Units & Measurement

Physics is a quantitative science — every result is a number paired with a unit. Saying a distance is "12" is meaningless without specifying whether you mean meters, feet, or light-years. A force of "500" could describe the tension in a guitar string or the thrust of a rocket engine, depending on the unit. Units are not optional notation — they are part of the physical quantity itself.

This fact is more powerful than it first appears. Because units are algebraic objects that cancel and combine like variables, you can use them to check any equation you write and to derive relationships between physical quantities without solving a single equation. This technique is called dimensional analysis, and it is one of the most immediately useful skills in all of physics.

The precision of a measurement matters equally. A result of $9.8\text{ m/s}^2$ and $9.80000\text{ m/s}^2$ are the same number to two significant figures, but the second form claims six-figure precision. Reporting the right number of significant figures communicates how much confidence to place in each digit.

In 1960, the General Conference on Weights and Measures (CGPM) established the Système International d'Unités — the International System of Units, abbreviated SI worldwide. It defines seven base quantities, each assigned a single standard unit. Every other physical unit in existence is derived from these seven through multiplication and division.

Derived units are built from products and ratios of base units. For example, from Newton's second law $F = ma$, force has units of mass × acceleration, so $1\text{ newton} = 1\text{ N} = 1\text{ kg}\cdot\text{m/s}^2$. Similarly, $1\text{ joule} = 1\text{ J} = 1\text{ N}\cdot\text{m} = 1\text{ kg}\cdot\text{m}^2\text{/s}^2$, and $1\text{ watt} = 1\text{ W} = 1\text{ J/s} = 1\text{ kg}\cdot\text{m}^2\text{/s}^3$.

How the Base Units Are Defined

Since 2019, all seven SI base units are defined by fixing exact numerical values of fundamental constants of nature. The units are no longer tied to physical objects and can in principle be reproduced in any laboratory anywhere.

• The meter is defined by fixing the speed of light to exactly $c = 299{,}792{,}458\text{ m/s}$. One meter is the distance light travels in $1/299{,}792{,}458$ of a second.

• The second is defined by the cesium-133 hyperfine transition: one second equals exactly $9{,}192{,}631{,}770$ oscillations of the microwave radiation emitted by a cesium-133 atom. Atomic clocks realize this to better than 1 part in $10^{16}$.

• The kilogram is defined by fixing Planck's constant to exactly $h = 6.62607015 \times 10^{-34}\text{ kg}\cdot\text{m}^2\text{/s}$. This replaced the old platinum–iridium cylinder kept in a vault near Paris, which had been the world's mass standard since 1889.

Physical quantities span an enormous range — from the diameter of a proton ($\sim 10^{-15}$ m) to the observable universe ($\sim 10^{26}$ m), a ratio of $10^{41}$. The metric system handles this through prefixes that scale any SI unit by an exact power of ten.

Watch out for the kilogram: it is the SI base unit of mass, yet it already contains the "kilo" prefix. When attaching other prefixes to mass, you attach them to gram, not kilogram. So 1 milligram = $10^{-3}\text{ g} = 10^{-6}\text{ kg}$.

Every unit conversion is multiplication by 1. If $1\text{ km} = 1000\text{ m}$, then the ratio $\frac{1000\text{ m}}{1\text{ km}}$ equals exactly 1 — multiplying any quantity by it changes nothing physically, only the units in which we express the result. Chain-link conversion strings together as many such identity fractions as needed.

The golden rule: write the units explicitly in every fraction, verify that unwanted units cancel before touching the arithmetic, and only compute after the algebra is clean.

Example — Converting speed

Convert 60 miles per hour to meters per second. ($1\text{ mi} = 1609\text{ m}$, $1\text{ hr} = 3600\text{ s}$):

The "mi" and "hr" each cancel between numerator and denominator, leaving only m/s. A useful shortcut: to convert any speed from km/h to m/s, divide by 3.6, because $1000/3600 = 1/3.6$.

Example — Converting area and volume

A common error is forgetting to raise conversion factors to the appropriate power. To convert $1\text{ cm}^2$ to $\text{m}^2$:

Every physical quantity has a dimension — an expression in terms of the fundamental types: length $[L]$, mass $[M]$, time $[T]$, electric current $[I]$, temperature $[\Theta]$, amount $[N]$, and luminous intensity $[J]$. Dimensions are more fundamental than units: velocity has dimension $[LT^{-1}]$ whether you measure it in m/s, km/h, or furlongs per fortnight.

Checking Equations

A valid physical equation must be dimensionally consistent — every term on both sides must carry identical dimensions. This rule can immediately expose errors. Consider the kinematic equation $x = x_0 + v_0 t + \frac{1}{2}at^2$. Checking term by term:

All three terms have dimension $[L]$. If any term had, say, dimension $[L^2]$, the equation would be provably wrong — no further analysis needed.

Deriving Formulas from Dimensions

Suppose you want to know how the period of a simple pendulum depends on its length $\ell$ and gravity $g$. The period has dimension $[T]$. The only power-law combination of $\ell\sim[L]$ and $g\sim[LT^{-2}]$ that produces $[T]$ is $\sqrt{\ell/g}$:

Dimensional analysis predicts $T \propto \sqrt{\ell/g}$. The exact result is $T = 2\pi\sqrt{\ell/g}$; the factor of $2\pi$ is dimensionless and cannot be found by this method. But we've derived the dependence on $\ell$ and $g$ without solving a differential equation.

Physical quantities span from the Planck length ($\sim 10^{-35}$ m) to the diameter of the observable universe ($\sim 10^{26}$ m) — a ratio of $10^{61}$. Standard decimal notation is unworkable at these extremes. Instead every number is expressed in scientific notation:

Examples: the speed of light $c = 2.998 \times 10^8\text{ m/s}$; the electron mass $m_e = 9.109 \times 10^{-31}\text{ kg}$; Avogadro's number $N_A = 6.022 \times 10^{23}\text{ mol}^{-1}$.

Orders of Magnitude

The order of magnitude of a number is the power of 10 closest to it. The mass of Earth is $5.97 \times 10^{24}\text{ kg}$, so its order of magnitude is 25. Before calculating an answer precisely, every physicist first estimates the answer to the nearest power of 10. A computed answer that differs from the estimate by more than a factor of a few is a warning sign of an error.

Fermi Estimation

Enrico Fermi (1901–1954) was famous for deriving surprisingly accurate answers from rough estimates and order-of-magnitude arithmetic. His method: decompose the unknown quantity into a product of sub-quantities you can estimate, then multiply. Errors tend to partially cancel when you multiply many independent estimates.

Classic example — piano tuners in Chicago: Population $\approx 3 \times 10^6$; households $\approx 1.5 \times 10^6$; 1 in 20 owns a piano $\Rightarrow 75{,}000$ pianos; each tuned once per year; a tuner services 4 pianos/day × 250 days/year = 1000 pianos/year. Estimate: $75{,}000/1000 = 75$ tuners. Real answer: $\approx 80$.

The significant figures (sig figs) of a measured number are the digits that carry meaningful information about the measurement's precision. When you write a result, the number of sig figs communicates your confidence in each digit.

Rules for Counting Sig Figs

1. Non-zero digits are always significant: 1234 → 4 sig figs.

2. Zeros between non-zero digits are significant: 1007 → 4 sig figs.

3. Leading zeros (before the first non-zero digit) are never significant: 0.0045 → 2 sig figs (the 4 and 5).

4. Trailing zeros after a decimal point are significant: 3.800 → 4 sig figs. The trailing zeros declare that the measurement was made to that precision.

5. Trailing zeros in an integer without a decimal point are ambiguous: 3000 could be 1, 2, 3, or 4 sig figs. Use scientific notation to eliminate ambiguity: $3 \times 10^3$ (1 sig fig) vs $3.000 \times 10^3$ (4 sig figs).

Sig Figs Through Calculations

Multiplication and division: the result has as many sig figs as the least precise input.

Addition and subtraction: the result has as many decimal places as the input with the fewest decimal places.

No measurement is exact. Every measured value differs from the true value by some amount — the measurement uncertainty. A result reported without its uncertainty is incomplete. We write:

where $x$ is the best estimate and $\delta x > 0$ is the absolute uncertainty. The relative (fractional) uncertainty is $\delta x / x$, often expressed as a percentage.

Random vs Systematic Uncertainty

Random uncertainty arises from unpredictable fluctuations — thermal noise, vibration, the limit of reading an analog scale. Repeated measurements scatter around the true value. Averaging $N$ independent measurements reduces random uncertainty by $\sqrt{N}$:

Systematic uncertainty is a consistent bias that shifts every measurement the same way — a miscalibrated instrument, an uncorrected background, a zero offset. No amount of averaging removes it. Identifying and eliminating systematic errors is the hardest part of experimental physics.

Accuracy vs Precision

Accuracy: how close to the true value. Precision: how reproducible — how tightly clustered repeated measurements are. A precise instrument can be inaccurate (systematic error); an accurate result on average can be imprecise (large scatter). Both must be minimized. In everyday speech these words are used interchangeably, but in science they have distinct meanings.

Propagating Uncertainties

When computing a derived quantity from measured values, the uncertainties propagate. The two fundamental rules:

Sums and differences $q = x \pm y$: absolute uncertainties add:

Products and quotients $q = xy$ or $x/y$: relative uncertainties add in quadrature:

Example: A rectangle has $L = 12.0 \pm 0.2$ cm and $W = 5.0 \pm 0.1$ cm. The relative uncertainty in area $A = LW$ is:

Since $A = 60.0\text{ cm}^2$, the absolute uncertainty is $0.026 \times 60.0 \approx 1.6\text{ cm}^2$, giving $A = 60.0 \pm 1.6\text{ cm}^2$.