Physical quantities, SI base and derived units, prefixes and standard form, homogeneity of equations, and estimating and handling uncertainties.

What this dot point is asking

CCEA wants you to know the SI base quantities and units, build derived units from them, use prefixes and standard form, check whether an equation is homogeneous by comparing units on each side, and estimate and combine uncertainties in measured quantities. Homogeneity proofs and uncertainty combination are tested almost every series.

The answer

Base and derived units

A derived unit is a combination of base units. For example, the newton is $1\ \text{N} = 1\ \text{kg m s}^{-2}$ from $F = ma$, and the joule is $1\ \text{J} = 1\ \text{kg m}^2\,\text{s}^{-2}$. Use prefixes (such as $\text{k}$ for $10^{3}$, $\text{m}$ for $10^{-3}$, $\mu$ for $10^{-6}$) and standard form to keep numbers manageable.

Homogeneity of equations

For example, in $v = u + at$ each term has units of $\text{m s}^{-1}$, so the equation is homogeneous and could be correct.

Uncertainties

The absolute uncertainty is the $\pm$ range on a single measurement, often taken as half the smallest scale division or the spread of repeated readings.

Worked example: percentage uncertainty in a density

Examples in context

Example 1. Designing an experiment to minimise uncertainty. Because percentage uncertainties add, a CCEA student measuring the Young modulus picks the quantity with the largest percentage uncertainty (often the small extension) and improves it first, by using a longer wire so the extension is larger relative to its reading error.

Example 2. Sanity-checking a derived formula. Faced with a half-remembered equation in an exam, a quick homogeneity check on the base units catches a slip such as writing $v^2 = u^2 + 2at$ (units of $\text{m}^2\,\text{s}^{-2}$ versus $\text{m s}^{-1}$) instead of the correct $v^2 = u^2 + 2as$, saving marks before any number is substituted.

Try this

Q1. Show that the unit of energy, the joule, is equivalent to $\text{kg m}^2\,\text{s}^{-2}$. [2 marks]

• Cue. Use $E_k = \tfrac{1}{2}mv^2$, so units are $\text{kg} \times (\text{m s}^{-1})^2 = \text{kg m}^2\,\text{s}^{-2}$.

Q2. A length is measured as $0.250\ \text{m}$ with an absolute uncertainty of $0.005\ \text{m}$. State its percentage uncertainty. [1 mark]

• Cue. $\frac{0.005}{0.250} \times 100 = 2\%$.

Q3. A current of $0.50\ \text{A} \pm 2\%$ flows through a resistor of $12\ \Omega \pm 3\%$. Find the percentage uncertainty in the power dissipated, given $P = I^2 R$. [2 marks]

• Cue. $P = I^2 R$, so add $2 \times 2\% + 3\% = 7\%$.