Physical quantities and units
A physical quantity is anything we can measure, such as length, mass, time or temperature.
To measure is to compare a quantity with a fixed amount called a unit. The answer always has two parts: a number (the magnitude) and a unit. "7" alone means nothing; "7 cm" tells us the size.
Old units like the hand span, foot or cubit changed from person to person. So in 1960 scientists agreed on the International System of Units (SI), used in almost every country.
Some quantities need a direction too. A scalar has only size (mass 5 kg, time 10 s). A vector has size and direction (a force of 10 N downwards, a velocity of 20 m/s north).
SI base units and derived units
| Quantity | SI unit | Symbol |
|---|---|---|
| length | metre | m |
| mass | kilogram | kg |
| time | second | s |
| electric current | ampere | A |
| temperature | kelvin | K |
| amount of substance | mole | mol |
| brightness of light | candela | cd |
All other units are derived from these. Area = length × length, so its unit is m². Volume is m³. Speed = distance ÷ time, so m/s. Density = mass ÷ volume, so kg/m³.
Rules for writing units: small letters for most symbols (m, kg, s), capital when named after a person (A for Ampère, K for Kelvin, N for Newton); no full stop after the symbol; a space between number and unit (5 kg).
Prefixes and converting units
| Prefix | Symbol | Means |
|---|---|---|
| kilo | k | × 1000 |
| centi | c | ÷ 100 |
| milli | m | ÷ 1000 |
| micro | µ | ÷ 1 000 000 |
1 km = 1000 m · 1 m = 100 cm · 1 cm = 10 mm · 1 kg = 1000 g · 1 L = 1000 mL.
Rule: changing to a smaller unit, the number gets bigger (multiply). Changing to a bigger unit, the number gets smaller (divide).
Area and volume
Careful: 1 m = 100 cm, but 1 m² = 100 × 100 = 10 000 cm² and 1 m³ = 1 000 000 cm³.
Time
Time is not in tens: 1 min = 60 s, 1 h = 60 min = 3600 s.
Measuring length and time well
The least count of an instrument is the smallest value it can read. For a normal ruler it is 1 mm; for a stopwatch, often 0.01 s.
- Start from the zero mark, or subtract the starting reading (end − start).
- Keep your eye straight above the mark. Looking from the side gives parallax error.
- To measure a curved line, lay a thread along it and then measure the thread.
- To measure something very thin, like one sheet of paper, measure 100 sheets and divide by 100.
- For short times, time many swings of a pendulum (say 20) and divide, which reduces the error from your reaction time.
- Repeat a reading and take the average. Record results in a table with units in the column headings.
Accuracy, rounding, bounds and estimation
Accuracy is how close a reading is to the true value. Precision is how close repeated readings are to each other, or how fine the scale is.
Every measurement is rounded. If a length is 7.4 cm to the nearest 0.1 cm, the true value lies within half a step either side:
7.35 cm ≤ L < 7.45 cm
7.35 is the lower bound and 7.45 the upper bound. This range is called the error interval.
Significant figures show how precise a number is: 0.0520 has 3 significant figures (5, 2 and the last 0). Your answer should not have more significant figures than your least precise measurement.
Estimation means getting a sensible rough value: a door is about 2 m tall, an apple about 0.1 kg, a heartbeat about 1 s. Estimating first helps you spot mistakes in calculations.
Key formulas and definitions
- Measurement = number × unit (e.g. 7 × 1 cm = 7 cm)
- 1 km = 1000 m; 1 m = 100 cm; 1 cm = 10 mm; 1 kg = 1000 g; 1 L = 1000 mL
- 1 m² = 10 000 cm²; 1 m³ = 1 000 000 cm³ = 1000 L
- 1 h = 60 min = 3600 s
- Length = end reading − start reading
- Bounds: value ± half the rounding unit (lower ≤ true value < upper)
- Average = sum of readings ÷ number of readings
Worked examples
1. Convert 2.5 km to metres and to centimetres.
2.5 km × 1000 = 2500 m. 2500 m × 100 = 250 000 cm.
2. Convert 3600 g to kilograms.
Going to a bigger unit, divide: 3600 ÷ 1000 = 3.6 kg.
3. A pencil lies from 2.0 cm to 9.6 cm on a ruler. How long is it, in cm and mm?
Length = 9.6 − 2.0 = 7.6 cm = 76 mm.
4. A pendulum makes 20 swings in 32.0 s. Find the time for one swing.
32.0 ÷ 20 = 1.6 s. Timing 20 swings makes your reaction-time error 20 times smaller per swing.
5. A rope is 12 m long, to the nearest metre. Write its error interval.
Half of 1 m is 0.5 m. So 11.5 m ≤ L < 12.5 m.
6. A room is 4 m by 3 m, both to the nearest metre. Find the largest possible area.
Use the upper bounds: 4.5 m × 3.5 m = 15.75 m². (The smallest possible area is 3.5 × 2.5 = 8.75 m².)
Common mistakes
- Writing a number with no unit. "Mass = 5" is incomplete; write 5 kg.
- Multiplying when you should divide. Going to a bigger unit (g → kg), the number gets smaller.
- Using 1 m² = 100 cm². It is 100 × 100 = 10 000 cm².
- Measuring from the end of the ruler instead of the zero mark, or reading from the side (parallax).