveling $1$ meter per $1$ second (when I say $1$, even though there isn't any place that explicitly puts a "$1$" in front of the $\textrm{m}$ and the $\textrm{s}$, it is implied, because $1$ multiplied by anything is just itself). Another derived unit is the Newton, or N, and its derived form is this: $\frac{\textrm{kg} \cdot \textrm{m}}{\textrm{s}^2}$. Whew! That's a lot of units, right? If you break it down, though, it becomes more apparent. You'll learn this later on, but since Newtons is the unit to measure force, it tells you that $1$ Newton is the force needed to give a mass of $1 ~ \textrm{kg}$ with an acceleration of $1 \frac{\textrm{m}}{\textrm{s}^2}$! Isn't that cool?

Conclusion

The second part is analysis. We won't focus too much on the analysis part, since that requires some more advanced mathematical knowledge. Regardless, it is important to understand that in physics, the unit gives meaning to a number. Throughout lessons, we will introduce to you units and derived units, which are all important to know. Are you ready for some "real" physics now? Then let's get going!

Analysis

The second part is analysis. Analysis is a bit more interesting. When you do analysis, you are basically doing math that requires converting or changing units. For example, you might need to do dimensional analysis to convert units, or because you're solving an equation. This is the algebra-based version, so we assume you know anything divided by itself is equal to one. The same goes for units.

So if we want to convert $1 ~ \textrm{L}$ to $\textrm{mL}$, and we know $1 ~ \textrm{L} = 1000 ~ \textrm{mL}$, we can do something like this: $$1 ~ \cancel{\textrm{L}} ~ \cdot ~ 1000 ~ \frac{\textrm{mL}}{\cancel{\textrm{L}}}$$ You notice how units are able to cancel out, just like you would do with variables or numbers!

Now let's see this in an equation. A simple equation you may already know is $\textrm{speed} \cdot \textrm{time} = \textrm{distance}$. Speed is measured in $\frac{\textrm{meters}}{\textrm{second}}$, and time is measured in $\textrm{seconds}$. Thus, let's use some arbitrary numbers and see it in action! $$3 ~ \frac{\textrm{m}}{\cancel{\textrm{s}}} ~ \cdot ~ 9 ~ \cancel{\textrm{s}} = 27 ~ \textrm{m}$$ Do we see how this works? Hopefully you understand how to do math with units now!

>Now, there are many other practical uses and applications of dimensional analysis. One important use is to establish a relationship between two quantities. This is going a little beyond the scope of this lesson, but say we wanted to relate displacement (unit: $\textrm{m}$) and acceleration (unit: $\frac{\textrm{m}}{\textrm{s}^2}$). From what we already know about cancelling units, if we multiply acceleration by $\textrm{s}^2$, you can get our desired result.
IMPORTANT! These letters are not referring to variables, but units! Don't forget that when you do math with units!

Time has the dimension of $\textrm{s}$, so multiplying by time squared will give us displacement!Thus we craft an equation like so:
($\Delta x$ = displacement, $\Delta t$ = time, $a$ = acceleration)

$$\Delta x \sim a \Delta t^2$$
By realizing we can create relationships between two quantities via equations is a powerful tool. The funning wavy looking symbol in the middle is simply used to denote that the dimensions are equal, minus the possibility of extra constants, or coefficients. If we used an equal sign instead, it could potentially be an incorrect statement since we may have missed out on some constants or coefficients. Speaking of constants, some constants will have dimensions.

Again, you will learn these equations much later on, but this is just for reference. We know the force that gravity exerts on you can be given through this equation: $$F_g = mg$$ Where $g$ is a constant (its the acceleration due to gravity, again, you'll learn about this later on). Since the unit for force is given by $\frac{\textrm{kg} \cdot \textrm{m}}{\textrm{s}^2}$, and mass is given by $\textrm{kg}$, we can use dimensional analysis to solve for this constant's units: $$\frac{\frac{\cancel{\textrm{kg}} \cdot \textrm{m}}{\textrm{s}^2}}{\cancel{\textrm{kg}}}$$ This leaves us with $\frac{\textrm{m}}{\textrm{s}^2}$, which is the exact same as acceleration!

Now you see how the exact same concept can be used to apply to constants because not all constants are dimensionless scalars!

Conclusion

As you get more and more familiar or experienced with the math being done, you can simply omit the units for the sake of time and space. However, when just starting out, which we assume you are, we highly recommend adding units, because it gives you an idea of how varying units can work together to produce a different unit as a result. By using units, you see what is really going on when you input values into an equation, and not just inputting them because you were told to do so or blindly doing so.

Now after reading that lesson, can you see why it is important to understand how the unit gives meaning to a number? Throughout lessons, we will introduce to you many different units and derived units, which are all important to know. Are you ready for some "real" physics now? Then let's get going!