1 Units and Measurement

1.2 Units and Standards

Learning Objectives

By the end of this section, you will be able to:

  • Describe how SI base units are defined.
  • Describe how derived units are created from base units.
  • Express quantities given in SI units using metric prefixes.

As we saw previously, the range of objects and phenomena studied in physics is immense. From the incredibly short lifetime of a nucleus to the age of Earth, from the tiny sizes of subnuclear particles to the vast distance to the edges of the known universe, from the force exerted by a jumping flea to the force between Earth and the Sun, there are enough factors of 10 to challenge the imagination of even the most experienced scientist. Giving numerical values for physical quantities and equations for physical principles allows us to understand nature much more deeply than qualitative descriptions alone. To comprehend these vast ranges, we must also have accepted units in which to express them. We shall find that even in the potentially mundane discussion of meters, kilograms, and seconds, a profound simplicity of nature appears: all physical quantities can be expressed as combinations of only seven base physical quantities.

We define a physical quantity either by specifying how it is measured or by stating how it is calculated from other measurements. For example, we might define distance and time by specifying methods for measuring them, such as using a meter stick and a stopwatch. Then, we could define average speed by stating that it is calculated as the total distance traveled divided by time of travel.

Measurements of physical quantities are expressed in terms of units, which are standardized values. For example, the length of a race, which is a physical quantity, can be expressed in units of meters (for sprinters) or kilometers (for distance runners). Without standardized units, it would be extremely difficult for scientists to express and compare measured values in a meaningful way ((Figure)).

A drawing of a person looking at a map that has the distance scale labeled as 1 cable, and wondering how big is a cable.
Figure 1.7 Distances given in unknown units are maddeningly useless.

Two major systems of units are used in the world: SI units (for the French Système International d’Unités), also known as the metric system, and English units (also known as the customary or imperial system). English units were historically used in nations once ruled by the British Empire and are still widely used in the United States. English units may also be referred to as the foot–pound–second (fps) system, as opposed to the centimeter–gram–second (cgs) system. You may also encounter the term SAE units, named after the Society of Automotive Engineers. Products such as fasteners and automotive tools (for example, wrenches) that are measured in inches rather than metric units are referred to as SAE fasteners or SAE wrenches.

Virtually every other country in the world (except the United States) now uses SI units as the standard. The metric system is also the standard system agreed on by scientists and mathematicians.

SI Units: Base and Derived Units

In any system of units, the units for some physical quantities must be defined through a measurement process. These are called the base quantities for that system and their units are the system’s base units. All other physical quantities can then be expressed as algebraic combinations of the base quantities. Each of these physical quantities is then known as a derived quantity and each unit is called a derived unit. The choice of base quantities is somewhat arbitrary, as long as they are independent of each other and all other quantities can be derived from them. Typically, the goal is to choose physical quantities that can be measured accurately to a high precision as the base quantities. The reason for this is simple. Since the derived units can be expressed as algebraic combinations of the base units, they can only be as accurate and precise as the base units from which they are derived.

Based on such considerations, the International Standards Organization recommends using seven base quantities, which form the International System of Quantities (ISQ). These are the base quantities used to define the SI base units. (Figure) lists these seven ISQ base quantities and the corresponding SI base units.

ISQ Base Quantities and Their SI Units
ISQ Base Quantity SI Base Unit
Length meter (m)
Mass kilogram (kg)
Time second (s)
Electrical current ampere (A)
Thermodynamic temperature kelvin (K)
Amount of substance mole (mol)
Luminous intensity candela (cd)

You are probably already familiar with some derived quantities that can be formed from the base quantities in (Figure). For example, the geometric concept of area is always calculated as the product of two lengths. Thus, area is a derived quantity that can be expressed in terms of SI base units using square meters

    \[(\text{m}\,×\,\text{m}={\text{m}}^{2}).\]

Similarly, volume is a derived quantity that can be expressed in cubic meters

    \[({\text{m}}^{3}).\]

Speed is length per time; so in terms of SI base units, we could measure it in meters per second (m/s). Volume mass density (or just density) is mass per volume, which is expressed in terms of SI base units such as kilograms per cubic meter (kg/m3). Angles can also be thought of as derived quantities because they can be defined as the ratio of the arc length subtended by two radii of a circle to the radius of the circle. This is how the radian is defined. Depending on your background and interests, you may be able to come up with other derived quantities, such as the mass flow rate (kg/s) or volume flow rate (m3/s) of a fluid, electric charge

    \[(\text{A}·\text{s}),\]

mass flux density

    \[\text{[kg/}({\text{m}}^{2}·\text{s)],}\]

and so on. We will see many more examples throughout this text. For now, the point is that every physical quantity can be derived from the seven base quantities in (Figure), and the units of every physical quantity can be derived from the seven SI base units.

For the most part, we use SI units in this text. Non-SI units are used in a few applications in which they are in very common use, such as the measurement of temperature in degrees Celsius

    \[(\text{°}\text{C}),\]

the measurement of fluid volume in liters (L), and the measurement of energies of elementary particles in electron-volts (eV). Whenever non-SI units are discussed, they are tied to SI units through conversions. For example, 1 L is

    \[{10}^{-3}{\,\text{m}}^{3}.\]

Check out a comprehensive source of information on SI units at the National Institute of Standards and Technology (NIST) Reference on Constants, Units, and Uncertainty.

Units of Time, Length, and Mass: The Second, Meter, and Kilogram

The initial chapters in this textbook are concerned with mechanics, fluids, and waves. In these subjects all pertinent physical quantities can be expressed in terms of the base units of length, mass, and time. Therefore, we now turn to a discussion of these three base units, leaving discussion of the others until they are needed later.

The second

The SI unit for time, the second (abbreviated s), has a long history. For many years it was defined as 1/86,400 of a mean solar day. More recently, a new standard was adopted to gain greater accuracy and to define the second in terms of a nonvarying or constant physical phenomenon (because the solar day is getting longer as a result of the very gradual slowing of Earth’s rotation). Cesium atoms can be made to vibrate in a very steady way, and these vibrations can be readily observed and counted. In 1967, the second was redefined as the time required for 9,192,631,770 of these vibrations to occur ((Figure)). Note that this may seem like more precision than you would ever need, but it isn’t—GPSs rely on the precision of atomic clocks to be able to give you turn-by-turn directions on the surface of Earth, far from the satellites broadcasting their location.

A top view of an atomic fountain.
Figure 1.8 An atomic clock such as this one uses the vibrations of cesium atoms to keep time to a precision of better than a microsecond per year. The fundamental unit of time, the second, is based on such clocks. This image looks down from the top of an atomic fountain nearly 30 feet tall. (credit: Steve Jurvetson)

The meter

The SI unit for length is the meter (abbreviated m); its definition has also changed over time to become more precise. The meter was first defined in 1791 as 1/10,000,000 of the distance from the equator to the North Pole. This measurement was improved in 1889 by redefining the meter to be the distance between two engraved lines on a platinum–iridium bar now kept near Paris. By 1960, it had become possible to define the meter even more accurately in terms of the wavelength of light, so it was again redefined as 1,650,763.73 wavelengths of orange light emitted by krypton atoms. In 1983, the meter was given its current definition (in part for greater accuracy) as the distance light travels in a vacuum in 1/299,792,458 of a second ((Figure)). This change came after knowing the speed of light to be exactly 299,792,458 m/s. The length of the meter will change if the speed of light is someday measured with greater accuracy.

A drawing of a meter stick and a flashlight shining a beam of light. An arrow indicates that the beam spans the length of the meter stick. The drawing is labeled “ light travels a distance of 1 meter in 1 over 299,792,458 seconds.
Figure 1.9 The meter is defined to be the distance light travels in 1/299,792,458 of a second in a vacuum. Distance traveled is speed multiplied by time.

The kilogram

The SI unit for mass is the kilogram (abbreviated kg); it is defined to be the mass of a platinum–iridium cylinder kept with the old meter standard at the International Bureau of Weights and Measures near Paris. Exact replicas of the standard kilogram are also kept at the U.S. National Institute of Standards and Technology (NIST), located in Gaithersburg, Maryland, outside of Washington, DC, and at other locations around the world. Scientists at NIST are currently investigating two complementary methods of redefining the kilogram (see (Figure)). The determination of all other masses can be traced ultimately to a comparison with the standard mass.

There is currently an effort to redefine the SI unit of mass in terms of more fundamental processes by 2018. You can explore the history of mass standards and the contenders in the quest to devise a new one at the website of the Physical Measurement Laboratory.

Figure a is a photograph of the U S national institute of Standards and technology’s watt balance. Figure b is a photograph of a highly polished silicon sphere.
Figure 1.10 Redefining the SI unit of mass. Complementary methods are being investigated for use in an upcoming redefinition of the SI unit of mass. (a) The U.S. National Institute of Standards and Technology’s watt balance is a machine that balances the weight of a test mass against the current and voltage (the “watt”) produced by a strong system of magnets. (b) The International Avogadro Project is working to redefine the kilogram based on the dimensions, mass, and other known properties of a silicon sphere. (credit a and credit b: National Institute of Standards and Technology)

Metric Prefixes

SI units are part of the metric system, which is convenient for scientific and engineering calculations because the units are categorized by factors of 10. (Figure) lists the metric prefixes and symbols used to denote various factors of 10 in SI units. For example, a centimeter is one-hundredth of a meter (in symbols, 1 cm = 10–2 m) and a kilometer is a thousand meters (1 km = 103 m). Similarly, a megagram is a million grams (1 Mg = 106 g), a nanosecond is a billionth of a second (1 ns = 10–9 s), and a terameter is a trillion meters (1 Tm = 1012 m).

Metric Prefixes for Powers of 10 and Their Symbols
Prefix Symbol Meaning Prefix Symbol Meaning
yotta- Y 1024 yocto- y 10–24
zetta- Z 1021 zepto- z 10–21
exa- E 1018 atto- a 10–18
peta- P 1015 femto- f 10–15
tera- T 1012 pico- p 10–12
giga- G 109 nano- n 10–9
mega- M 106 micro-

    \[\mu\]

10–6
kilo- k 103 milli- m 10–3
hecto- h 102 centi- c 10–2
deka- da 101 deci- d 10–1

The only rule when using metric prefixes is that you cannot “double them up.” For example, if you have measurements in petameters (1 Pm = 1015 m), it is not proper to talk about megagigameters, although

    \[{10}^{6}\,×\,{10}^{9}={10}^{15}.\]

In practice, the only time this becomes a bit confusing is when discussing masses. As we have seen, the base SI unit of mass is the kilogram (kg), but metric prefixes need to be applied to the gram (g), because we are not allowed to “double-up” prefixes. Thus, a thousand kilograms (103 kg) is written as a megagram (1 Mg) since

    \[{10}^{3}\text{kg}={10}^{3}\,×\,{10}^{3}\text{g}={10}^{6}\text{g}=1\,\text{Mg}\text{.}\]

Incidentally, 103 kg is also called a metric ton, abbreviated t. This is one of the units outside the SI system considered acceptable for use with SI units.

As we see in the next section, metric systems have the advantage that conversions of units involve only powers of 10. There are 100 cm in 1 m, 1000 m in 1 km, and so on. In nonmetric systems, such as the English system of units, the relationships are not as simple—there are 12 in. in 1 ft, 5280 ft in 1 mi, and so on.

Another advantage of metric systems is that the same unit can be used over extremely large ranges of values simply by scaling it with an appropriate metric prefix. The prefix is chosen by the order of magnitude of physical quantities commonly found in the task at hand. For example, distances in meters are suitable in construction, whereas distances in kilometers are appropriate for air travel, and nanometers are convenient in optical design. With the metric system there is no need to invent new units for particular applications. Instead, we rescale the units with which we are already familiar.

Example

Using Metric Prefixes

Restate the mass

    \[1.93\,×\,{10}^{13}\text{kg}\]

using a metric prefix such that the resulting numerical value is bigger than one but less than 1000.

Strategy

Since we are not allowed to “double-up” prefixes, we first need to restate the mass in grams by replacing the prefix symbol k with a factor of 103 (see (Figure)). Then, we should see which two prefixes in (Figure) are closest to the resulting power of 10 when the number is written in scientific notation. We use whichever of these two prefixes gives us a number between one and 1000.

Solution

Replacing the k in kilogram with a factor of 103, we find that

    \[1.93\,×\,{10}^{13}\text{kg}=1.93\,×\,{10}^{13}\,×\,{10}^{3}\text{g}=1.93\,×\,{10}^{16}\text{g}\text{.}\]

From (Figure), we see that 1016 is between “peta-” (1015) and “exa-” (1018). If we use the “peta-” prefix, then we find that

    \[1.93\,×\,{10}^{16}\text{g}=1.93\,×\,{10}^{1}\text{Pg},\]

since

    \[16=1+15.\]

Alternatively, if we use the “exa-” prefix we find that

    \[1.93\,×\,{10}^{16}\text{g}=1.93\,×\,{10}^{-2}\text{Eg},\]

since

    \[16=-2+18.\]

Because the problem asks for the numerical value between one and 1000, we use the “peta-” prefix and the answer is 19.3 Pg.

Significance

It is easy to make silly arithmetic errors when switching from one prefix to another, so it is always a good idea to check that our final answer matches the number we started with. An easy way to do this is to put both numbers in scientific notation and count powers of 10, including the ones hidden in prefixes. If we did not make a mistake, the powers of 10 should match up. In this problem, we started with

    \[1.93\,×\,{10}^{13}\text{kg,}\]

so we have 13 + 3 = 16 powers of 10. Our final answer in scientific notation is

    \[1.93\,×\,{10}^{1}\]

Pg, so we have 1 + 15 = 16 powers of 10. So, everything checks out.

If this mass arose from a calculation, we would also want to check to determine whether a mass this large makes any sense in the context of the problem. For this, (Figure) might be helpful.

Check Your Understanding

Restate

    \[4.79\,×\,{10}^{5}\text{kg}\]

using a metric prefix such that the resulting number is bigger than one but less than 1000.

[reveal-answer q=”fs-id1168326860727″]Show Solution[/reveal-answer]

[hidden-answer a=”fs-id1168326860727″]

    \[4.79\,×\,{10}^{2}\]

Mg or 479 Mg
[/hidden-answer]

Summary

  • Systems of units are built up from a small number of base units, which are defined by accurate and precise measurements of conventionally chosen base quantities. Other units are then derived as algebraic combinations of the base units.
  • Two commonly used systems of units are English units and SI units. All scientists and most of the other people in the world use SI, whereas nonscientists in the United States still tend to use English units.
  • The SI base units of length, mass, and time are the meter (m), kilogram (kg), and second (s), respectively.
  • SI units are a metric system of units, meaning values can be calculated by factors of 10. Metric prefixes may be used with metric units to scale the base units to sizes appropriate for almost any application.

Conceptual Questions

Identify some advantages of metric units.

[reveal-answer q=”fs-id1168329020784″]Show Solution[/reveal-answer]

[hidden-answer a=”fs-id1168329020784″]

Conversions between units require factors of 10 only, which simplifies calculations. Also, the same basic units can be scaled up or down using metric prefixes to sizes appropriate for the problem at hand.

[/hidden-answer]

What are the SI base units of length, mass, and time?

What is the difference between a base unit and a derived unit? (b) What is the difference between a base quantity and a derived quantity? (c) What is the difference between a base quantity and a base unit?

[reveal-answer q=”fs-id1168329316244″]Show Solution[/reveal-answer]

[hidden-answer a=”fs-id1168329316244″]

a. Base units are defined by a particular process of measuring a base quantity whereas derived units are defined as algebraic combinations of base units. b. A base quantity is chosen by convention and practical considerations. Derived quantities are expressed as algebraic combinations of base quantities. c. A base unit is a standard for expressing the measurement of a base quantity within a particular system of units. So, a measurement of a base quantity could be expressed in terms of a base unit in any system of units using the same base quantities. For example, length is a base quantity in both SI and the English system, but the meter is a base unit in the SI system only.

[/hidden-answer]

For each of the following scenarios, refer to (Figure) and (Figure) to determine which metric prefix on the meter is most appropriate for each of the following scenarios. (a) You want to tabulate the mean distance from the Sun for each planet in the solar system. (b) You want to compare the sizes of some common viruses to design a mechanical filter capable of blocking the pathogenic ones. (c) You want to list the diameters of all the elements on the periodic table. (d) You want to list the distances to all the stars that have now received any radio broadcasts sent from Earth 10 years ago.

Problems

The following times are given using metric prefixes on the base SI unit of time: the second. Rewrite them in scientific notation without the prefix. For example, 47 Ts would be rewritten as

    \[4.7\,×\,{10}^{13}\text{s.}\]

(a) 980 Ps; (b) 980 fs; (c) 17 ns; (d)

    \[577\,\mu \text{s}.\]

The following times are given in seconds. Use metric prefixes to rewrite them so the numerical value is greater than one but less than 1000. For example,

    \[7.9\,×\,{10}^{-2}\text{s}\]

could be written as either 7.9 cs or 79 ms. (a)

    \[9.57\,×\,{10}^{5}\text{s;}\]

(b) 0.045 s; (c)

    \[5.5\,×\,{10}^{-7}\text{s;}\]

(d)

    \[3.16\,×\,{10}^{7}\text{s.}\]

[reveal-answer q=”fs-id1168326839305″]Show Solution[/reveal-answer]

[hidden-answer a=”fs-id1168326839305″]

a. 957 ks; b. 4.5 cs or 45 ms; c. 550 ns; d. 31.6 Ms

[/hidden-answer]

The following lengths are given using metric prefixes on the base SI unit of length: the meter. Rewrite them in scientific notation without the prefix. For example, 4.2 Pm would be rewritten as

    \[4.2\,×\,{10}^{15}\text{m.}\]

(a) 89 Tm; (b) 89 pm; (c) 711 mm; (d)

    \[0.45\,\mu \text{m}\text{.}\]

The following lengths are given in meters. Use metric prefixes to rewrite them so the numerical value is bigger than one but less than 1000. For example,

    \[7.9\,×\,{10}^{-2}\text{m}\]

could be written either as 7.9 cm or 79 mm. (a)

    \[7.59\,×\,{10}^{7}\text{m;}\]

(b) 0.0074 m; (c)

    \[8.8\,×\,{10}^{-11}\text{m;}\]

(d)

    \[1.63\,×\,{10}^{13}\text{m.}\]

[reveal-answer q=”fs-id1168326798286″]Show Solution[/reveal-answer]

[hidden-answer a=”fs-id1168326798286″]

a. 75.9 Mm; b. 7.4 mm; c. 88 pm; d. 16.3 Tm

[/hidden-answer]

The following masses are written using metric prefixes on the gram. Rewrite them in scientific notation in terms of the SI base unit of mass: the kilogram. For example, 40 Mg would be written as

    \[4\,×\,{10}^{4}\text{kg.}\]

(a) 23 mg; (b) 320 Tg; (c) 42 ng; (d) 7 g; (e) 9 Pg.

The following masses are given in kilograms. Use metric prefixes on the gram to rewrite them so the numerical value is bigger than one but less than 1000. For example,

    \[7\,×\,{10}^{-4}\text{kg}\]

could be written as 70 cg or 700 mg. (a)

    \[3.8\,×\,{10}^{-5}\text{kg;}\]

(b)

    \[2.3\,×\,{10}^{17}\text{kg;}\]

(c)

    \[2.4\,×\,{10}^{-11}\text{kg;}\]

(d)

    \[8\,×\,{10}^{15}\text{kg;}\]

(e)

    \[4.2\,×\,{10}^{-3}\text{kg.}\]

[reveal-answer q=”fs-id1168326821936″]Show Solution[/reveal-answer]

[hidden-answer a=”fs-id1168326821936″]

a. 3.8 cg or 38 mg; b. 230 Eg; c. 24 ng; d. 8 Eg e. 4.2 g

[/hidden-answer]

Glossary

base quantity
physical quantity chosen by convention and practical considerations such that all other physical quantities can be expressed as algebraic combinations of them
base unit
standard for expressing the measurement of a base quantity within a particular system of units; defined by a particular procedure used to measure the corresponding base quantity
derived quantity
physical quantity defined using algebraic combinations of base quantities
derived units
units that can be calculated using algebraic combinations of the fundamental units
English units
system of measurement used in the United States; includes units of measure such as feet, gallons, and pounds
kilogram
SI unit for mass, abbreviated kg
meter
SI unit for length, abbreviated m
metric system
system in which values can be calculated in factors of 10
physical quantity
characteristic or property of an object that can be measured or calculated from other measurements
second
the SI unit for time, abbreviated s
SI units
the international system of units that scientists in most countries have agreed to use; includes units such as meters, liters, and grams
units
standards used for expressing and comparing measurements

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