Units and measurement systems

What a unit must be. A measurement is always (numerical value) times (unit) — $5\ \text{m}$ means "5 times the metre." For a unit to be usable across labs and centuries, it must be well-defined, invariable (independent of time, place, or the person defining it), easily reproducible, accessible for comparison, and of convenient size — not so large or small that everyday numbers become unwieldy.

Systems that led to SI. Different eras and regions settled on different base sets before the world agreed on one:

• CGS — centimetre, gram, second
• FPS — foot, pound, second
• MKS — metre, kilogram, second
• MKSA — metre, kilogram, second, ampere, adding the ampere so electrical quantities had a base
• SI (Système International d'Unités) — adopted in 1971, the modern extension of MKSA, built on seven independent base units

where: the numerical value depends entirely on which unit you chose, while the physical quantity itself does not change — this is why $5\ \text{m}$ and $500\ \text{cm}$ describe the identical length. The seven SI base units are:

Derived and supplementary units. Every other unit — for velocity, force, energy, pressure — is a derived unit, built by combining base units according to the defining equation of that quantity. Velocity is length divided by time, so its unit is $\text{m/s}$; force is mass times acceleration, so its unit ($\text{kg}\,\cdot\,\text{m/s}^2$) is later named the newton. Two units historically kept in a separate "supplementary" class are the radian (plane angle) and steradian (solid angle); modern SI treats both as derived, dimensionless units, but they matter throughout rotational-motion and radiation formulae later in the course.

Prefixes for scale. Because measured quantities range from subatomic to cosmological, SI multiplies or divides a unit by an exact power of ten using standard prefixes, from atto ($a$, $10^{-18}$) to tera ($T$, $10^{12}$):

Each prefix is an exact multiplier, never an approximation — $1\ \text{km}$ is exactly $1000\ \text{m}$, not "roughly a thousand."

Special units for extreme scales. Beyond prefixed SI units, physics keeps a handful of named units convenient for particular size regimes:

• $1\ \text{fermi} = 1\ \text{fm} = 10^{-15}\ \text{m}$ — nuclear sizes
• $1\ \text{angstrom} = 1\ \text{Å} = 10^{-10}\ \text{m}$ — atomic sizes
• $1\ \text{light year} = 9.46\times10^{15}\ \text{m}$ — distance light covers in vacuum in one year
• $1\ \text{astronomical unit (AU)} = 1.496\times10^{11}\ \text{m}$ — mean Earth–Sun distance
• $1\ \text{parsec} = 3.08\times10^{16}\ \text{m}$ — a stellar-distance unit, about $3.26$ light years
• $1$ unified atomic mass unit $(u) = 1.66\times10^{-27}\ \text{kg}$ — mass scale for atoms

Problem. A hydrogen atom has a nuclear radius of about $1.2\ \text{fm}$ and an atomic radius of about $0.53\ \text{Å}$. Find the ratio of atomic radius to nuclear radius.

Step 1. Convert both radii to the same unit, metres, since fermi and angstrom are defined against different powers of ten and cannot be compared directly. $1.2\ \text{fm} = 1.2\times10^{-15}\ \text{m}$ and $0.53\ \text{Å} = 0.53\times10^{-10}\ \text{m} = 5.3\times10^{-11}\ \text{m}$.

Step 2. Divide atomic radius by nuclear radius: $\dfrac{\text{atomic radius}}{\text{nuclear radius}} = \dfrac{5.3\times10^{-11}}{1.2\times10^{-15}} \approx 4.4\times10^{4}$.

Answer: the atom's radius is roughly 44,000 times its nucleus's radius — which is why an atom is mostly empty space, with almost all its mass packed into a nucleus occupying a tiny fraction of its volume.

Competitive stretch

How many astronomical units fit inside one parsec? Using $1\ \text{parsec} = 3.08\times10^{16}\ \text{m}$ and $1\ \text{AU} = 1.496\times10^{11}\ \text{m}$: $\dfrac{3.08\times10^{16}}{1.496\times10^{11}} \approx 2.06\times10^{5}$. A parsec is about $206{,}000$ astronomical units — one more reminder that each special unit exists because it keeps the numbers of its own domain, from nuclear to galactic, close to a convenient single digit times a small power of ten.

• Confusing base units with derived units — only the seven listed quantities have base units; force, energy, pressure, velocity and everything else is derived from them.
• Treating a prefix like kilo or milli as an approximate factor rather than an exact power of ten — $1\ \text{km}$ is exactly $1000\ \text{m}$, never "about a thousand."
• Mixing unit systems inside one calculation, such as using centimetres for length while keeping seconds and kilograms for the rest — convert everything to one consistent system, usually SI, before calculating.
• Dismissing the radian and steradian as unimportant because they seem dimensionless — they are supplementary units used consistently in later rotational-motion and radiation formulae, and dropping them silently causes unit-mismatch errors.

Ask the tutor: want to practice converting between more of these special units, like turning a parsec into light years or a mass in atomic mass units into kilograms?