Chapter 2: Units and Measurements

Physics is a quantitative science — every equation links measurable quantities. Before we can write equations sensibly, we must agree on what to measure, how to measure it, what units to use, and how to handle the inevitable errors in our measurements. This chapter develops the entire toolkit you will use in every subsequent chapter: the SI system, dimensional analysis (a remarkably powerful sanity-check), significant figures, and error propagation.

A famous failure illustrates why this matters: in 1999 NASA lost the Mars Climate Orbiter — a \125$ million spacecraft — because one team used pound-seconds and the other used newton-seconds. The probe burned up in Mars's atmosphere. Units are not bookkeeping; they are physics.

Key Concepts

2.1 Physical Quantities and Units

A physical quantity is anything that can be measured. Examples: length, mass, time, force, current, temperature.

A unit is a standardized reference amount of a physical quantity. To express a measurement we write: $Q = n \, u$ where $n$ is the numerical value and $u$ is the chosen unit. The same length $1.7$ m can equally be written as $170$ cm or $0.0017$ km — but $n \, u$ is invariant: $n_1 u_1 = n_2 u_2 \quad \Longrightarrow \quad n_2 = n_1 \frac{u_1}{u_2}.$ Key consequence: if the unit is large, the numerical value is small, and vice versa. A car's mass is $1000$ kg, or equivalently $1$ tonne — same physical quantity.

2.2 The International System of Units (SI)

The SI (Système International d'Unités), adopted in 1971 and refined since (most recently in 2019), uses seven base units from which all other units are derived.

Two supplementary units complete the SI:

Derived units combine base units. A few you must know:

Standard SI prefixes (memorise these — every chapter uses them):

2.3 Measurement of Length, Mass and Time — Direct and Indirect

Length. Range of length measurements is enormous, from $\sim 10^{-15}$ m (proton) to $\sim 10^{26}$ m (observable universe).

• Direct: metre scale ($10^{-3}$ m), vernier calipers ($10^{-4}$ m), screw gauge ($10^{-5}$ m).
• Indirect: parallax method (astronomical distances), echo / RADAR / LIDAR (large terrestrial), electron microscopy (sub-micron).

Parallax method: to find the distance $D$ to a faraway star, observe it from two ends of Earth's orbit (basis $b = 2 \times 1\, \text{AU}$). The parallax angle $\theta$ (in radians) satisfies $D = \frac{b}{\theta}.$

Common astronomical length units:

• $1$ astronomical unit (AU) = $1.496 \times 10^{11}$ m (mean Earth-Sun distance).
• $1$ light-year (ly) = $9.46 \times 10^{15}$ m.
• $1$ parsec (pc) = $3.08 \times 10^{16}$ m $\approx 3.26$ ly. (Distance at which $1$ AU subtends $1$ arc-second.)

Mass. From $\sim 10^{-30}$ kg (electron) to $\sim 10^{53}$ kg (universe).

• Direct: common / beam / electronic balance.
• Indirect: gravitational / inertial methods. Mass of Earth from $g$ and the law of gravitation; mass of a planet from a satellite's orbit; atomic masses by mass spectrometer.
• $1$ atomic mass unit (u) = $1.66 \times 10^{-27}$ kg, defined as $1/12$ of the mass of $^{12}$C.

Time. From $\sim 10^{-24}$ s (nuclear) to $\sim 10^{17}$ s (age of universe).

• Atomic clocks (Cs-133): accurate to $1$ part in $10^{13}$ — drift of about $1$ s in $300\,000$ years.
• Optical / strontium lattice clocks: $1$ part in $10^{18}$ (research).

2.4 Accuracy, Precision and Errors

• Accuracy: closeness to the true value.
• Precision: closeness of repeated measurements to each other (i.e., reproducibility).

A rifle that always hits the same spot but not the bullseye is precise but inaccurate; one whose shots scatter around the bullseye is accurate on average but imprecise.

Sources of error:

• Systematic errors — repeatable bias due to instrument zero-error, calibration drift, environmental conditions, or experimenter's habit. Reduced by correcting / recalibrating, not by averaging.
• Random errors — unpredictable fluctuations from many small unknown causes. Reduced by taking many readings and averaging.
• Gross errors / blunders — recording, arithmetic, or instrument-reading mistakes. Removed by repeating the measurement.
• Least-count error — limit of the resolution of the instrument; treated like a random error.

2.5 Absolute, Relative and Percentage Errors

Suppose a quantity $a$ is measured $n$ times, giving readings $a_1, a_2, \ldots, a_n$. The arithmetic mean is taken as the best estimate of the true value: $\bar{a} = \frac{1}{n} \sum_{i=1}^{n} a_i.$

• Absolute error of the $i$-th reading: $\Delta a_i = \vert \bar{a} - a_i\vert$.
• Mean absolute error: $\Delta \bar{a} = \dfrac{1}{n} \sum_{i=1}^{n} \Delta a_i$.
• Relative error: $\delta a = \dfrac{\Delta \bar{a}}{\bar{a}}$ (dimensionless).
• Percentage error: $\delta a \times 100 \%$.

The final result is reported as $a = \bar{a} \pm \Delta \bar{a}.$

2.6 Propagation of Errors (Combinations)

If $A = \bar{a} \pm \Delta a$ and $B = \bar{b} \pm \Delta b$, then for various combinations the errors add as follows:

(a) Sum or difference. $Z = A + B$ or $Z = A - B$. The absolute errors add: $\Delta Z = \Delta A + \Delta B.$

(b) Product or quotient. $Z = A B$ or $Z = A / B$. The relative errors add: $\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}.$

(c) Power. $Z = A^p B^q / C^r$. The relative errors multiply by the exponents and add: $\frac{\Delta Z}{Z} = |p| \frac{\Delta A}{A} + |q| \frac{\Delta B}{B} + |r| \frac{\Delta C}{C}.$

Important: errors always add (never subtract) when propagating, because we assume the worst case. This is why subtraction of nearly equal quantities is dangerous: the absolute error stays the same but the result shrinks, so the relative error blows up. Always design experiments to avoid taking the difference of two nearly equal large numbers.

Derivation sketch for the product rule. Take logarithms: $\ln Z = \ln A + \ln B$. Differentiating: $dZ/Z = dA/A + dB/B$. Replacing differentials by errors and adding magnitudes gives the rule.

2.7 Significant Figures

The significant figures of a measurement are the digits known reliably plus the first uncertain digit.

Examples: $1.23$ has 3 s.f.; $1.230$ has 4 s.f. (trailing zero after the decimal matters); $1230$ has $3$ or $4$ s.f. (ambiguous — write $1.23 \times 10^3$ or $1.230 \times 10^3$).

Rules for counting significant figures:

1. All non-zero digits are significant. ($3456$ has 4 s.f.)
2. Zeros between non-zero digits are significant. ($3005$ has 4 s.f.)
3. Leading zeros are NOT significant. ($0.0034$ has 2 s.f.)
4. Trailing zeros after a decimal are significant. ($3.400$ has 4 s.f.)
5. Trailing zeros in a whole number without a decimal point are ambiguous. Express in scientific notation. ($4500$ may be 2, 3, or 4 s.f.)
6. Exact / counting numbers ($N = 23$ students, $\pi$, $2$ in $E = \frac{1}{2}mv^2$) have infinite significant figures.

Rules for arithmetic with s.f.:

• Addition / subtraction — the result has the same number of decimal places as the operand with the fewest decimal places. Example: $2.34 + 0.012 = 2.35$ (2 d.p., dropped to least).
• Multiplication / division — the result has the same number of significant figures as the operand with the fewest s.f. Example: $5.74 \times 0.4 = 2$ (1 s.f.).
• Rounding off: if the digit to be dropped is
  • less than $5$ — drop it (round down);
  • greater than $5$ — round up;
  • exactly $5$ — round to even (banker's rounding): $1.25 \to 1.2$ but $1.35 \to 1.4$.

2.8 Dimensions of Physical Quantities

The dimension of a physical quantity is the way it depends on the seven base quantities. The base quantities are denoted:

$\text{[L] length}, \quad \text{[M] mass}, \quad \text{[T] time}, \quad \text{[A] current}, \quad \text{[K] temperature}, \quad \text{[mol]}, \quad \text{[cd]}.$

For mechanics, only $\text{L, M, T}$ usually matter.

Dimensional formula of a quantity expresses it as $[\text{M}^a \text{L}^b \text{T}^c]$.

Examples:

Dimensionless quantities: angle ($\theta$), strain, refractive index, relative density, $\pi$, exponents, trigonometric values, logarithms — all carry $[\text{M}^0 \text{L}^0 \text{T}^0]$.

2.9 Principle of Homogeneity (Dimensional Consistency)

Every term in a physically valid equation must have the same dimensions.

You cannot add quantities of different dimensions, just as you cannot add metres to seconds. This single rule is the engine behind dimensional analysis.

Use 1 — Checking equations. Take $s = ut + \tfrac{1}{2} a t^2$.

• $[s] = \text{L}$.
• $[ut] = (\text{L T}^{-1})(\text{T}) = \text{L}$. $\checkmark$
• $[at^2] = (\text{L T}^{-2})(\text{T}^2) = \text{L}$. $\checkmark$ All terms agree — the equation is dimensionally consistent. (Dimensional consistency does not prove correctness; it only rules out gross errors.)

Use 2 — Converting units. Convert $1$ joule into ergs. $[\text{J}] = [\text{M L}^2 \text{T}^{-2}]$. SI: kg, m, s. CGS: g, cm, s. $1\, \text{J} = (1\, \text{kg})(1\, \text{m})^2 (1\, \text{s})^{-2} = (10^3\, \text{g})(10^2\, \text{cm})^2 (1\, \text{s})^{-2} = 10^3 \times 10^4\, \text{g cm}^2 \text{s}^{-2} = 10^7\, \text{erg}.$

Use 3 — Deriving relationships. Suppose the period $T$ of a simple pendulum depends only on its length $l$, mass $m$, and gravitational acceleration $g$. Write $T = k\, l^a m^b g^c$ with $k$ a dimensionless constant. Equating dimensions: $\text{T}^1 = (\text{L})^a (\text{M})^b (\text{L T}^{-2})^c = \text{M}^b \text{L}^{a+c} \text{T}^{-2c}.$ Comparing: $b = 0$, $a + c = 0$, $-2c = 1 \Rightarrow c = -1/2$, $a = 1/2$. Hence $T = k \sqrt{l/g}.$ Dimensional analysis cannot find $k = 2\pi$ — but it has correctly predicted that the period is independent of mass, an experimental fact.

2.10 Limitations of Dimensional Analysis

Dimensional analysis is powerful but not all-powerful:

1. Cannot determine dimensionless constants (the $2\pi$ in the pendulum, the $\tfrac{1}{2}$ in $\tfrac{1}{2} m v^2$).
2. Fails when a quantity depends on more than three M, L, T variables — you don't have enough equations.
3. Fails for trigonometric, exponential, or logarithmic relations (e.g. $y = A \sin(\omega t)$ — sin is dimensionless, no help in determining $A$).
4. Cannot decide between $s = ut + \tfrac{1}{2} at^2$ and $s = ut + at^2$ — both are dimensionally consistent. Other physics is needed.
5. Cannot distinguish two quantities with identical dimensions — e.g., torque and energy both $[\text{M L}^2 \text{T}^{-2}]$; impulse and momentum both $[\text{M L T}^{-1}]$.

2.11 Order-of-Magnitude Estimation

A quantity $a$ written as $a = b \times 10^n$ with $0.5 \le b < 5$ is said to have order of magnitude $10^n$. (Some books use $1 \le b < 10$.) Order-of-magnitude reasoning lets you check whether an answer is reasonable: if you compute the mass of an electron and get $10^{-3}$ kg, you have made a 27-orders-of-magnitude error.

Worked Examples

Example 2.1 — Convert units using dimensions. The value of $G$ in SI is $6.67 \times 10^{-11}$ N m$^2$/kg$^2$. Find its value in CGS (dyne cm$^2$/g$^2$).

Solution. $[G] = [\text{M}^{-1} \text{L}^3 \text{T}^{-2}]$. The conversion factor from SI to CGS for $G$ is: $\left(\frac{\text{kg}}{\text{g}}\right)^{-1} \left(\frac{\text{m}}{\text{cm}}\right)^{3} \left(\frac{\text{s}}{\text{s}}\right)^{-2} = (10^3)^{-1} (10^2)^3 (1)^{-2} = 10^{-3} \times 10^6 = 10^3.$ Hence $G_{\text{CGS}} = 6.67 \times 10^{-11} \times 10^3 = 6.67 \times 10^{-8}$ dyne cm$^2$/g$^2$.

Example 2.2 — Significant figures arithmetic. A rectangular sheet has length $16.2$ cm and breadth $10.1$ cm. Compute its area, reporting to the correct number of significant figures.

Solution. $16.2 \times 10.1 = 163.62$ cm$^2$. Both operands have 3 s.f., so the answer must have 3 s.f.: $A = 164$ cm$^2$.

Example 2.3 — Error in a formula. The period of a pendulum is determined from $T = 2\pi\sqrt{l/g}$. The length is measured as $l = 100.0 \pm 0.1$ cm and the time for $20$ oscillations is $40.2 \pm 0.1$ s. Find the percentage error in $g$.

Solution. From $T = 2\pi\sqrt{l/g}$, we get $g = 4\pi^2 l / T^2$. Hence $\frac{\Delta g}{g} = \frac{\Delta l}{l} + 2\frac{\Delta T}{T}.$ $\Delta l / l = 0.1 / 100 = 0.001 = 0.1\%$. $T = 40.2/20 = 2.01$ s, $\Delta T = 0.1/20 = 0.005$ s, $\Delta T / T = 0.005/2.01 = 0.249\%$. $\Delta g / g = 0.1\% + 2 \times 0.249\% = 0.6\%$ (approximately).

Lesson: the time error contributes much more than the length error, because (a) it enters with a factor of 2 (squared dependence) and (b) it is intrinsically larger. To improve, count more oscillations.

Example 2.4 — Dimensional analysis to derive a formula. The drag force $F$ on a sphere moving slowly through a viscous fluid depends on the radius $r$, the viscosity $\eta$ (units kg m$^{-1}$ s$^{-1}$) and the velocity $v$. Derive the form of the dependence.

Solution. Assume $F = k\, r^a \eta^b v^c$. Equating dimensions: $[\text{M L T}^{-2}] = (\text{L})^a (\text{M L}^{-1} \text{T}^{-1})^b (\text{L T}^{-1})^c = \text{M}^b \text{L}^{a-b+c} \text{T}^{-b-c}.$ $b = 1$, $-b - c = -2 \Rightarrow c = 1$, $a - b + c = 1 \Rightarrow a = 1$. So $F = k \eta r v$. (Stokes' law gives $k = 6\pi$.)

Example 2.5 — Mean and error. Five measurements of a time period give $2.63, 2.56, 2.42, 2.71, 2.80$ s. Find the mean, the mean absolute error, the relative error and the percentage error.

Solution. Mean $\bar{T} = (2.63+2.56+2.42+2.71+2.80)/5 = 13.12/5 = 2.624$ s. Absolute errors: $\vert 0.006\vert , \vert 0.064\vert , \vert 0.204\vert , \vert 0.086\vert , \vert 0.176\vert$ — i.e., $0.006, 0.064, 0.204, 0.086, 0.176$. Mean absolute error: $(0.006+0.064+0.204+0.086+0.176)/5 = 0.536/5 = 0.107 \approx 0.11$ s. Relative error: $0.11/2.62 = 0.042$. Percentage error: $4.2\%$. Reported: $T = (2.62 \pm 0.11)$ s, or $T = 2.62$ s $\pm 4.2\%$.

Example 2.6 — Order of magnitude. Estimate the order of magnitude of the mass of air in a $10$ m $\times 10$ m $\times 3$ m room. ($\rho_{\text{air}} \approx 1.2$ kg/m$^3$.)

Solution. $V = 300$ m$^3$, $m = \rho V = 1.2 \times 300 = 360$ kg. Order of magnitude $10^2$ kg.

Common Traps

• "Light-year is a unit of time." It is a unit of length (distance light travels in one year, $9.46 \times 10^{15}$ m).
• "AU = atomic unit of mass." No: AU = astronomical unit of length. Atomic mass unit is u.
• "Subtraction reduces error." Wrong. Errors in $A - B$ are $\Delta A + \Delta B$ — they add. Worse, the value $A-B$ may be small, so the relative error explodes.
• In powers, exponent in error formula is taken with absolute value. Even for negative powers — $\Delta Z / Z = \vert p\vert \, \Delta A / A$, never with a minus sign.
• Ignoring the $2$ when the formula contains $T^2$ or $r^2$. Errors propagate through exponents.
• Mixing decimal places and significant figures rules. Add/subtract uses decimal places. Multiply/divide uses significant figures.
• Treating trailing zeros without a decimal as significant. $4500$ is ambiguous; write $4.5 \times 10^3$ (2 s.f.) or $4.500 \times 10^3$ (4 s.f.).
• "Dimensional analysis proves a formula correct." It only proves dimensional consistency — necessary, not sufficient.
• Forgetting that angle is dimensionless. $[\theta] = [\text{M}^0 \text{L}^0 \text{T}^0]$ — even though it has the unit radian.
• Confusing the dimensions of torque and energy. Both are $[\text{M L}^2 \text{T}^{-2}]$, but torque is a vector and energy a scalar. The two are physically very different.
• Ignoring the difference between accuracy and precision. Repeating a faulty thermometer's reading $1000$ times gives high precision but the same systematic inaccuracy.
• Using $g = 9.8$ m/s$^2$ to 3 s.f. but then quoting an answer to 6 s.f. A result is no more precise than its least-precise input.

Quick Recap

• SI has 7 base units (m, kg, s, A, K, mol, cd) + 2 supplementary (rad, sr).
• Modern SI definitions (post-2019) are anchored to fundamental constants ($c, h, e, k_B, N_A$).
• Important indirect length techniques: parallax (astronomy), echo / RADAR (terrestrial).
• Accuracy ≠ precision. Errors are systematic (correctable) or random (averaged out).
• For a series of $n$ measurements, the best estimate is the mean; the spread is captured by the mean absolute error.
• Error propagation:
  • Sum / difference: absolute errors add.
  • Product / quotient: relative errors add.
  • Powers: relative errors multiply by exponent and add.
• Significant figures:
  • Add/sub $\to$ decimal-place rule.
  • Mul/div $\to$ s.f. rule.
• Dimensional formula uses $[\text{M}^a \text{L}^b \text{T}^c]$. The principle of homogeneity says every term has the same dimensions.
• Three classical uses of dimensions: checking equations, converting units, guessing formulas (up to a dimensionless constant).
• Dimensional analysis fails for dimensionless multipliers, for trig/exp/log, and for $> 3$ unknowns.

Formula Summary

With this measurement toolkit in hand, the next chapter dives into the first real physics problem: describing the motion of a particle along a straight line.