1.1 SI units and prefixes — Practise with help

Work each problem first. Open a hint only if you are stuck, and the full working only to check.

Warm-up: remember

Write 3500 m in kilometres.
Write 0.045 s in milliseconds.
Work out $0.5 \times 1200 \times 15^2$ .

Answers

$3.5\,\text{km}$ .
$45\,\text{ms}$ .
$0.5 \times 1200 \times 225 = 135000$ .

If question 1 or 2 goes wrong, look again at the prefix table in Section 2 of the Learn page. Kilo means $10^{3}$ and milli means $10^{-3}$ . If question 3 goes wrong, work out the power before you multiply.

Practise with help

1. Write $25\,\text{cm}^2$ in $\text{m}^2$ .

Hint

The prefix applies to the metre, and the whole unit is squared. Change the centimetre to a metre first, then square the result.

Next step

Write $1\,\text{cm} = 10^{-2}\,\text{m}$ , so $1\,\text{cm}^2 = (10^{-2})^2\,\text{m}^2$ .

Full working

\begin{aligned} 25\,\text{cm}^2 &= 25 \times (10^{-2})^2\,\text{m}^2 \\ &= 25 \times 10^{-4}\,\text{m}^2 \\ &= 2.5 \times 10^{-3}\,\text{m}^2 \end{aligned}

The answer is much smaller than 25, which is what a square centimetre should be.

2. A capacitor is marked $4.7\,\mu\text{F}$ . Write its capacitance in farads.

Hint

Find the symbol $\mu$ in the prefix table and write its factor in place of the symbol.

Next step

The factor for micro is $10^{-6}$ , so replace $\mu$ by $10^{-6}$ .

Full working

4.7\,\mu\text{F} = 4.7 \times 10^{-6}\,\text{F}

There is no power on the unit here, so the factor is used once.

3. Show that $s = ut + \tfrac{1}{2}at^2$ is homogeneous.

Hint

There are two terms on the right. Each term must have the same base units as the left side.

Next step

Write the base units of $ut$ and of $at^2$ separately before you compare them with $s$ .

Full working

The left side, $s$ , is a distance, so its base units are $\text{m}$ .

\begin{aligned} ut &= \text{m}\,\text{s}^{-1} \times \text{s} = \text{m} \\ at^2 &= \text{m}\,\text{s}^{-2} \times \text{s}^2 = \text{m} \end{aligned}

Both terms are $\text{m}$ , the same as $s$ , so the equation is homogeneous. The $\tfrac{1}{2}$ is a pure number and has no unit.

4. The energy of a photon is $E = hf$ . Use base units to find the unit of the Planck constant $h$ .

Hint

Rearrange for $h$ first. The frequency $f$ is measured in hertz, and $1\,\text{Hz} = 1\,\text{s}^{-1}$ .

Next step

Dividing by $f$ gives $h = \dfrac{E}{f}$ , so write the unit of a joule divided by the unit of a hertz.

Full working

\begin{aligned} h &= \frac{E}{f} \\ &= \frac{\text{kg}\,\text{m}^2\,\text{s}^{-2}}{\text{s}^{-1}} \\ &= \text{kg}\,\text{m}^2\,\text{s}^{-1} \end{aligned}

This is the same as a joule second, $\text{J}\,\text{s}$ .

5. A solid has a density of $7.8\,\text{g}\,\text{cm}^{-3}$ . Write the density in $\text{kg}\,\text{m}^{-3}$ .

Hint

Change the gram to a kilogram and the cubic centimetre to a cubic metre. Do the two changes in separate steps so neither is lost.

Next step

Write $7.8\,\text{g} = 7.8 \times 10^{-3}\,\text{kg}$ and $1\,\text{cm}^3 = 10^{-6}\,\text{m}^3$ .

Full working

\begin{aligned} 7.8\,\text{g}\,\text{cm}^{-3} &= \frac{7.8 \times 10^{-3}\,\text{kg}}{10^{-6}\,\text{m}^3} \\ &= 7.8 \times 10^{3}\,\text{kg}\,\text{m}^{-3} \end{aligned}

So the density is $7800\,\text{kg}\,\text{m}^{-3}$ . The cubic centimetre is a million times smaller than the cubic metre, so the number becomes much larger.

6. The force in a spring is $F = kx$ , where $x$ is the extension. One student writes the unit of the spring constant $k$ as $\text{N}\,\text{m}$ . Another writes $\text{N}\,\text{m}^{-1}$ . Decide which student is correct, and explain your choice.

Hint

Rearrange the equation to make $k$ the subject before you decide.

Next step

Dividing by $x$ gives $k = \dfrac{F}{x}$ , so $k$ is a force divided by an extension.

Full working

k = \frac{F}{x} = \frac{\text{N}}{\text{m}} = \text{N}\,\text{m}^{-1}

The second student is correct. A spring with a large $k$ needs a large force for a small extension, so the unit is newtons per metre. In base units this is $\text{kg}\,\text{s}^{-2}$ .