 So now that we've introduced the basic concept dimensions and we've looked at the dimensions for three of our basic things we're going to be dealing with in mechanics, length, mass, and time. Let's take a look at a little bit more complex tool called dimensional analysis. In physics, we're always working with equations and these equations have different variables and constants in them. For example, one of the famous ones, force is equal to mass times acceleration. Well, each one of these quantities, force, mass, and acceleration has some dimensions that are associated with it. So we need to learn and understand a little bit about how those dimensions interact with each other. We've got some basic rules that we have to follow. The first one is that no matter what you've got, the dimensions on both sides of the equations must be equal to each other. So for a generic equation like a equals b, whatever dimensions a has, b has to have the exact same dimensions. And the next rule that we have to follow is that when we've got an equation where two things are added in terms of two terms, b and c, they have to have the same dimensions. b has to have the same dimensions as c in order to be able to add them up. And when you add them, they retain that same set of dimensions, which is what you're going to have on the other side of the equation for a. Now, if I've got things that are multiplied, I follow the general rules of algebra. So whatever dimensions b has and whatever dimensions c have, which don't have to be the same now if I'm multiplying them, when I multiply them, the product of those dimensions is the same dimensions as I've got over here for a. If I were to do some algebra on these equations, like divide both sides of the equation by c, then b has to have the same dimensions as a divided by the dimensions of c. So when I work with our dimensions in algebra, they've got to follow those same basic rules. So now let's look at how I can give you some examples. So for example, volume. Here's an equation for volume of a rectangular box, length times width times height. And as we discussed when we were first talking about dimensions of length, whether it's a length or a width or a height, as a variable, it has the same type of dimensions of length, length times length times length in terms of its dimensions. And using our rules of algebra when I multiply those three dimensions, I end up with a length cubed. So every time I've got a measurement of volume, it must have a dimension of length cubed. Now here's another example, velocity and acceleration. It's a little bit more complicated, but velocities and accelerations, the dimensions of those things, are given in your first chapter of your textbook. So here's one of the equations that we'll get in chapter 2, and it's for the average acceleration. And we find out that it's related to the difference between two velocities divided by the difference between two times. If I look at this just in terms of dimensions, what I see is that each of the velocities has a length per time type of dimensions. Ending to the times is a basic fundamental thing, so it has dimension of just time. When I do the subtraction, just like if I was doing an addition, both of those terms had to have the same dimension. And when I do the subtraction, I don't get zero like these were numbers, instead it retains the same dimension. So I have simply length over time divided by time for my dimensions. And if I do the algebra on that quantity, what I get is the length per time squared is the type of dimension I have for acceleration. Now we can use this in what we call dimensional analysis to help solve for known dimensions in a particular equation. So let's say we've got mass is equal to density times volume. Well from my previous work, I know that mass is a basic dimension, capital M. Volumes I've just seen have a dimension of length cubed. But what dimension do I have for density? Well I can do my algebra on here and recognize that I have to divide both sides of the equation by L cubed. And that leaves me with a mass per length cubed. Now some students will look at this and simply say, well yeah, that makes sense. What I have to have in the blank here is a mass per length cubed, so that when I multiply it by length cubed, I'm left with just mass. We can also use dimensional analysis to check an equation. So let's say we're given this equation here. Position is equal to velocity times time plus one-half the acceleration times time squared. That's one we'll see in the next chapter as well. Each one of these quantities has a dimension associated with it. Position is just length, velocity is a length per time, time is just time. The factor of one-half is a pure number, so it doesn't have dimensions associated with it. But the acceleration is length per time squared and time squared is time squared. When I do the algebra here, I see that the time on the bottom cancels out with the time that I'm multiplying it with, similarly here for the time squared. So this reduces down to just length plus length equals length, which is our rule that the dimensions when they're added must be equal to each other and it retains that same dimension across the equal sign. Now this will let you be able to check to see if dimensionally the equation is correct. You still have to go back and figure out if it's exactly correct because factors like my one-half, they don't work in. So maybe it was one-third, maybe it was one-quarter, but I know at least I have my dimensions correct here. Now that you've seen a few examples of dimensional analysis, you'll be able to use these as we move throughout the physics course.