Physics 1.1 - 1.4 - Units, prefixes and quantities

Quantities and SI units

A physical quantity is a property that can be measured, such as time, force or electric
current. A measurement needs both a numerical value and a unit. The number tells us how much; the unit tells us what scale it uses.

For example, $12.0\,\mathrm{s}$ communicates a duration; $12.0$ on its own does not. The unit tells us
what reference scale was used. The International System of Units, abbreviated SI, gives
scientists a common set of references, so a measurement made in one laboratory can be
compared meaningfully with a measurement made elsewhere.

Physics 1.1 refers to the units listed in Appendix 5 of the Combined Science specification.
The Appendix 5 unit list is:

Unit symbols are case-sensitive. $\mathrm{W}$ means watt, while $w$ is not its SI symbol. Symbols do
not gain a plural $\mathrm{s}$: write $5\,\mathrm{kg}$, not “5 kgs”. Leave a space between the value and the
unit symbol, and write $300\,\mathrm{K}$, not “300 °K”; the kelvin symbol has no degree sign. Unit
names are lower-case in ordinary writing even when named after a person: newton, volt and
tesla, but $\mathrm{N}$, $\mathrm{V}$ and $\mathrm{T}$.

Prefixes as scale factors

The same physical quantity may be awkward to express in an unprefixed unit. A radio
frequency can involve billions of hertz, while a short duration can involve billionths of a
second. An SI prefix attached to a unit symbol replaces a power-of-ten multiplier.

The prefix and unit symbol are joined, with no space between them. Therefore:

• $5.2\,\mathrm{MW} = 5.2 \times 10^{6}\,\mathrm{W}$
• $7.0\,\mathrm{ns} = 7.0 \times 10^{-9}\,\mathrm{s}$
• $3.4\,\mathrm{cm} = 3.4 \times 10^{-2}\,\mathrm{m}$

Case carries meaning. $M$ is mega, but $\mathrm{m}$ is milli: $1\,\mathrm{MW}$ is one million watts, whereas
$1\,\mathrm{mW}$ is one thousandth of a watt. Context also matters. In $\mathrm{m}$, the letter is the unit
symbol for metre; in $\mathrm{mm}$, the first $\mathrm{m}$ is the milli prefix and the second $\mathrm{m}$ is metre.
The printed micro symbol may appear as $μ$ or $µ$; both represent micro, but ordinary $u$
is not the SI prefix symbol.

The kilogram needs special care because it is the SI base unit for mass even though its name
already contains kilo. Useful mass relationships are $1\,\mathrm{kg} = 1000\,\mathrm{g}$, $1\,\mathrm{g} = 1000\,\mathrm{mg}$, and
therefore $1\,\mathrm{mg} = 10^{-6}\,\mathrm{kg}$.

Changing the prefix does not change the physical quantity. $1.20\,\mathrm{km}$ and $1200\,\mathrm{m}$
describe the same length. A smaller unit is needed more times, so expressing a quantity in
a smaller unit gives a larger numerical value.

Standard form for extreme values

Long strings of zeros are hard to read and easy to mistype. Standard form writes a
number as:

The coefficient $a$ contains the significant digits. The exponent $n$ records the scale. A
positive exponent makes a number greater than or equal to 10; a negative exponent makes a
positive number smaller than 1. A negative exponent does not mean that the quantity itself
is negative.

Worked example: a large frequency

Write $58200000\,\mathrm{Hz}$ in standard form.

1. Move the decimal point left until the coefficient is between 1 and 10:
   $58200000 \to 5.82$.
2. The decimal point moved 7 places, so the exponent is $+7$.
3. Answer: $5.82 \times 10^{7}\,\mathrm{Hz}$.
4. Sense-check: $10^{7}$ means move the decimal point 7 places right, which recreates
   $58200000$.

Worked example: a small duration

Write $0.00000470\,\mathrm{s}$ in standard form.

1. The first non-zero digit is 4, giving the coefficient $4.70$.
2. The decimal point must move 6 places right to reach that coefficient, so the original
   number uses exponent $-6$.
3. Answer: $4.70 \times 10^{-6}\,\mathrm{s}$.
4. Sense-check: the negative exponent is appropriate because the duration is less than
   one second.

The reverse route uses the exponent to restore the decimal. For example,
$6.4 \times 10^{-5}\,\mathrm{N} = 0.000064\,\mathrm{N}$. Standard form also makes a prefixed measurement's scale
visible: $2.8\,\mathrm{GHz} = 2.8 \times 10^{9}\,\mathrm{Hz}$ and $2.8\,\mathrm{ns} = 2.8 \times 10^{-9}\,\mathrm{s}$.