In this lecture I'll go over the four main proportional relationships that we deal with
in IB physics, which are linear, quadratic, inverse, and inverse square relationships.
I'll start off with the notation that we use. That symbol that looks kind of like
a fish means "is proportional to," so if you see notation like what I have below,
Y, that notation means "Y is proportional to X." What it means for something to be proportional to
something else is that that one thing is equal to a constant multiplied by the other thing,
so here "Y is proportional to X" means that Y is equal to some constant multiplied by X. We
use the symbol when we don't know the value of the constant and are really just worried about
the relationship between Y and X. One implication of this definition is if Y is proportional to X,
X is proportional to Y. because if Y is equal to a constant times X, that means that X is equal to
1 over that constant times Y, and this is just another constant. 1 over a constant is itself a
constant, so that means that X is also equal to a constant times Y. So we could say that X is
proportional to Y. So if Y is proportional to X, X is proportional to Y. These are the four
main proportional relationships that we worry about in IB physics. There are others as well
but we mainly focus on these when we're analyzing proportionality. In a linear relationship between
two variables Y is proportional to X, and you can see that results in a straight line linear
graph when we graph Y against X. In a quadratic relationship Y is proportional to X squared, which
means that Y is equal to some constant times X squared, and you can see the shape of a quadratic
equation that happens when we graph Y vs. X in that situation. In an inverse relationship Y is
proportional to 1 over X and you can see the graph for that, and in an inverse-square relationship Y
is proportional to 1 over X squared and you can see we have the shape of that graph as well. So
one thing knowing these patterns allows us to do is to take the relationship between 2 variables in
an equation and understand exactly what the graph of those two variables would look like if we held
everything else constant. I'll give you a bunch of examples of what using these physically looks like
in problem solving. To identify the proportional relationship between 2 variables in an equation,
hold all other variables constant and match the resulting equation to one of the four
relationships. If both variables appear on one side of the equation, rearrange the equation so
that one variable is on each side. In this first example, it's asking: Given this equation what is
the relationship between force and acceleration, and what is the shape of a force vs. acceleration
graph? Because we're only worried about force and acceleration, the rule says to hold all other
variables constant, so m is going to be a constant here. I'll treat F as my Y and a as my X. This
fits the form Y = some constant times X, and that implies that Y is proportional to X, so in the
same way we can understand that F is proportional to a. Force is proportional to acceleration. So
what we get from this pattern is that force has a linear relationship with acceleration,
because if we go back to this chart we can see that if two variables are proportional to just
each other with no square or no denominator, they exist in a linear relationship with each other. So
force has a linear relationship to acceleration. We can also see, based on the equation, that force
is proportional to acceleration, and because this fits a linear pattern I know that the graph of
force versus acceleration is also going to be linear, so just using the shape of the equation
I can automatically understand what the graph of those two variables will look like. So that's why
we care about these proportional relationships. We can talk about the relationship between two
variables and understand how the resulting graph of the two will look. Example number two says
"Given this equation, what is the relationship between mass and acceleration, and what is the
shape of a mass versus acceleration graph?" So here we're going to hold force constant,
so we're imagining we have a constant force in this situation, so I'll isolate mass by
dividing both sides by acceleration, and when I do that I find that the equation fits this form:
Y is equal to a constant times 1 over X, which means that Y is proportional to 1 over X,
so I can see that the mass is proportional to 1 over acceleration and that tells us that this
relationship is inverse. Mass has an inverse relationship with acceleration, and mass is
proportional to 1 over acceleration, and because this is an inverse pattern I know automatically
that the graph of mass versus acceleration is going to look like this. It will have the shape of
an inverse graph when we plot those two variables. Example problem 3 says "Given this equation, what
is their relationship between the gravitational field and the distance from the source?" I can
see that gravitational field is lowercase g and the distance from the source is r, so I'm going to
have to hold the other two things constant in that equation. I'll rearrange this just a little bit
just to draw out one over r squared, and when I do that I can see that this fits the form Y equals
some constant times 1 over X squared, which means that Y is proportional to 1 over X squared, which
means that the gravitational field is proportional to 1 over the distance from the source squared,
so what this tells us is that the gravitational field has an inverse-square relationship with
the distance and the gravitational field is proportional to 1 over distance from the source
squared, and because this is an inverse-square relationship we know automatically that the
graph of gravitational field versus the distance from the source will take this general shape.
Example 4 says "Given this equation, what is the relationship between the gravitational
field and the mass?" so I'll hold capital G and r squared constant. I'll rearrange the equation
just to get the mass by itself. I can see that the gravitational field is proportional to the mass,
which tells us that the gravitational field has a linear relationship with mass,
and the gravitational field is proportional to mass, and this is what a graph of gravitational
field versus mass would look like. Finally, Example 5 says "Given this equation, what is
the relationship between the mass and the distance from the source?" so now we need to isolate m and
r squared, so I'll hold lowercase and uppercase G constant. So I'll rearrange the equation to get
one of those variables by itself. I'll just get an X by itself. When I generalize this,
I see that this fits the form Y equals a constant times X squared, so that means that Y is
proportional to X squared and mass is proportional to r squared. So what we get from this is the mass
has a quadratic relationship with distance, and the mass is proportional to distance squared,
so this is the mass versus distance from source graph, and it fits a quadratic pattern. We can
also find the mathematical relationship between variables by looking at the pattern on a graph of
both variables, assuming other variables are held constant. As an example, I can see that
momentum has a linear relationship with velocity, which means that momentum is proportional to just
velocity without anything done to it, so this is the information I can get just looking at
that graph. When I look at the kinetic energy versus velocity graph, I can see that that's a
quadratic pattern, so I know that kinetic energy is proportional to velocity squared. So that's the
information I can get from that graph. When you look at resistance versus current, I can see that
that relationship is inverse, so therefore the resistance is proportional to 1 over the current,
and that's the information I have and when I look at a graph of an electric field over distance
that's an inverse-square relationship. So I know that the electric field is proportional to 1 over
the distance squared, and the electric field has an inverse square relationship with distance. So
that's how you use proportional relationships along the patterns that show up in graphs to
get more information about the relationships between different variables and physics.