Units and Dimensions

1. Why Physical Quantities Need Units

A number by itself almost never describes a physical quantity. If a drawing tells you that a length is 2, you cannot build anything from it. Two meters, two feet, and two hand spans are three different lengths, and only one of them is what the designer meant. The number answers how many, and the unit answers how many of what. Both are part of the answer, and dropping either one makes the statement meaningless.¹

The Cost of an Implicit Unit: The Mars Climate Orbiter

In 1999 the Mars Climate Orbiter was lost because one team supplied thrust impulse data in pound-force seconds while the receiving software expected newton seconds. The numbers were right. The units were not communicated, and a $125 million spacecraft entered the Martian atmosphere at the wrong altitude.

2. Standardizing the Units

It is not enough for everyone to use units. Everyone must use the same units, defined the same way. When an engineer in Boston specifies a shaft of mass 4.5 kg, a supplier in Osaka must understand exactly the same mass.²

The push for a single worldwide system led to the Metre Convention of 1875 and eventually, in 1960, to the International System of Units, abbreviated SI from the French Système International d'Unités.

Evolution of defined units from physical objects to definition based on nature's constants

The first standards were physical objects. A platinum-iridium bar held near Paris was the meter, and a platinum-iridium cylinder kept beside it was the kilogram. Every national laboratory received a certified copy, and every instrument in every factory was traceable, through a chain of comparisons, back to those two artifacts.

Artifacts have an unavoidable flaw. They change. Metal expands with temperature, absorbs contaminants from the air, and loses a little material every time it is handled or cleaned. Over a century the international prototype kilogram and its official copies drifted apart from one another by several tens of micrograms. Worse, there was no way to tell which one had drifted, because the prototype defined the kilogram and so was, by construction, always exactly right.

The solution was to stop defining units by objects and start defining them by constants of nature, which do not age. The transition happened one unit at a time:

• 1960: the meter was redefined in terms of the wavelength of a krypton-86 emission line, retiring the platinum-iridium bar.
• 1967: the second was redefined by the hyperfine transition frequency of the cesium-133 atom.
• 1983: the meter was redefined again, this time by fixing the speed of light at exactly $c=299792458\mathrm{m}/\mathrm{s}$. One meter is therefore the distance light travels in vacuum in $1/299792458$ of a second.
• 2019: the kilogram, the last unit still tied to a physical artifact, was redefined by fixing the Planck constant at exactly $h=6.62607015\times {10}^{-34}\mathrm{J}\mathrm{s}$. The ampere, kelvin, and mole were redefined in the same revision.

Since May 20, 2019, every SI unit is defined by a fixed numerical value of a physical constant. The practical consequence for you is that the definitions no longer change, and a measurement made in any properly equipped laboratory in the world is comparable to any other.

3. The Seven Base Quantities

There are thousands of physical quantities, and defining a separate independent standard for each one would be impossible to maintain and impossible to keep mutually consistent. Scientists recognized long ago that this is unnecessary. Every physical and chemical quantity can be expressed in terms of just seven base quantities, with all remaining quantities following from physical laws that relate them.

Base quantity	Quantity symbol	Dimension symbol	SI base unit	Unit symbol
Length	$l,x,r,d$	$𝖫$	meter	m
Mass	$m$	$𝖬$	kilogram	kg
Time	$t$	$𝖳$	second	s
Thermodynamic temperature	$T$	$\mathsf{\Theta }$	kelvin	K
Electric current	$I,i$	$𝖨$	ampere	A
Amount of substance	$n$	$𝖭$	mole	mol
Luminous intensity	$I_v$	$𝖩$	candela	cd

In mechanics we work almost entirely with length, mass, and time, and we add temperature when thermal effects enter. The last three rows belong to electrical engineering, chemistry, and photometry.

A note on typography

Following the international standard ISO 80000-1, dimension symbols are set in upright sans-serif capitals, $𝖬$, $𝖫$, $𝖳$, $\mathsf{\Theta }$, $𝖨$, $𝖭$, $𝖩$, while symbols for physical quantities are set in italic. The distinction is not decoration. In a single chapter $T$ may denote a period, a temperature, a torque, or a kinetic energy, whereas $𝖳$ always means the dimension of time. Likewise $𝖨$ is the dimension of electric current, while $I$ is usually the second moment of area. Train your eye on the difference now and a great deal of later confusion disappears.

You will find older textbooks, especially in structural and mechanical engineering, that treat force rather than mass as a base dimension and write dimensions in an $𝖥$, $𝖫$, $𝖳$ system. That choice is not wrong, and it was once convenient because engineers measure force with a scale far more often than they measure mass directly. It is no longer standard, and it interacts badly with the pound, as Section 9 explains. This book uses the $𝖬$, $𝖫$, $𝖳$, $\mathsf{\Theta }$ system throughout.

SI Prefixes

Rather than inventing new unit names for very large and very small values, SI attaches a prefix to an existing unit. The prefix multiplies the unit by a power of ten. The ones you will actually use are:

Prefix	Symbol	Factor	Prefix	Symbol	Factor
tera	T	${10}^{12}$	centi	c	${10}^{-2}$
giga	G	${10}^9$	milli	m	${10}^{-3}$
mega	M	${10}^6$	micro	$\mu$	${10}^{-6}$
kilo	k	${10}^3$	nano	n	${10}^{-9}$
			pico	p	${10}^{-12}$

Three prefixes dominate engineering practice. Stresses (force divided by area) are quoted in MPa or GPa, dimensions on drawings in mm, and structural loads in kN. The complete list runs from quetta (${10}^{30}$) to quecto (${10}^{-30}$), the four outermost prefixes having been added by the General Conference on Weights and Measures in November 2022. See the NIST table of SI prefixes for the full set.

Two rules are worth memorizing now.

• First, prefixes do not stack: write nanometer, never millimicrometer.
• Second, the kilogram is the only base unit whose name already contains a prefix, so additional prefixes attach to the gram, not the kilogram. One millionth of a kilogram is a milligram (mg), not a microkilogram.

4. Dimension Versus Unit

The dimension of a quantity records which base quantities it is built from, and in what powers. Writing $\dim Q$ for "the dimension of $Q$," every mechanical quantity has the form

\bbox[8px, #E6F0FA, border: 3px solid #0066CC]{\dim Q = \mathsf{M}^{a}\,\mathsf{L}^{b}\,\mathsf{T}^{c}\,\mathsf{\Theta}^{d}}

where the exponents $a,b,c,d$ are fixed rational numbers, determined by the physics.

Some books write $[Q]$ in place of $\dim Q$; the two notations mean the same thing, and we use $\dim Q$ here.

Once the dimension of a quantity is known, its SI unit follows immediately by replacing each base dimension with its base unit.

Table of Common Derived Quantities

Quantity	Defining relation	Dimension	SI unit	Special name
Area	$A=\ell w$	${𝖫}^2$	${\mathrm{m}}^2$
Volume	$V=\ell wh$	${𝖫}^3$	${\mathrm{m}}^3$
Velocity	$v=\mathrm{\Delta }x/\mathrm{\Delta }t$	$𝖫{𝖳}^{-1}$	$\mathrm{m}/\mathrm{s}$
Acceleration	$a=\mathrm{\Delta }v/\mathrm{\Delta }t$	$𝖫{𝖳}^{-2}$	$\mathrm{m}/{\mathrm{s}}^2$
Angular velocity	$\omega =\mathrm{\Delta }\theta /\mathrm{\Delta }t$	${𝖳}^{-1}$	$\mathrm{rad}/\mathrm{s}$
Frequency	$f=1/T_{\text{period}}$	${𝖳}^{-1}$	${\mathrm{s}}^{-1}$	hertz (Hz)
Density	$\rho =m/V$	$𝖬{𝖫}^{-3}$	$\mathrm{kg}/{\mathrm{m}}^3$
Momentum	$p=mv$	$𝖬𝖫{𝖳}^{-1}$	$\mathrm{kg}\cdot \mathrm{m}/\mathrm{s}$
Force	$F=ma$	$𝖬𝖫{𝖳}^{-2}$	$\mathrm{kg}\cdot \mathrm{m}/{\mathrm{s}}^2$	newton (N)
Moment, torque	$M_O=Fd$	$𝖬{𝖫}^2{𝖳}^{-2}$	$\mathrm{N}\cdot \mathrm{m}$
Pressure, stress	$\sigma =F/A$	$𝖬{𝖫}^{-1}{𝖳}^{-2}$	$\mathrm{N}/{\mathrm{m}}^2$	pascal (Pa)
Work, energy	$W=Fd$	$𝖬{𝖫}^2{𝖳}^{-2}$	$\mathrm{N}\cdot \mathrm{m}$	joule (J)
Power	$P=W/t$	$𝖬{𝖫}^2{𝖳}^{-3}$	$\mathrm{J}/\mathrm{s}$	watt (W)
Dynamic viscosity	$\tau =\mu dv/dy$ ($\tau$ is shear stress and $v$ is velocity)	$𝖬{𝖫}^{-1}{𝖳}^{-1}$	$\mathrm{Pa}\cdot \mathrm{s}$

The full list of SI derived units with special names is given in the BIPM SI Brochure and, in a more compact form, on the NIST SI units page.

5. Units Outside SI That Survive in Practice

SI is the reference system, but a number of non-SI units remain in daily use because they are convenient, entrenched, or both. You should be able to recognize and convert them.

Quantity	Unit	Equivalent
Temperature	degree Celsius (${}^{\circ }$C)	$T[\mathrm{K}]=t{[}^{\circ }\mathrm{C}]+273.15$; a difference of $1{}^{\circ }$C equals $1$ K
Energy	electronvolt (eV)	$1\mathrm{eV}=1.602176634\times {10}^{-19}$ J
Energy	calorie (cal)	$1\mathrm{cal}=4.184$ J
Energy	British thermal unit (Btu)	$1\mathrm{Btu}\approx 1055$ J
Power	horsepower (hp)	$1\mathrm{hp}=745.7$ W
Pressure	bar	$1\mathrm{bar}={10}^5$ Pa
Pressure	atmosphere (atm)	$1\mathrm{atm}=101325$ Pa
Volume	liter (L)	$1\mathrm{L}={10}^{-3}{\mathrm{m}}^3$
Time	hour (h)	$1\mathrm{h}=3600$ s
Mass	tonne (t)	$1\mathrm{t}=1000$ kg

The Celsius scale deserves particular care. Celsius and kelvin degrees are the same size, so temperature differences convert one-to-one, but the two scales have different zeros. In any formula containing an absolute temperature, such as the ideal gas law $pV=nRT$, you must use kelvin. Substituting Celsius there is a common and expensive error.

6. The Principle of Dimensional Homogeneity

The reasoning is simple. You cannot add a length to a mass any more than you can add three apples to four Tuesdays. If an equation says $A=B+C$, then $A$, $B$, and $C$ must all be the same kind of quantity, so

$$\dim A=\dim B=\dim C.$$

Two corollaries follow, and both are used constantly:

1. Arguments of transcendental functions must be dimensionless. In $\sin \theta$, $e^x$, $\ln x$, and $\tanh x$, the argument carries no dimension, and neither does the result. You can see why from the series $e^x=1+x+x^2/2!+\cdots$: for those terms to be addable, $x$ must be a pure number. So in $x(t)=A{e}^{-\lambda t}$, the constant $\lambda$ must have dimension ${𝖳}^{-1}$.
2. Both sides of an equation must have the same dimension. This gives you a free error check on every result you derive.

This is a preview of a much more powerful method. Homogeneity is a checking tool, used after the fact to catch errors. The same principle can be turned around and used as a predictive tool, one that extracts the form of an unknown physical relationship, reduces the number of experiments needed to characterize a system, and tells an engineer how to scale results from a small model to a full-size structure.

Working Rules for Writing Units

Units carry meaning, so they follow a grammar:

• Leave a space between the number and the unit: $5\mathrm{kg}$, not $5\mathrm{kg}$. The exception is the degree symbol for angles, as in ${30}^{\circ }$.
• Unit symbols are never pluralized and take no period: $10\mathrm{kg}$, not $10\mathrm{kgs}$ or $10\mathrm{kg}.$
• Unit symbols derived from a person's name are capitalized (N, Pa, J, W, K, A), but the spelled-out name is not (newton, pascal, joule, watt, kelvin, ampere).
• Write K, not ${}^{\circ }$K. The degree sign was dropped from the kelvin in 1967.
• Use at most one solidus in a compound unit, or none at all: $\mathrm{J}/(\mathrm{kg}\cdot \mathrm{K})$ or $\mathrm{J}{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$, never $\mathrm{J}/\mathrm{kg}/\mathrm{K}$.
• Carry units through the entire calculation rather than restoring them at the end. Treating units as algebraic factors that cancel is the surest way to catch a mistake early.

7. The US Customary System

The US Customary System (USCS) is the system of feet, pounds, and seconds still in everyday use in the United States. Its units descend from the English units of the colonial period.

US Customary System vs Imperial System

It is often called "the imperial system," and that name is not quite right. The British Imperial system was defined by an Act of Parliament in 1824, after the United States had already standardized on the older English units. The two systems therefore diverged, and they still differ for some quantities. A US gallon is 3.785 L; an imperial gallon is 4.546 L. A US fluid ounce and an imperial fluid ounce are also different. For length, mass, and force the two agree, because of an international treaty described below, but it is safer to call the American system US customary and reserve imperial for the British one.

USCS units are defined in terms of SI units

One point often surprises students: since the International Yard and Pound Agreement of 1959, USCS units are themselves defined in terms of SI units. By international agreement,

$$1\mathrm{yd}=0.9144\mathrm{m}\text{(exactly)},1\mathrm{lb}=0.45359237\mathrm{kg}\text{(exactly)}$$

So there is no competing physical standard for the foot or the pound. There is one system of standards, SI, and USCS is a set of conversion factors on top of it.

Metrication

Metrication: Why the US and Other Nations Use Mixed Unit Systems

Through the twentieth century most countries converted to SI. Many were moving from the older cgs system (centimeter, gram, second) or from national metric variants, so the change was a rescaling rather than a conceptual break.

The United Kingdom and Canada converted only partially. Both use SI officially, yet British road signs give distances in miles and speeds in miles per hour, beer is sold in pints, and body weight is commonly quoted in stones. Canadian road signs are metric while lumber, plumbing fittings, and much construction practice remain in inches and feet, largely because of trade with the United States.

In the United States, SI has been legal since 1866 and preferred for federal agencies since 1988, but it never displaced USCS in daily life. The outcome is a split along industry lines. Scientific research, pharmaceuticals, medicine, the military, and the automotive and aerospace industries have largely moved to SI. Construction, consumer products, and the trades have not. The reason is practical rather than ideological: an entire supply chain is built around the existing sizes. A carpenter asks for a "two by four," a plumber specifies $3\frac{1}{8}$ inch pipe, and every tool, fitting, fastener, and building code in the chain is dimensioned to match. Converting one link of that chain in isolation gains nothing and breaks everything.

As an engineer working in the United States you should expect to move between the two systems and should be fluent in both. Carrying units explicitly through every calculation is what makes that safe.

8. Base Units of the USCS

The USCS in engineering is usually set up as a force-length-time system:

Base quantity	Unit	Symbol
Force	pound-force	lbf
Length	foot	ft
Time	second	s

Mass is then a derived quantity. This is the reverse of SI, where mass is a base quantity and force is derived, and it is exactly the historical situation mentioned in Section 3.

9. Pound-Force, Pound-Mass, and the Slug

This section addresses the single most common source of numerical errors in American engineering practice. Read it slowly.

Where the Trouble Starts

Historically the pound was a unit of mass. It was also used as a unit of weight, meaning the gravitational force on that mass, because a merchant weighing goods on a balance does not care about the distinction. Both usages survive, so the word "pound" now denotes two different quantities:

• pound-mass (lbm): a mass, exactly $0.45359237$ kg.
• pound-force (lbf): a force, defined as the weight of one pound-mass at standard gravity $g_0=32.174\mathrm{ft}/{\mathrm{s}}^2=9.80665\mathrm{m}/{\mathrm{s}}^2$. Numerically, $1\mathrm{lbf}=4.4482$ N.

The definition was chosen so that one pound-mass weighs one pound-force at the Earth's surface. That is convenient at the grocery store and disastrous in Newton's second law. Substituting directly,

$$F=ma⟶1\mathrm{lbf}\stackrel{?}{=}(1\mathrm{lbm})(32.174\mathrm{ft}/{\mathrm{s}}^2)=32.174\mathrm{lbm}\cdot \mathrm{ft}/{\mathrm{s}}^2$$

The left and right sides are numerically unequal. The set {lbf, lbm, ft, s} is not a consistent set of units, because the unit of force is not equal to (unit of mass) $\times$ (unit of acceleration). SI has no such problem: the newton was defined as $\mathrm{kg}\cdot \mathrm{m}/{\mathrm{s}}^2$, so $F=ma$ holds with no extra factor.

Three Ways Out

Engineers solve this in one of three ways.

Option 1: Introduce a conversion constant $g_c$. Keep both lbm and lbf and write Newton's second law as

\bbox[8px, #E6F0FA, border: 3px solid #0066CC]{F = \frac{ma}{g_{c}}, \qquad g_{c} = 32.174\ \frac{\mathrm{lbm\cdot ft}}{\mathrm{lbf\cdot s^{2}}}}

The constant $g_c$ is a pure unit conversion, dimensionally equivalent to 1 in the same way that $12\mathrm{in}/\mathrm{ft}$ is, not a physical quantity, and in particular it is not the local gravitational acceleration even though it shares the number 32.174. You will see this form in thermodynamics and heat transfer texts, where mass flow rates are naturally expressed in lbm/s.

Option 2: Define a consistent mass unit, the slug. Take lbf, ft, and s as the base units and define the mass unit so that $F=ma$ works unchanged:

\bbox[8px, #E6F0FA, border: 3px solid #0066CC]{1\ \mathrm{slug} = 1\ \frac{\mathrm{lbf\cdot s^{2}}}{\mathrm{ft}}, \qquad 1\ \mathrm{slug} = 32.174\ \mathrm{lbm} = 14.594\ \mathrm{kg}}

A one-slug mass accelerates at $1\mathrm{ft}/{\mathrm{s}}^2$ under $1\mathrm{lbf}$. This is the standard choice in statics, dynamics, and fluid mechanics, and it is the convention this book follows when USCS units are used.

Option 3: Define a consistent force unit, the poundal. Take lbm, ft, and s as base and define $1\mathrm{pdl}=1\mathrm{lbm}\cdot \mathrm{ft}/{\mathrm{s}}^2=0.1383$ N. The poundal is rarely used today and appears mainly in older British texts.

The general rule: {lbf, slug, ft, s} is consistent, and {lbf, lbm, ft, s} is not. In a consistent set you may use $F=ma$ directly. In an inconsistent set you must carry $g_c$.

Weight

For an object of mass $m$ at a location where the gravitational acceleration is $g$, the weight is

\bbox[8px, #E6F0FA, border: 3px solid #0066CC]{W = mg \quad \text{(SI and slug-based USCS)}, \qquad W = \frac{mg}{g_{c}} \quad \text{(lbm-based USCS)}}

Weight depends on location; mass does not. On the Moon, where $g\approx 1.62\mathrm{m}/{\mathrm{s}}^2$, a 70 kg astronaut still has a mass of 70 kg but weighs only 113 N instead of 687 N.

Useful Conversions

Quantity	Conversion
Length	$1\mathrm{in}=25.4$ mm (exact); $1\mathrm{ft}=0.3048$ m (exact); $1\mathrm{mi}=1.609$ km
Mass	$1\mathrm{lbm}=0.45359237$ kg (exact); $1\mathrm{slug}=14.594$ kg
Force	$1\mathrm{lbf}=4.4482$ N; $1\mathrm{kip}=1000\mathrm{lbf}=4.4482$ kN
Pressure	$1\mathrm{psi}=6.8948$ kPa; $1\mathrm{ksi}=6.8948$ MPa
Energy	$1\mathrm{ft}\cdot \mathrm{lbf}=1.3558$ J; $1\mathrm{Btu}=1055.1$ J
Power	$1\mathrm{hp}=550\mathrm{ft}\cdot \mathrm{lbf}/\mathrm{s}=745.7$ W
Temperature	$\mathrm{\Delta }T[\mathrm{K}]=\frac{5}{9}\mathrm{\Delta }T{[}^{\circ }\mathrm{R}]$; $T{[}^{\circ }\mathrm{F}]=\frac{9}{5}t{[}^{\circ }\mathrm{C}]+32$

The absolute temperature scale paired with Fahrenheit is the Rankine scale, $T{[}^{\circ }\mathrm{R}]=T{[}^{\circ }\mathrm{F}]+459.67$. Use it wherever an absolute temperature is required in USCS work, just as you would use kelvin in SI.

9. Best Practices for Converting Units

Every conversion in this chapter follows one mechanical rule. Take an equality between units, such as $1\mathrm{ft}=0.3048\mathrm{m}$, and rewrite it as a fraction equal to 1:

$$\frac{0.3048\mathrm{m}}{1\mathrm{ft}}=1\text{or}\frac{1\mathrm{ft}}{0.3048\mathrm{m}}=1$$

Multiplying a quantity by 1 does not change it, so you may multiply by either fraction freely. Choose the one that places the old unit in the opposite position from where it sits in the quantity: if the old unit is in the numerator, put it in the denominator of the fraction, and vice versa. The old unit then cancels and the new one remains. Squared or cubed units need the fraction squared or cubed as well.

Summary

• A physical quantity is a number together with a unit. Only dimensionless quantities such as strain, friction coefficients, and the Reynolds number are exceptions.
• SI defines all units through fixed values of physical constants. Since the 2019 redefinition of the kilogram, no SI unit depends on a physical artifact.
• Seven base quantities (length, mass, time, temperature, electric current, amount of substance, luminous intensity) generate every other quantity through physical laws.
• The dimension of a quantity, written ${𝖬}^a{𝖫}^b{𝖳}^c{\mathsf{\Theta }}^d{𝖨}^e{𝖭}^f{𝖩}^g$, states what kind of quantity it is. A unit is the particular standard used to measure it. The two are not the same thing.
• Every term in a valid equation has the same dimension, and arguments of transcendental functions are always dimensionless. Use this to check your work on every problem, and see the next section for how the same principle predicts the form of unknown relationships.
• USCS units are defined in terms of SI units. In USCS, use the consistent set {lbf, slug, ft, s} and apply $F=ma$ directly, or keep lbm and carry $g_c=32.174\mathrm{lbm}\cdot \mathrm{ft}/(\mathrm{lbf}\cdot {\mathrm{s}}^2)$. Never mix lbf and lbm in $F=ma$ without $g_c$.

References

1. Bureau International des Poids et Mesures, The International System of Units (SI), 9th edition, 2019. https://www.bipm.org/en/publications/si-brochure
2. NIST, Guide for the Use of the International System of Units (SI), Special Publication 811. https://www.nist.gov/pml/special-publication-811
3. NIST, SI Units and Prefixes reference tables. https://physics.nist.gov/cuu/Units/units.html and https://physics.nist.gov/cuu/Units/prefixes.html
4. NIST, The International System of Units (SI), Special Publication 330, 2019 edition. https://nvlpubs.nist.gov/nistpubs/SpecialPublications/NIST.SP.330-2019.pdf

Footnotes

1. A small number of quantities are genuinely dimensionless and carry no unit. Ratios of two quantities of the same kind fall in this class: strain (length per length), Poisson's ratio (a material property that quantifies the proportional contraction in width relative to the extension in length), the coefficient of friction (${\mu }_s$ in $f_s\le {\mu }_{s}N$ relating the static friction force to the normal force, with equality at impending slip), and specific gravity. So do the named groups you will meet in fluid mechanics and heat transfer, such as the Reynolds, Mach, Prandtl, and Nusselt numbers. Plane angle is a special case. The radian is defined as a ratio of arc length to radius, so it is dimensionless, yet we still write "rad" to make clear that an angle, and not some other pure number, is intended. Counts behave the same way: the number of bolts in a flange is 12, not 12 of anything. ↩

2. Be careful with everyday language here. In casual speech people say "my weight is 70 kilograms," but the kilogram is a unit of mass, not weight. Weight is the gravitational force on an object, so its SI unit is the newton. A 70 kg person weighs about 687 N at the Earth's surface. ↩