Measurement
Units, standards, and the language every physical quantity speaks
- Why every measurement is a number paired with a unit, anchored to a standard.
- The seven SI base quantities and how all other units are built from them.
- Scientific notation, SI prefixes, and confident unit conversion.
- How the metre, second, and kilogram are defined today.
- Density, and how to make fast order-of-magnitude estimates.
What Is Physics?
Physics is the search for the basic rules that govern matter, energy, space, and time. Almost everything in physics begins with one humble act: measurement. To test an idea, we must compare it against the world by measuring something — a length, a time, a temperature — and seeing whether the numbers agree.
Because measurement is so central, physicists obsess over making it accurate. A familiar example: the Global Positioning System (GPS) in your phone works only because atomic clocks keep time to within billionths of a second. Sloppy timekeeping would translate into position errors of kilometres, and GPS would be useless.
Measuring Things
To measure a physical quantity is to compare it with a standard. Three words do all the work here:
The property being measured — length, time, mass, temperature, and so on.
The agreed name for one measure of that quantity — for example, the metre (m) for length.
A physical reference that is exactly 1.0 unit. Every ruler ultimately traces back to it.
There are far too many physical quantities to give each its own independent standard. Luckily they are related: speed, for instance, is just a length divided by a time. So scientists agree on a small set of base quantities (such as length and time), assign a standard to each, and then define every other quantity in terms of these. Quantities built this way are called derived quantities.
The International System of Units (SI)
In 1971 an international conference settled on seven base quantities, forming the International System of Units — abbreviated \(SI\) from the French Système International, and known popularly as the metric system. Almost the entire scientific world uses it.
| Base quantity | Unit name | Symbol |
|---|---|---|
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric current | ampere | A |
| Temperature | kelvin | K |
| Amount of substance | mole | mol |
| Luminous intensity | candela | cd |
The early chapters of this book mostly need just the first three — length, mass, and time. Everything else is a derived unit, assembled by multiplying and dividing base units. Two you will meet soon:
Scientific Notation & SI Prefixes
Physics spans the unimaginably large and the vanishingly small. Writing all those zeros is error-prone, so we use scientific notation — a number between 1 and 10 multiplied by a power of ten:
A second convenience is the prefix: a short syllable that stands for a power of ten and is attached directly to a unit. For example \(1.27 \times 10^{9} W = 1.27 GW\) (gigawatts), and \(2.35 \times 10^{-9} s = 2.35 ns\) (nanoseconds).
| Factor | Prefix | Symbol | Factor | Prefix | Symbol |
|---|---|---|---|---|---|
| 1012 | tera- | T | 10−2 | centi- | c |
| 109 | giga- | G | 10−3 | milli- | m |
| 106 | mega- | M | 10−6 | micro- | µ |
| 103 | kilo- | k | 10−9 | nano- | n |
| 102 | hecto- | h | 10−12 | pico- | p |
| 101 | deka- | da | 10−15 | femto- | f |
Changing Units
We constantly need to switch units. The safe, foolproof method is chain-link conversion: multiply by a conversion factor, which is simply a ratio that equals 1. Since 1 min and 60 s are the same interval,
The trick is to pick the version of the factor that cancels the unit you don't want. To convert 2 minutes into seconds:
Carry units through every line of a calculation and cancel them just like variables. If the final units are wrong, the calculation is wrong — units are a free, built-in error check.
Length
The story of the metre shows how standards march toward ever-greater invariability:
- 1792 — France defines the metre as one ten-millionth of the distance from the north pole to the equator.
- 1800s — Replaced by a platinum–iridium bar kept near Paris, with copies sent worldwide.
- 1960 — Redefined as 1 650 763.73 wavelengths of orange-red light from krypton-86, reproducible in any lab.
- 1983 — Redefined in terms of the speed of light, where it remains today.
This fixes the speed of light at exactly \(c = 299 792 458 m/s\) — a defined constant, not a measured one.
| Measurement | Length (m) |
|---|---|
| Distance to the first galaxies | 2 × 1026 |
| Distance to Proxima Centauri | 4 × 1016 |
| Radius of Earth | 6 × 106 |
| Height of Mt. Everest | 9 × 103 |
| Thickness of this page | 1 × 10−4 |
| Radius of a hydrogen atom | 5 × 10−11 |
| Radius of a proton | 1 × 10−15 |
Time
Time answers two questions: "When did it happen?" and "How long did it last?" Any repeating phenomenon can serve as a clock. Earth's rotation worked for centuries, but it drifts with tides and winds — too unreliable for modern science. The answer was the atomic clock, which counts the rock-steady vibrations of an atom.
Two cesium clocks would have to run for about 6000 years to drift apart by even one second. Newer optical clocks are far better still.
| Measurement | Interval (s) |
|---|---|
| Age of the universe | 5 × 1017 |
| Human life expectancy | 2 × 109 |
| Length of a day | 9 × 104 |
| Time between heartbeats | 8 × 10−1 |
| Lifetime of the muon | 2 × 10−6 |
| The Planck time | 1 × 10−43 |
Mass & Density
For over a century the kilogram was a literal object: a platinum–iridium cylinder near Paris, nicknamed "Le Grand K." The trouble with a physical object is that it can pick up or lose a few atoms over time — not very invariable.
Since 20 May 2019, the kilogram is no longer tied to a metal cylinder. It is fixed by assigning the Planck constant an exact value, \(h = 6.626 070 15 \times 10^{-34} J\cdot s\). Today all seven SI base units are defined through unchanging constants of nature rather than physical artefacts — the ultimate "accessible and invariable" standards.
For atoms, comparing against a kilogram is clumsy. Instead we use the atomic mass unit (u), defined so that one carbon-12 atom has a mass of exactly 12 u:
Density tells us how much mass is packed into a given volume. It is mass per unit volume:
| Object | Mass (kg) |
|---|---|
| Known universe | 1 × 1053 |
| Sun | 2 × 1030 |
| Ocean liner | 7 × 107 |
| Elephant | 5 × 103 |
| Speck of dust | 7 × 10−10 |
| Uranium atom | 4 × 10−25 |
| Electron | 9 × 10−31 |
Putting It to Work
Problem. The world's largest ball of string is about 2 m in radius. To the nearest power of ten, how long is the string?
Idea. We don't need an exact answer, just the order of magnitude, so we estimate. Treat the string's cross-section as a tiny square of edge \(d \approx 4 \mathrm{mm}\) and total length \(L\). Its volume \(d^{2}L\) fills the ball's volume, which is roughly \(4R^{3}\) (using π ≈ 3 in \((4/3)\pi R^{3}\)).
So the ball holds roughly 1000 km of string — no calculator required. That is the power of order-of-magnitude thinking.
Problem. In an earthquake, loosely packed sand can behave like quicksand. The looseness is captured by the void ratio \(e = V_{\text{voids}} / V_{\text{grains}}\). If liquefaction sets in at \(e = 0.80\), what sand density does that correspond to? Solid silica has density \(\rho _{SiO_{2}} = 2.600 \times 10^{3} \mathrm{kg}/m^{3}\).
Solution. The grains' mass spread over the total volume gives
Below about \(1.4 \times 10^{3} \mathrm{kg}/m^{3}\) the ground can liquefy, and a building may sink several metres.
Chapter Summary
A measurement is a number + a unit, fixed by an accessible, invariable standard. Base quantities get standards directly; derived quantities are built from them.
Seven SI base units underpin everything. Scientific notation and prefixes tame very large and very small numbers.
Multiply by conversion factors equal to 1, chosen so unwanted units cancel. Units obey algebra.
Metre — light in 1/299 792 458 s. Second — 9 192 631 770 cesium oscillations. Kilogram — fixed via the Planck constant (since 2019).
\(\rho = m/V\), in kg/m3 or g/cm3; water ≈ 1000 kg/m3.
Order-of-magnitude reasoning gives quick, useful answers when precision isn't needed.
Problems
Try these with chain-link conversions and scientific notation. Difficulty rises down the list.
- Earth is roughly a sphere of radius \(6.37 \times 10^{6} m\). Find its (a) circumference in km, (b) surface area in km2, and (c) volume in km3.
- The micrometre (1 µm) is also called the micron. (a) How many microns are in 1.0 km? (b) What fraction of a centimetre is 1.0 µm? (c) How many microns are in 1.0 yard?
- A fortnight is exactly 2.0 weeks. How many microseconds are in a fortnight?
- A plant grew 3.7 m in 14 days. Express its average growth rate in micrometres per second.
- Antarctica is roughly a semicircle of radius 2000 km with an ice cover 3000 m thick. How many cubic centimetres of ice does it hold? (Ignore Earth's curvature.)
- Gold has density 19.32 g/cm3. A 27.63 g sample is pressed into a leaf 1.000 µm thick. What is the leaf's area?
- Assuming water's density is exactly 1 g/cm3, find the mass (in kg) of one cubic metre of water.
- A hydrogen atom has a mass of 1.0 u. Using \(1 u = 1.66 \times 10^{-27} \mathrm{kg}\), how many hydrogen atoms make up 1.0 kg?
- One astronomical unit (AU) is about \(1.50 \times 10^{8} \mathrm{km}\) and light travels at about \(3.0 \times 10^{8} m/s\). Express the speed of light in AU per minute.
- A cumulus cloud of radius 1.0 km and height 3.0 km contains about 50–500 droplets per cm3, each of radius 10 µm. Estimate the lower and upper values for the total mass of water it carries (water density 1000 kg/m3).