Units and the SI System: Quick Revision

A brief note on Units and the SI System. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• A measurement is a number times a unit; the unit is part of the answer.
• The SI has seven base units: m, kg, s, A, K, mol, cd. Every other unit is derived through a formula.
• Since 2019 the base units are fixed by constants of nature (the speed of light, Planck's constant and others).
• A prefix is a power of ten, and it goes with the unit's power: $1\ \text{cm}^2 = 10^{-4}\ \text{m}^2$.
• Scientific notation: $a \times 10^{n}$ with $1 \le a < 10$. Order of magnitude: $10^{n}$ if $a \le 5$, else $10^{n+1}$.
• Far distances by parallax, $D = b/\theta$; molecules by an oil film, thickness $=$ volume $\div$ area.

The facts at a glance

A worked example

A laser pointer's light has a wavelength of $640\ \text{nm}$. How many waves fit in $1\ \text{mm}$?

1. Same unit. $1\ \text{mm} = 10^{-3}\ \text{m}$ and $640\ \text{nm} = 6.4 \times 10^{-7}\ \text{m}$.
2. Divide. $\dfrac{10^{-3}\ \text{m}}{6.4 \times 10^{-7}\ \text{m}} = 1.56 \times 10^{3}$, about $1560$ waves.
3. Order of magnitude. $1.56 < 5$, so $10^{3}$.

Practice with new numbers

A program to try

The program converts lengths through metres, and shows how an area changes with the square of the factor.

Mistakes to avoid

• Grams as the base unit. The base unit of mass is the kilogram.
• A prefix without its power. $1\ \text{cm}^3$ is $10^{-6}\ \text{m}^3$, not $10^{-2}\ \text{m}^3$.
• Multiplying km/h by 3.6 to get m/s. A metre per second is the bigger unit, so the number gets smaller.
• Degrees in $D = b/\theta$. The angle must be in radians.
• "Kg", "kgs", "5kg". Write $5\ \text{kg}$: small k, no s, a space.

Check yourself

1. Write the unit of momentum, $mv$, in SI base units.

2. A wire is $0.35\ \text{mm}$ thick. Write this in metres in scientific notation.

3. What is the order of magnitude of $7.2 \times 10^{-6}$?

4. A garden is $12\ \text{m}$ by $8\ \text{m}$. What is its area in square centimetres?

5. A tower is $3.0\ \text{km}$ away, and two observers see it with a parallax of $0.010\ \text{rad}$. How far apart are they?