Classical Mechanics · Chapter 1

Measurement, Units, and Dimensional Analysis

Before we can do physics, we need a language for describing the physical world precisely. That language is mathematics, and its vocabulary begins with units of measurement.

Learning Goals

Identify the seven SI base units and express derived quantities in terms of them.
Apply dimensional analysis to check equations and derive the functional form of physical laws.
Convert measurements between unit systems using appropriate conversion factors.
Use significant figures correctly in multiplication, division, addition, and subtraction.
Perform order-of-magnitude Fermi estimates by combining rough numerical facts.

1.1 What is Physics?

Physics is the science of matter, energy, space, and time — and, more ambitiously, of the fundamental principles that govern all of them. Its central method is the same as it has been since Galileo1600s · Measurement becomes physicsGalileo did not just watch falling bodies. He slowed motion down with ramps, timed it, and looked for mathematical regularities. That shift from argument to measurement is one reason modern physics begins with him.: observe nature, construct a mathematical model, derive consequences from that model, and test those consequences against new observations. When the model fails, revise it. When it succeeds, extend it.

What distinguishes physics from other natural sciences is its commitment to generality. A physicist who understands Newton's second law F = ma can immediately analyze a falling apple, a planet's orbit, the vibration of a guitar string, and the collision of billiard balls — because all of these are instances of the same principle. This pursuit of the fewest, most general laws has been spectacularly successful. Four equations (Maxwell's) describe every electric and magnetic phenomenon ever observed. One equation (Schrödinger's) governs the behavior of every atom and molecule. The same inverse-square law that pulls an apple to the Earth guides the trajectory of the Voyager spacecraft, now more than 20 billion kilometers away.

Physics also provides the foundation for all other physical sciences and for engineering. Chemistry rests on quantum mechanics. Geology depends on thermodynamics and fluid mechanics. Neuroscience relies on electromagnetism and statistical physics. The tools you will develop here are genuinely universal.

1.2 The Scale of the Universe

One of the first things to appreciate about physics is the extraordinary range of scales it must address. The observable universe is about 10²⁶ meters across. The Planck length — the scale at which quantum gravity effects become important — is about 10⁻³⁵ meters. Between these two extremes lies 61 orders of magnitude. Physics operates fluently across all of it.

Figure 1.1. Orders of magnitude from the Planck length to the observable universe. Drag or scroll to travel through the scales. Notice how

vastly different regimes of physics apply at different scales: quantum mechanics below 10^{-}

The ability to think in orders of magnitude — to say confidently that an atom is roughly 10⁻¹⁰ m across and a human cell is roughly 10⁻⁵ m — is one of the most valuable habits a physicist can develop. We will practice it throughout this curriculum.

1.3 The SI System of Units

Any quantitative description of nature requires units of measurement. The Système International d'Unités (SI) is the international standard used in all scientific work. It is built on seven base units, from which every other unit can be derived:

Definition 1.1 — The Seven SI Base Units

Quantity	Unit	Symbol
Length	meter	m
Mass	kilogram	kg
Time	second	s
Electric current	ampere	A
Temperature	kelvin	K
Amount of substance	mole	mol
Luminous intensity	candela	cd

All other units are combinations of these seven. A newton (N) = kg·m/s². A joule (J) = kg·m²/s². A pascal (Pa) = kg/(m·s²). Understanding how derived units are constructed is the key to dimensional analysis.

The SI prefix system allows us to express any quantity without resorting to scientific notation in everyday contexts:

Definition 1.2 — SI Prefixes

pico (p)10⁻¹²

nano (n)10⁻⁹

micro (\mu

10⁻⁶

milli (m)10⁻³

centi (c)10⁻²

kilo (k)10³

mega (M)10⁶

giga (G)10⁹

tera (T)10¹²

peta (P)10¹⁵

1.4 Dimensional Analysis

Dimensional analysis is the most powerful elementary tool in physics. The core principle is simple: in any valid physical equation, every term must have the same dimensions. You cannot add meters to kilograms. You cannot set a force equal to a velocity. Checking dimensions catches errors, guides derivations, and can even tell you the answer before you do the calculation.

Definition 1.3 — Dimensional Consistency

In any physically valid equation, all ter

ms must have identical dimensions. Dimensions are denoted with square brackets: [length] =

Example: [velocity] = L/T \qquad [force] = MLT^{-2} \qquad [energy] = ML^{2}T^{-2}

Example 1.1 — Checking the Kinematic Equation

$Verify that x = x_{0} + v_{0}t + \frac{1}{2}at^{2} is dimensionally consistent$ .

[x] = [x₀] =[L] — displacement, meters. ✓

[v₀t] =[

LT^{-1}][T] = [L] ✓

[at²] =[

LT^{-2}][T^{2}] = [L] ✓

Conclusion:Every term has dimension L. The equation is dimensionally consistent.

Dimensional analysis can also derive the form of equations. If you know a result depends on certain variables, and you know their dimensions, you can often determine the answer up to a dimensionless constant.

Example 1.2 — Deriving the Pendulum Period by Dimensional Analysis

A pendulum's period T might depend on length L, mass m, and gravitational acceleration g. Find T.

Write the ansatz:

T = C_{i} Lᵃ mᵇ gᶜ for some powers a, b, c and constant C

Dimension equation:[

T] = Lᵃ Mᵇ (LT^{-2})ᶜ \to T^{1} = L^{a+c} M^b T^(-2c

Match powers:

T: 1 = -2c \to c = -\frac{1}{2} \qquad M: 0 = b \to b = 0 \qquad L: 0 = a+c \to a = \frac{1}{2}

Result:

T = C \sqrt(L/g). The mass drops out entirely. Experiment gives C = 2\pi

1.5 Scientific Notation and Significant Figures

Physics routinely deals with numbers spanning many orders of magnitude. Scientific notation expresses any number as a × 10ⁿ where 1 ≤ a < 10:

602{,}214{,}076{,}000{,}000{,}000{,}000{,}000 = 6.022 \times 10^{23} \qquad \text{(Avogadro's number)}

(1.1)

Significant figures express measurement precision. A measurement of 3.45 m has three significant figures, meaning uncertainty in the last digit: the true value is between 3.445 and 3.455 m. When multiplying or dividing, the result has as many significant figures as the least-precise input. When adding or subtracting, align decimal places.

In this curriculum, intermediate calculations retain extra digits; final answers are reported to three or four significant figures unless otherwise specified.

1.6 Order-of-Magnitude Estimation

A Fermi estimate is a rapid calculation to within a factor of 10 using only rough knowledge and dimensional reasoning. Fermi himself famously estimated the yield of the first atomic bomb by dropping scraps of paper during the Trinity test and watching how far they drifted. The ability to make rapid, reliable order-of-magnitude estimates is essential in every branch of physics.

Example 1.3 — How many piano tuners are there in Chicago?

This is the original Fermi problem, asked in the University of Chicago physics PhD entrance exam.

Population:

Chicago \approx 3 million people \approx 3\times10^{6}

Pianos:~

1 piano per 20 households, ~2.5 people/household \to ~60,000 pianos

Tunings per year:

Each piano tuned ~1/year \to 60,000 tunings/year

Tuner capacity:

Tuner works ~250 days/year, ~8 jobs/day \to 2,000 tunings/tuner/year

Result:60,000 / 2,000 ≈ 30 piano tuners. (Yellow Pages listed ~50.)

Definition 1.4 — Common Traps

Units are part of the number: 5 meters and 5 seconds are not interchangeable just because both contain 5.
Dimensional consistency is necessary, not sufficient: it catches many errors but does not prove an equation is correct.
Prefixes attach to units: $1 km^{2} means (1000 m)^{2}, not 1000 m^{2}$
Significant figures track measurement precision: do not report more precision than the data support.
Fermi estimates should be honest approximations: one significant digit is usually the right level of confidence.

Exercises — 1.1–1.6 Measurement and Units

Convert 5.5 km to meters.

Straightforward

Convert 13 km/h to m/s.

m/s

Straightforward

Determine the power n in the drag force formula F = C\rhoAv^n by dimensional analysis

(dimensionless power)

Intermediate

4.Fermi estimate: If all 8 billion humans were laid en

d to end, how many times would they stretch from Earth to the Moon? (Earth–Moon distance \approx

Intermediate

5.Use dimensional analysis to derive the form o

f the orbital period T of a satellite as a function of orbital radius r, planetary mass M_

0^{-11} m^{3}/(kg\cdots^{2}

Challenging

Key Takeaways

Physics seeks the fewest general principles that explain the widest range of phenomena.
SI base units: meter, kilogram, second, ampere, kelvin, mole, candela — all others are derived.
Every valid physical equation is dimensionally consistent — dimensions are an algebraic tool.
Dimensional analysis can derive the form of physical laws before doing any detailed calculation.
Order-of-magnitude estimation is a core skill: 30 piano tuners is as good as the exact answer.
The universe spans 61 orders of magnitude; physics operates fluently across all of them.