Scientific Conventions (VCE SSCE Physics): Revision Notes

Scientific Conventions

Introduction

Scientists need to communicate measurements and data in a standardised way so that everyone can understand the size and precision of values being discussed. Whether measuring something as small as a bacterium or as large as the distance to the Sun, scientific conventions ensure clear communication.

Key definitions:

• Magnitude: The size or numerical value of a quantity without sign (positive or negative) or direction
• SI unit: An accepted standard unit used for measuring a quantity
• Significant figures: All digits quoted, starting with the first non-zero digit, giving an indication of the confidence in a measurement

Units of measurement

Units of measurement provide a standard reference for the magnitude of different quantities. This allows us to compare different physical objects and processes in a consistent way.

Base SI units

The International System of Units (SI) has seven base units. These are defined using physical constants or processes. For example, the metre is defined with reference to the speed of light, and the second is defined by the frequency of energy transitions in caesium-133.

All measurements in physics should ultimately be expressible using these seven fundamental units.

Derived SI units

Derived SI units are formed by multiplying or dividing the base SI units. These units have special names to make them easier to work with. Here are some common examples:

For instance, force is measured in newtons ($\text{N}$), which is equivalent to $\text{kg m s}^{-2}$. This tells us that force involves mass, length, and time.

Quantities without dedicated units

Some quantities don't have their own special unit names. Instead, they use combinations of base and derived SI units:

These combinations tell us about the relationship between quantities. For example, velocity is measured in $\text{m s}^{-1}$, which shows it's a distance per unit time.

SI prefixes

Prefixes are added to SI units to indicate different orders of magnitude. This makes it easier to work with very large or very small numbers:

For example, $9.0$ picometres can be written as $9.0 \text{ pm} = 9.0 \times 10^{-12} \text{ m}$.

Significant figures

When we write down a measurement, the number of digits we use communicates how confident we are in that value. This is where significant figures become important.

Understanding significant figures

Significant figures indicate the degree of confidence we have in a measured value. They help us avoid claiming more precision than our measurements actually have.

Rules for counting significant figures

There are four key rules for determining how many significant figures a number contains:

Let's look at some examples:

• $0.32$ has two significant figures (leading zeros don't count)
• $53.2$ has three significant figures (all non-zero digits count)
• $53.00$ has four significant figures (trailing zeros count)
• $302$ has three significant figures (zeros between digits count)

Rules for calculations

When performing calculations, we need to ensure our answer reflects the appropriate level of precision:

For addition and subtraction: Use the least number of decimal places from your starting values. For example, $34.477 + 2.31 = 36.78$ (rounded to 2 decimal places because $2.31$ has only 2 decimal places).

For multiplication and division: Use the least number of significant figures from your starting values. For example, $34.477 \times 2.31 = 79.6$ (rounded to 3 significant figures because $2.31$ has only 3 significant figures).

Scientific notation

Scientific notation provides a standardised way to express numbers while clearly showing the number of significant figures. Numbers are written in the form:

$m \times 10^n$

where $m$ is a number with an absolute value between $1$ and $10$ (such as $5$ or $4.56$), and $n$ is an integer. All the digits in $m$ are significant.

We use scientific notation for two main reasons:

1. To write very large or very small numbers compactly: For example, $6.67 \times 10^{-11}$ is much easier to work with than $0.0000000000667$.
2. To correctly show our level of certainty: If we say a building is $20 \text{ m}$ tall, we're indicating we measured to the nearest metre. However, if we write $2.00 \times 10^1 \text{ m}$, we're showing we measured to the nearest $10 \text{ cm}$ (three significant figures instead of one).

Converting to scientific notation

To convert from standard notation to scientific notation:

1. Move the decimal point so it comes after the first significant digit
2. Count how many places you moved the decimal point - this becomes $n$
3. If you moved the decimal point to the left, $n$ is positive
4. If you moved the decimal point to the right, $n$ is negative

Examples of scientific notation

Notice how scientific notation makes the number of significant figures completely clear. Each example shows exactly which digits are significant.

Worked example: unit conversion with significant figures

Remember!