Why do we measure?
Physics tries to explain nature with numbers. To get a number, we measure. Measuring means comparing a thing with a fixed amount of the same kind. That fixed amount is called a unit.
So every measured value has two parts: a number and a unit. "The rope is 5" means nothing. "The rope is 5 m" is complete.
A quantity we can measure is called a physical quantity, for example length, mass, time, speed and force.
A good unit must be: well defined, the same everywhere, not changing with time or place, easy to copy, and of a handy size. A hand span fails, because every hand is different.
Systems of units: CGS, FPS, MKS and SI
A system of units is a full set of units that fit together. Older systems chose different base units for length, mass and time:
| System | Length | Mass | Time |
|---|---|---|---|
| CGS | centimetre | gram | second |
| FPS (British) | foot | pound | second |
| MKS | metre | kilogram | second |
In 1971 the world agreed on the SI (Système International). It grew out of MKS and adds four more base units. SI is decimal (everything goes in tens), so converting is easy.
SI base units and derived units
The 7 base (fundamental) units
| Quantity | Unit | Symbol |
|---|---|---|
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric current | ampere | A |
| Temperature | kelvin | K |
| Amount of substance | mole | mol |
| Luminous intensity | candela | cd |
Since 2019 all seven are fixed by exact values of nature's constants (like the speed of light and Planck's constant), not by a metal bar or a lump of metal. So anyone, anywhere, can rebuild them.
Two supplementary units are used for angles: the radian (rad) for a plane angle and the steradian (sr) for a solid angle. Angle in radians = arc ÷ radius.
Derived units
A derived unit is made by multiplying or dividing base units. Examples: area m², volume m³, speed m s⁻¹, acceleration m s⁻², force kg m s⁻² = newton (N), work N m = joule (J), power J s⁻¹ = watt (W), pressure N m⁻² = pascal (Pa).
Rules for writing units
- Symbols have no full stop and no plural: 5 kg, not 5 kgs.
- A unit named after a person has a small first letter in words (newton) but a capital symbol (N).
- Leave a space between number and unit: 20 m.
Prefixes and very large or small lengths
A prefix in front of a unit multiplies it by a power of ten.
| Prefix | Symbol | Value |
|---|---|---|
| giga | G | 10⁹ |
| mega | M | 10⁶ |
| kilo | k | 10³ |
| centi | c | 10⁻² |
| milli | m | 10⁻³ |
| micro | µ | 10⁻⁶ |
| nano | n | 10⁻⁹ |
| pico | p | 10⁻¹² |
| femto | f | 10⁻¹⁵ |
Special units: 1 fermi = 10⁻¹⁵ m (size of a nucleus), 1 angstrom (Å) = 10⁻¹⁰ m (size of an atom), 1 astronomical unit (AU) ≈ 1.5 × 10¹¹ m (Sun to Earth), 1 light year ≈ 9.46 × 10¹⁵ m, 1 parsec ≈ 3.08 × 10¹⁶ m.
Big distances are found by the parallax method: look at a far object from two points a known distance b apart. If the angle between the two sight lines is θ (in radians), distance D ≈ b ÷ θ.
Try it at home
Hold a finger at arm's length. Close one eye, then the other. The finger jumps against the wall. That jump is parallax. Bring the finger closer and the jump gets bigger.
Key formulas and definitions
- Measured value = n × u (number × unit); n₁u₁ = n₂u₂
- Angle (rad) = arc length ÷ radius; 1 rad ≈ 57.3°
- Parallax: D = b ÷ θ (θ in radians)
- 1 N = 1 kg m s⁻²; 1 J = 1 N m; 1 W = 1 J s⁻¹; 1 Pa = 1 N m⁻²
- 1 Å = 10⁻¹⁰ m; 1 fermi = 10⁻¹⁵ m; 1 ly ≈ 9.46 × 10¹⁵ m; 1 AU ≈ 1.5 × 10¹¹ m
Worked examples
1. Write 3.5 km in metres and in centimetres.
1 km = 10³ m, so 3.5 km = 3.5 × 10³ m = 3500 m. 1 m = 100 cm, so 3500 m = 3.5 × 10⁵ cm.
2. A bottle holds 750 mL. Write this in m³. (1 L = 10⁻³ m³)
750 mL = 0.750 L = 0.750 × 10⁻³ m³ = 7.5 × 10⁻⁴ m³.
3. Change 72 km/h into m/s.
72 km/h = 72 × 1000 m ÷ 3600 s = 20 m/s. Shortcut: multiply km/h by 5/18.
4. Write the SI unit of pressure in base units.
Pressure = force ÷ area = N ÷ m² = (kg m s⁻²) ÷ m² = kg m⁻¹ s⁻². This is called the pascal (Pa).
5. An arc of 10 cm is on a circle of radius 5 cm. Find the angle in radians and degrees.
θ = arc ÷ radius = 10 ÷ 5 = 2 rad. In degrees: 2 × 57.3 ≈ 114.6°.
6. Two observers 1000 km apart see the Moon. The angle between their sight lines is 0.0026 rad. Find the Moon's distance.
D = b ÷ θ = (1.0 × 10⁶ m) ÷ 0.0026 ≈ 3.8 × 10⁸ m. That is about 3,80,000 km.
7. Light travels 3 × 10⁸ m every second. How far does it go in one year (3.15 × 10⁷ s)?
Distance = speed × time = 3 × 10⁸ × 3.15 × 10⁷ = 9.45 × 10¹⁵ m. This is one light year. It is a unit of distance, not time.
Common mistakes
- Writing a number without its unit, like "the length is 12".
- Thinking a light year is a unit of time. It is a distance.
- Writing 5 Kg or 5 kgs. Correct: 5 kg (small k, no s).
- Mixing units in one sum, e.g. adding 2 m and 30 cm as 32. Convert first: 2.30 m.