Proportionality SPH4U

Introduction In physics, we are often interested in how one variable affects another.

Introduction In physics, we are often interested in how one variable affects another. Example: If you double the voltage supplied to a circuit, what happens to the current?

Voltage and Current Given:

Voltage and Current We can see that if this doubles Given:

Voltage and Current We can see that if this doubles Given: this also doubles

The “Fishy” We can say that: “Current is directly proportional to voltage.”

The “Fishy” We can say that: “Current is directly proportional to voltage.” To turn an equation into a proportionality, set everything else in the equation = 1.

Let’s show this algebraically: Initial: Final:

Let’s show this algebraically: Initial: Final: Note that resistance is constant. It is our controlled variable.

Let’s show this algebraically: Initial: Final: Substitute for what’s changed

Let’s show this algebraically: Initial: Final: Rearrange the expression until a part of it equals the original

Let’s show this algebraically: Initial: Final:

Experimentally How can you answer the question: If you double the voltage supplied to a circuit, what happens to the current? experimentally?

The Graph Set up your circuit, change your voltage (independent variable), measure your current (dependent variable), and graph your data.

The Graph Set up your circuit, change your voltage (independent variable), measure your current (dependent variable), and graph your data.

The Graph Set up your circuit, change your voltage (independent variable), measure your current (dependent variable), and graph your data. What is the slope of this line?

Slope Until we determine the relationship between current and all other variables, we just call it k, the “proportionality constant.”

Slope Until we determine the relationship between current and all other variables, we just call it k, the “proportionality constant.” (In this example, k should equal .)

An Inverse Relationship Consider now the question: If you double the resistance of a circuit, what happens to the current?

The “Fishy” We can say that: “Current is inversely proportional to resistance.”

An Inverse Relationship Initial: Final: Note that this time, voltage is our controlled variable.

An Inverse Relationship Initial: Final:

An Inverse Relationship Initial: Final:

An Inverse Relationship Initial: Final: If the resistance is doubled, the current is halved.

The Graph The graph of this relationship would look like:

And what’s our proportionality constant k? The Graph The graph of this relationship would look like: How do we know this is actually 1/R and not some similar relationship like 1/R2? And what’s our proportionality constant k?

The Revised Graph We graph I vs. 1/R and see if it’s a straight line:

Inverse Square An inverse square relationship looks similar.

You might also see powers For example, if you were measuring the distance Dd an object travels while accelerating from rest in time Dt : Slope = ½ a

You might also see roots For example, if you were measuring the work W required to accelerate a mass to a speed v: