Chapter 1. Physical quantities, units and measurement FREE
1.1 Physical quantities and SI units
In Technical Sciences we measure the world with physical quantities. A physical quantity always has two parts: a numerical value and a unit. If you say a table is \( 2 \) long, it is unclear; if you say it is \( 2 \text{ m} \) long, it carries meaning. Without the unit the number is meaningless.
The world uses the SI system (Systeme International) of units. There are seven base units, but in Grade 10 we work mainly with three:
- Length is measured in metres (\( \text{m} \)).
- Mass is measured in kilograms (\( \text{kg} \)).
- Time is measured in seconds (\( \text{s} \)).
Other units are derived units that are built up from the base units. Speed, for example, is length divided by time, so the unit is \( \text{m·s}^{-1} \).
1.2 Prefixes
We use prefixes to write very large or very small numbers neatly. Important ones are kilo (\( \times 1000 \)), centi (\( \div 100 \)), milli (\( \div 1000 \)). For example, \( 3 \text{ km} = 3000 \text{ m} \) and \( 250 \text{ mm} = 0{,}25 \text{ m} \).
1.3 Scalars and vectors
We divide quantities into two groups. A scalar has only magnitude (magnitude alone), for example mass, time, distance, speed and energy. A vector has both magnitude and direction, for example displacement, velocity, acceleration and force.
The difference is important: if a car travels at \( 60 \text{ km·h}^{-1} \), that is its speed (scalar). If you say it travels \( 60 \text{ km·h}^{-1} \) north, that is its velocity (vector).
Example 1: A learner walks \( 40 \text{ m} \) east and then \( 30 \text{ m} \) west. Determine (a) the total distance and (b) the displacement.
- Distance is a scalar, so we just add the path lengths: \( 40 + 30 = 70 \text{ m} \).
- Displacement is a vector. Take east as positive: \( +40 \text{ m} + (-30 \text{ m}) = +10 \text{ m} \).
- The displacement is therefore \( 10 \text{ m} \) east.
1.4 Measurement and significant figures
Every measuring instrument has a precision. If a ruler is marked in millimetres, you can measure to the nearest millimetre. The significant figures in a measurement show how precise it is. The rules are: all non-zero digits are significant, zeros between non-zero digits are significant, and leading zeros (as in \( 0{,}0032 \)) are not significant.
So \( 4{,}50 \text{ m} \) has three significant figures, while \( 0{,}045 \text{ m} \) has only two.
Example 2: Write down the number of significant figures in \( 0{,}00840 \text{ kg} \) and round \( 12{,}367 \text{ m} \) to three significant figures.
- In \( 0{,}00840 \) the leading zeros do not count; the significant figures are \( 8 \), \( 4 \) and the last \( 0 \), so three significant figures.
- To round \( 12{,}367 \) to three significant figures, we look at the fourth digit (\( 6 \)); it is \( 5 \) or more, so we round up: \( 12{,}4 \text{ m} \).
When you calculate answers from measurements, the answer must not have more significant figures than the measurements themselves. This keeps your answer honest about how precise the original measurement was.