1.1 SI units and prefixes — Practise

Work through these on your own. Show the base units at every step, so a mistake in the units is visible before it reaches the answer.

The questions and the answers below come from a set that is checked by computer every time the book is built.

A wire has a cross-sectional area of $2.5\,\text{mm}^2$ . Write this area in $\text{m}^2$ .
A capacitor is marked $4.7\,\mu\text{F}$ . Write its capacitance in farads.
A train travels at $72\,\text{km}\,\text{h}^{-1}$ . Write this speed in $\text{m}\,\text{s}^{-1}$ .
Show that the kinetic energy equation $E_k = \tfrac{1}{2}mv^2$ is homogeneous. Use the base units $\text{kg}$ , $\text{m}$ and $\text{s}$ .
Show that the equation of motion $v^2 = u^2 + 2as$ is homogeneous.
The energy of a photon is $E = hf$ , where $f$ is the frequency. Use base units to find the unit of the Planck constant $h$ .
The drag force on a small sphere moving slowly through a liquid is $F = 6\pi\eta rv$ , where $r$ is the radius and $v$ is the speed. Use base units to find the unit of the viscosity $\eta$ .
The pascal is one newton per square metre. Show that its base units are $\text{kg}\,\text{m}^{-1}\,\text{s}^{-2}$ .
The mass of a proton is $1.67 \times 10^{-27}\,\text{kg}$ . Write this mass in grams.
A student writes the density of aluminium as $2700\,\text{g}\,\text{m}^{-3}$ . Explain the mistake and write the density correctly in $\text{kg}\,\text{m}^{-3}$ .
Two students disagree about the unit of force. Student A says $\text{kg}\,\text{m}\,\text{s}^{-2}$ . Student B says $\text{kg}\,\text{m}\,\text{s}^{-1}$ . Decide which student is correct, and show how the base units follow from $F = ma$ .
The wavelength of a green light wave is $0.000\,000\,54\,\text{m}$ . Write this wavelength in standard form to two significant figures.

Answers and concise solutions

1. The area in square metres

Answer: $2.5\,\text{mm}^2 = 2.5 \times 10^{-6}\,\text{m}^2$ . The prefix is squared, so the factor is $1 \times 10^{-6}$ .

2. The capacitance in farads

Answer: $4.7\,\mu\text{F} = 4.7 \times 10^{-6}\,\text{F}$ .

3. The speed in metres per second

Answer: $72\,\text{km}\,\text{h}^{-1} = 72 \times \dfrac{1000}{3600}\,\text{m}\,\text{s}^{-1} = 20\,\text{m}\,\text{s}^{-1}$ .

4. A homogeneity check

Answer: The joule is $\text{kg}\,\text{m}^2\,\text{s}^{-2}$ . On the right, $mv^2$ has base units $\text{kg} \times (\text{m}\,\text{s}^{-1})^2 = \text{kg}\,\text{m}^2\,\text{s}^{-2}$ . Both sides match, so the equation is homogeneous.

5. A homogeneity check

Answer: The left side has base units $(\text{m}\,\text{s}^{-1})^2 = \text{m}^2\,\text{s}^{-2}$ . On the right, $u^2$ is also $\text{m}^2\,\text{s}^{-2}$ , and $as$ is $\text{m}\,\text{s}^{-2} \times \text{m} = \text{m}^2\,\text{s}^{-2}$ . All three terms match, so the equation is homogeneous.

6. The base units of the Planck constant

Answer: $h = \dfrac{E}{f}$ , so its unit is $\text{J}\,\text{s} = \text{kg}\,\text{m}^2\,\text{s}^{-1}$ .

7. The base units of the viscosity

Answer: The number $6\pi$ has no unit, so $\eta = \dfrac{F}{rv}$ . The unit is $\dfrac{\text{kg}\,\text{m}\,\text{s}^{-2}}{\text{m} \times \text{m}\,\text{s}^{-1}} = \text{kg}\,\text{m}^{-1}\,\text{s}^{-1}$ .

8. The base units of the pascal

Answer: One newton is $\text{kg}\,\text{m}\,\text{s}^{-2}$ , so a pascal is $\dfrac{\text{kg}\,\text{m}\,\text{s}^{-2}}{\text{m}^2} = \text{kg}\,\text{m}^{-1}\,\text{s}^{-2}$ .

9. The mass in grams

Answer: $1.67 \times 10^{-27}\,\text{kg} = 1.67 \times 10^{-24}\,\text{g}$ , because a gram is one thousandth of a kilogram.

10. The mistake and the correction

The density has been written with grams in place of the SI base unit for mass, the kilogram.
To change grams to kilograms, divide by 1000. The density is 2.7 kg m^-3.

11. The correct unit of force

Answer: Student A is correct. In $F = ma$ , the mass has base units $\text{kg}$ and the acceleration has base units $\text{m}\,\text{s}^{-2}$ . Their product is $\text{kg}\,\text{m}\,\text{s}^{-2}$ , which is one newton. Student B has divided by time once instead of twice.

12. The wavelength in standard form

Answer: $0.000\,000\,54\,\text{m} = 5.4 \times 10^{-7}\,\text{m}$ .