1.1 Physical Quantities and the International System of Units
Every measurement is a number times a unit. The SI builds every unit in physics from seven base units, scaled by prefixes in powers of ten.
On 23 July 1983 an Air Canada Boeing 767 ran out of fuel at about 12 000 m over Manitoba. Nothing had leaked. The ground crew had worked out the fuel load correctly — as a number — but multiplied by a density in pounds per litre when the aircraft's computer expected kilograms. The aircraft took off with less than half the fuel the crew believed they had, and the pilots glided it to a safe landing on a disused runway at Gimli. A number without the right unit is not a measurement: it is a trap.
Physics is built on measurement. Before we can say that a force is "large" or a speed is "dangerous", we must compare it with an agreed standard — a unit. This section sets up the language every later chapter uses: what a physical quantity is, the seven base units of the International System (SI), how all other units are built from them, how prefixes such as kilo- and micro- scale them, and how to convert reliably from one unit to another.
Physical quantities and units
Length, mass, time, speed, force, energy, temperature: each of these can be measured, and each measurement is a comparison with a chosen standard of the same kind. When we write "the corridor is $12\ \text{m}$ long" we mean that the corridor is twelve times as long as the standard called the metre.
For example, $Q = 1.5\ \text{km} = 1500\ \text{m}$: the kilometre is $1000$ times larger than the metre, so the numerical value is $1000$ times smaller. The relation $n_1u_1 = n_2u_2$ is the whole secret of unit conversion.
A unit system needs only a few independent standards. Once length, mass and time have units, speed (length ÷ time), acceleration (speed ÷ time) and force (mass × acceleration) all get units automatically.
The International System of Units (SI)
Several unit systems grew up historically — the CGS system (centimetre, gram, second), the FPS system (foot, pound, second) and the MKS system (metre, kilogram, second). Science and almost every country now use the Système Internationale d'Unités (French for "International System of Units"), abbreviated SI, agreed by the General Conference on Weights and Measures. It has seven base quantities.
| Base quantity | SI unit | Symbol | Fixed by (since 2019) |
|---|---|---|---|
| length | metre | m | speed of light $c$ |
| mass | kilogram | kg | Planck constant $h$ |
| time | second | s | caesium-133 frequency $\Delta\nu_{\text{Cs}}$ |
| electric current | ampere | A | elementary charge $e$ |
| thermodynamic temperature | kelvin | K | Boltzmann constant $k_B$ |
| amount of substance | mole | mol | Avogadro constant $N_A$ |
| luminous intensity | candela | cd | luminous efficacy $K_{\text{cd}}$ |
Until 2019 the kilogram was a platinum–iridium cylinder kept near Paris; if it gained a few atoms of dirt, the mass of everything else in the world changed on paper. Today every base unit is defined by giving an exact value to a constant of nature, so any well-equipped laboratory anywhere can realise it. In plain words:
- second: the time taken by $9\,192\,631\,770$ oscillations of the radiation from a particular transition in the caesium-133 atom (this is how atomic clocks keep time);
- metre: the distance light travels in vacuum in $\tfrac{1}{299\,792\,458}$ of a second, so $c = 299\,792\,458\ \text{m/s}$ exactly;
- kilogram: fixed by setting $h = 6.626\,070\,15\times10^{-34}\ \text{J s}$ exactly (it is realised with an instrument called a Kibble balance);
- ampere: fixed by $e = 1.602\,176\,634\times10^{-19}\ \text{C}$ exactly — one ampere is a flow of about $6.24\times10^{18}$ elementary charges per second;
- kelvin: fixed by $k_B = 1.380\,649\times10^{-23}\ \text{J/K}$ exactly;
- mole: exactly $6.022\,140\,76\times10^{23}$ elementary entities (atoms, molecules, ions…);
- candela: fixed by the luminous efficacy of green light of frequency $540\times10^{12}\ \text{Hz}$, $683$ lumens per watt.
Two more units measure angles. They are dimensionless derived units (once called "supplementary units"), and they appear constantly in rotation, waves and optics.
The plane angle $d\theta$ subtended at the centre of a circle of radius $r$ by an arc of length $ds$ is $d\theta=\dfrac{ds}{r}$. Its unit is the radian (rad): one radian is the angle subtended by an arc equal in length to the radius. A full circle is $\dfrac{2\pi r}{r}=2\pi\ \text{rad}=360^\circ$.
The solid angle $d\Omega$ subtended at the centre of a sphere of radius $r$ by a patch of its surface of area $dA$ is $d\Omega=\dfrac{dA}{r^2}$. Its unit is the steradian (sr). The whole sphere subtends $\dfrac{4\pi r^2}{r^2}=4\pi\ \text{sr}$.
- Unit symbols are upright and never take a plural "s" or a full stop: $5\ \text{kg}$, not "5 kgs." or "5 Kg".
- Leave a space between the number and the unit: $3.0\ \text{m}$. (Exception: the degree signs, $30^\circ$.)
- A unit named after a person has a capital symbol but a lower-case name: newton (N), joule (J), kelvin (K) — and temperature in kelvin is "$273\ \text{K}$", never "°K".
- Write products with a space or a centred dot ($\text{N m}$ or $\text{N·m}$) and quotients with a solidus or a negative power ($\text{m/s}$ or $\text{m s}^{-1}$). Use at most one solidus: $\text{m/s}^2$, not "m/s/s".
- Do not mix symbols and names ("km/hour"), and do not invent abbreviations ("sec", "hrs", "mps"). Watch the space: $\text{m s}^{-1}$ is metre per second but $\text{ms}^{-1}$ is "per millisecond".
Derived units, prefixes and scientific notation
A derived unit is obtained by following the defining equation of the quantity and replacing every quantity by its SI unit. Force is defined by $F=ma$, so its unit is $\text{kg}\times\text{m/s}^2=\text{kg m s}^{-2}$. This combination is used so often that it has its own name, the newton (N). The same idea gives every unit in this course.
| Quantity | Defining relation | Named unit | In base units |
|---|---|---|---|
| force | $F=ma$ | newton, N | $\text{kg m s}^{-2}$ |
| work, energy | $W=Fd$ | joule, J $=\text{N m}$ | $\text{kg m}^2\text{s}^{-2}$ |
| power | $P=W/t$ | watt, W $=\text{J/s}$ | $\text{kg m}^2\text{s}^{-3}$ |
| pressure, stress | $p=F/A$ | pascal, Pa $=\text{N/m}^2$ | $\text{kg m}^{-1}\text{s}^{-2}$ |
| frequency | $\nu=1/T$ | hertz, Hz | $\text{s}^{-1}$ |
| electric charge | $q=It$ | coulomb, C | $\text{A s}$ |
Physics deals with sizes from the proton ($\sim10^{-15}\ \text{m}$) to the observable universe ($\sim10^{26}\ \text{m}$). We handle them with scientific notation, $a\times10^{b}$ with $1\le a\lt10$, and with SI prefixes, each of which multiplies a unit by a power of ten.
| Prefix | Symbol | Factor | Prefix | Symbol | Factor | |
|---|---|---|---|---|---|---|
| deci | d | $10^{-1}$ | deca | da | $10^{1}$ | |
| centi | c | $10^{-2}$ | hecto | h | $10^{2}$ | |
| milli | m | $10^{-3}$ | kilo | k | $10^{3}$ | |
| micro | µ | $10^{-6}$ | mega | M | $10^{6}$ | |
| nano | n | $10^{-9}$ | giga | G | $10^{9}$ | |
| pico | p | $10^{-12}$ | tera | T | $10^{12}$ | |
| femto | f | $10^{-15}$ | peta | P | $10^{15}$ | |
| atto | a | $10^{-18}$ | exa | E | $10^{18}$ |
(The list continues in steps of $10^3$: zepto/zetta $10^{\mp21}$, yocto/yotta $10^{\mp24}$, and since 2022 ronto/ronna $10^{\mp27}$ and quecto/quetta $10^{\mp30}$.) Two warnings: the kilogram already carries a prefix, so prefixes for mass are attached to the gram ($1\ \text{mg}=10^{-6}\ \text{kg}$, not "1 µkg"); and a prefix belongs to the unit before any power is applied, so $1\ \text{cm}^3=(10^{-2}\ \text{m})^3=10^{-6}\ \text{m}^3$.
1. Forgetting to raise the conversion factor to a power. $1\ \text{m}^2=10^4\ \text{cm}^2$, not $100\ \text{cm}^2$; $1\ \text{L}=10^{-3}\ \text{m}^3$.
2. Prefixes on the kilogram. $1\ \text{g}=10^{-3}\ \text{kg}$ and $1\ \text{mg}=10^{-6}\ \text{kg}$. Always convert to $\text{kg}$ before using $F=ma$ or $K=\tfrac12mv^2$.
3. Substituting numbers in mixed units. Convert every quantity to SI before substituting; then the answer comes out in SI automatically.
Exercises
- 1 Classify as base or derived and give the SI unit: (a) volume, (b) electric current, (c) density, (d) temperature.
- 2 Express the newton in SI base units.
- 3 Express the joule in SI base units, starting from $W=Fd$.
- 4 Express the watt in SI base units.
- 5 Express the pascal in SI base units.
- 6 Momentum is mass × velocity. Find its SI unit and show that it equals $\text{N s}$.
- 7 Which are written correctly? "5 Kg", "12 sec", "3.0 m/s", "20 newtons", "273 K", "273 °K". Correct the others.
- 8 Why is the kilogram the only SI base unit whose name contains a prefix, and what does that mean for writing $10^{-6}\ \text{kg}$?
- 1 $54\ \text{km/h}$ in $\text{m/s}$.
- 2 $25\ \text{m/s}$ in $\text{km/h}$.
- 3 $2.5\ \text{g/cm}^3$ in $\text{kg/m}^3$.
- 4 One day in seconds.
- 5 One year of $365.25$ days in seconds.
- 6 $500\ \text{cm}^2$ in $\text{m}^2$.
- 7 $1.0\ \text{L}$ in $\text{m}^3$ (one litre is $1000\ \text{cm}^3$).
- 8 $g=9.8\ \text{m/s}^2$ in $\text{km/h}^2$.
- 1 Write with a suitable prefix: (a) $4.5\times10^{-6}\ \text{m}$, (b) $3.2\times10^{9}\ \text{Hz}$, (c) $2.0\times10^{-9}\ \text{s}$, (d) $6.8\times10^{5}\ \text{W}$.
- 2 Write in SI base units in scientific notation: (a) $15\ \text{mg}$, (b) $250\ \text{µs}$, (c) $40\ \text{nm}$.
- 3 How many square metres is $1\ \text{µm}^2$?
- 4 A point on the rim of a wheel of radius $0.35\ \text{m}$ moves $1.4\ \text{m}$ along the rim. Through what angle has the wheel turned, in radians and degrees?
- 5 Convert $60^\circ$ to radians and $2.5\ \text{rad}$ to degrees.
- 6 A detector of area $2.0\ \text{cm}^2$ faces a small source $0.50\ \text{m}$ away. Find the solid angle it subtends and the fraction of the source's radiation (emitted equally in all directions) that it receives.
- 1 Medicine. A drug is infused at $2.5\ \text{µg}$ per kilogram of body mass per minute into a $60\ \text{kg}$ patient for one hour. How many milligrams are given?
- 2 Sport. A marathon ($42.195\ \text{km}$) is run in $2\ \text{h}\ 1\ \text{min}\ 9\ \text{s}$. Find the runner's mean speed in m/s and km/h.
- 3 Electronics. Light travels at $3.00\times10^8\ \text{m/s}$. How far does it go in one nanosecond? Why does this limit the size of a fast computer chip?
- 4 Environment. $12\ \text{mm}$ of rain falls on a $1.5$-hectare field ($1\ \text{ha}=10^4\ \text{m}^2$). Find the volume of water in $\text{m}^3$ and in litres.
- 5 Transport. A car uses $6.0\ \text{L}$ of fuel per $100\ \text{km}$. Show that this quantity has the unit of an area, express it in $\text{m}^2$, and interpret it.
E1. Board · 3 marks (a) Distinguish between a base unit and a derived unit, with one example of each. (b) Name the SI base quantities whose units are the kelvin and the candela. (c) Express the pascal in terms of SI base units.
E2. NEET Which of the following is not an SI base unit? (a) ampere (b) kelvin (c) newton (d) mole
E3. JEE Main NEET Which pair of quantities has the same unit when written in SI base units? (a) work and power (b) pressure and energy per unit volume (c) momentum and force (d) impulse and power
E4. JEE Main (Numerical value) A $1.8\ \text{kW}$ hair-dryer is used for $10$ minutes every day for $30$ days. The energy used is $x\ \text{MJ}$. Find $x$.
E5. JEE Advanced (One or more correct) Which of the following are equal to $1\ \text{kWh}$? (A) $3.6\ \text{MJ}$ (B) $3.6\times10^6\ \text{Pa m}^3$ (C) $3.6\times10^9\ \text{mJ}$ (D) $3.6\times10^6\ \text{kg m s}^{-2}$
Research question. How often do unit mix-ups cause medication errors in hospitals, and which unit conventions make them less likely?
Why it matters. Patient-safety agencies in many countries list confusion between mg and µg, between mL and mL/h, and between doses "per kg" and total doses among the commonest causes of serious drug errors. A factor of $1000$ hidden in a prefix can be fatal; this is exactly the conversion skill of this section.
What to do.
- A child of mass $18\ \text{kg}$ is prescribed $15\ \text{mg/kg}$ of paracetamol per dose. The syrup contains $120\ \text{mg}$ in $5.0\ \text{mL}$. Find the dose in mg and the volume in mL.
- An infusion must deliver $0.10\ \text{µg}$ per kg per minute to a $70\ \text{kg}$ adult. The bag holds $4.0\ \text{mg}$ in $250\ \text{mL}$. Find the pump rate in mL/h.
- Redo step 2 as if a nurse had misread µg as mg. By what factor is the rate wrong?
- Look up the guidance of a patient-safety body (for example the WHO or a national pharmacy council) on writing "µg" versus "mcg" and on trailing zeros such as "5.0 mg". Explain the reasons in terms of this section.
Tools: Calculator; a published patient-safety guideline.
Hand in: Your calculations with every unit written and cancelled, and a half-page summary of three rules you would put on a hospital poster.
Going further. Compare with the fuel mix-up at Gimli (1983) and the loss of NASA's Mars Climate Orbiter (1999). What do the three cases have in common?
🧪 Self-check quiz
Q1. How many base units does the SI have?
Q2. The candela is the SI unit of
Q3. A speed of $72\ \text{km/h}$ equals
Q4. $1\ \text{µm}^2$ is equal to
Q5. Which is a correct way to write a speed of three metres per second?