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## Abstract

### Learning Outcomes

• Recognise proportionality.
• Recognise inverse proportionality.
• Create equations from proportional relationships.

### Keywords

• Proportional
• Inversely proportional
• Proportionality
• Constant of proportionality
• Page Contents

## Proportionality

In physics, we often talk about proportionality. This is a relationship between two quantities where they increase or decrease at the same rate. In other words, when quantity A changes by a certain factor, quantity B will change by the same factor. Take for example a bar of chocolate that you can buy for €1. If you wanted to buy two bars of chocolate, it would then cost you €2. Likewise, ten bars of chocolate would cost €10 etc. While most shops wouldn't allow it, you could also potentially buy half a bar of chocolate for €0.50. The relationship between the money you spend and the amount of chocolate you get is called a proportional relationship, and that the amount of chocolate you get is proportional to (or sometimes directly proportional to) the amount of money you spend. We use the symbol ∝ to say that quantities are proportional, i.e.: $A \propto B$ , or to use our previous example: $\text{amount of chocolate} \propto \text{price paid}$

## Inverse Proportionality

Some relationships don't increase or decrease at the same time, but are still related proportionally. Think about a long journey to see some family. The speed you travel at is related to the time the journey takes, but not in the same way as above. As you increase your speed, the journey time decreases instead. In fact travelling twice as fast will cut the journey time in half. We call this type of relationship inverse proportionality. If one quantity increases by a certain factor, the other quantity decreases by that same factor. We use the same symbol as for proportionality, but represent one quantity by its inverse, so: $A \propto \frac{1}{B}$ , or to use our previous example: $\text{speed} \propto \frac{1}{\text{journey time}}$.

## Proportion Equations

All proportional relationships can be rewritten in the form of an equation with a constant of proportionality. So our proportional relationship can be rewritten from: $A \propto B$ to $A = kB$

This constant is usually written as $$k$$ by default, but it may have other symbols (or combinations of symbols) in specific cases. If we look at our previous two examples: $\text{amount of chocolate} \propto \text{price paid}$ $\Rightarrow \text{amount of chocolate} = \text{price per chocolate} \times \text{price paid}$ and $\text{speed} \propto \frac{1}{\text{journey time}}$ $\Rightarrow \text{speed} = \text{distance}\times\frac{1}{\text{journey time}}$ , we see that the constant can have entirely different meanings depending on the context. We will use this fact in several parts of the course.

## Other Proportional Relationships

Proportional relationships can get more complicated than simply directly or inversely proportional. Very often quantities depend on the square root or cube of another quantity. One relationship we will see a number of times on the course is called inverse square proportionality. For example we will later see that the force between two masses due to gravity is inversely proportional to the square of the distance between the masses, or: $F \propto \frac{1}{d^2}$ We will also see that the fundamental frequency of a stretched string is proportional to the square root of its tension, or: $f \propto \sqrt{T}$ We will make note of these relationships as we come across them on the course.

## Detecting Proportionality

As explained in the introduction, physics and science in general are concerned not just with theory, but practice as well. As part of this course, you need to be able to recognise when relationships are proportional, particularly as a result of your experiments. You do this by drawing a graph of the relationship. The following graphs illustrate various types of proportionality: Other proportional relationships will have their own characteristic graphs, but can always be replotted to make a proportional graph. If we look at the frequency of a stretched string example from before ($$f \propto \sqrt{T}$$), plotting $$f$$ vs. $$T$$ will yield a curve characteristic of square roots, but plotting $$f$$ vs. $$\sqrt{T}$$ will yield the standard straight line through the origin graph.