Physical Quantities and Units
Figure 1 The distance from Earth to the Moon may seem immense, but it is just a tiny fraction of the distances from Earth to other celestial bodies. (credit: NASA).
By the end of this section, you will be able to:
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 sub-nuclear 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 does qualitative description alone. To comprehend these vast ranges, we must also have accepted units in which to express them. And 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 four fundamental physical quantities: length, mass, time, and electric current.
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 define distance and time by specifying methods for measuring them, whereas we define average speed by stating that it is calculated as 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. (See Figure 2.)
Figure 2 Distances given in unknown units are maddeningly useless.
There are two major systems of units used in the world: SI units (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. Virtually every other country in the world now uses SI units as the standard; the metric system is also the standard system agreed upon by scientists and mathematicians. The acronym “SI” is derived from the French Système International.
Choose the 4 fundamental magnitudes of the metric decimal system.
SI Units: Fundamental and Derived Units
Table 1 gives the fundamental SI units that are used throughout this textbook. This text uses non-SI units in a few applications where they are in very common use, such as the measurement of blood pressure in millimeters of mercury (mm Hg). Whenever non-SI units are discussed, they will be tied to SI units through conversions.
It is an intriguing fact that some physical quantities are more fundamental than others and that the most fundamental physical quantities can be defined only in terms of the procedure used to measure them. The units in which they are measured are thus called fundamental units. In this textbook, the fundamental physical quantities are taken to be length, mass, time, and electric charge. (Note that electric current will not be introduced until much later in this text.) All other physical quantities, such as force and electric current, can be expressed as algebraic combinations of length, mass, time, and current (for example, speed is length divided by time); these units are called derived units.
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 non-varying, or constant, physical phenomenon (because the solar day is getting longer due to very gradual slowing of the 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. (See Figure 1.) Accuracy in the fundamental units is essential, because all measurements are ultimately expressed in terms of fundamental units and can be no more accurate than are the fundamental units themselves.
Figure 1 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 is looking down from the top of an atomic fountain nearly 30 feet tall! (credit: Steve Jurvetson/Flickr)
The SI unit for length is the meter (abbreviated m); its definition has also changed over time to become more accurate and 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 present definition (partly for greater accuracy) as the distance light travels in a vacuum in 1/299,792,458 of a second. (See Figure 2.) This change defines the speed of light to be exactly 299,792,458 meters per second. The length of the meter will change if the speed of light is someday measured with greater accuracy.
Figure 2 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. Electric current and its accompanying unit, the ampere, will be introduced in Introduction to Electric Current, Resistance, and Ohm's Law when electricity and magnetism are covered. The initial modules in this textbook are concerned with mechanics, fluids, heat, and waves. In these subjects all pertinent physical quantities can be expressed in terms of the fundamental units of length, mass, and time.
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 United States' National Institute of Standards and Technology, or NIST, located in Gaithersburg, Maryland outside of Washington D.C., and at other locations around the world. The determination of all other masses can be ultimately traced to a comparison with the standard mass.
SI units are part of the metric system. The metric system is convenient for scientific and engineering calculations because the units are categorized by factors of 10. Table 2 gives metric prefixes and symbols used to denote various factors of 10.
Metric systems have the advantage that conversions of units involve only powers of 10. There are 100 centimeters in a meter, 1000 meters in a kilometer, and so on. In nonmetric systems, such as the system of U.S. customary units, the relationships are not as simple—there are 12 inches in a foot, 5280 feet in a mile, and so on. Another advantage of the metric system is that the same unit can be used over extremely large ranges of values simply by using an appropriate metric prefix. For example, distances in meters are suitable in construction, while distances in kilometers are appropriate for air travel, and the tiny measure of nanometers are convenient in optical design. With the metric system there is no need to invent new units for particular applications.
The term order of magnitude refers to the scale of a value expressed in the metric system. Each power of \(10\) in the metric system represents a different order of magnitude. For example, \(10^1\),\(10^2\),\(10^3\), and so forth are all different orders of magnitude. All quantities that can be expressed as a product of a specific power of \(10\) are said to be of the same order of magnitude. For example, the number \(800\) can be written as \(8 \times 10^2\), and the number \(450\) can be written as \(4.5\times 10^2\). Thus, the numbers \(800\) and \(450\) are of the same order of magnitude: \(10^2\). Order of magnitude can be thought of as a ballpark estimate for the scale of a value. The diameter of an atom is on the order of \(10^{-9}\) m, while the diameter of the Sun is on the order of \(10^9m\).
The fundamental units described in this chapter are those that produce the greatest accuracy and precision in measurement. There is a sense among physicists that, because there is an underlying microscopic substructure to matter, it would be most satisfying to base our standards of measurement on microscopic objects and fundamental physical phenomena such as the speed of light. A microscopic standard has been accomplished for the standard of time, which is based on the oscillations of the cesium atom.
The standard for length was once based on the wavelength of light (a small-scale length) emitted by a certain type of atom, but it has been supplanted by the more precise measurement of the speed of light. If it becomes possible to measure the mass of atoms or a particular arrangement of atoms such as a silicon sphere to greater precision than the kilogram standard, it may become possible to base mass measurements on the small scale. There are also possibilities that electrical phenomena on the small scale may someday allow us to base a unit of charge on the charge of electrons and protons, but at present current and charge are related to large-scale currents and forces between wires.
The vastness of the universe and the breadth over which physics applies are illustrated by the wide range of examples of known lengths, masses, and times in Table 1.3. Examination of this table will give you some feeling for the range of possible topics and numerical values. (See Figure 3 and Figure 4.)
Figure 3 Tiny phytoplankton swims among crystals of ice in the Antarctic Sea. They range from a few micrometers to as much as 2 millimeters in length. (credit: Prof. Gordon T. Taylor, Stony Brook University; NOAA Corps Collecti10−18ons).
Figure 4 Galaxies collide 2.4 billion light years away from Earth. The tremendous range of observable phenomena in nature challenges the imagination. (credit: NASA/CXC/UVic./A. Mahdavi et al. Optical/lensing: CFHT/UVic./H. Hoekstra et al.)
Which of the following is defined as the distance traveled by light during a precisely specified time interval?
Which is defined in terms of the oscillations of light emitted by an atomic (cesium-133) source?
How are base units combined to become derived units? Choose 2 correct options.
In the units of length, the prefix nano is equivalent to:
\(10^{-2}\) meters is equivalent to:
The names of the additional units in measurements are derived by adding a _________ at the beginning of a fundamental unit.
It is often necessary to convert from one type of unit to another. For example, if you are reading a European cookbook, some quantities may be expressed in units of liters and you need to convert them to cups. Or, perhaps you are reading walking directions from one location to another and you are interested in how many miles you will be walking. In this case, you will need to convert units of feet to miles.
Let us consider a simple example of how to convert units. Let us say that we want to convert 80 meters (m) to kilometers (km).
The first thing to do is to list the units that you have and the units that you want to convert to. In this case, we have units in meters and we want to convert to kilometers.
Next, we need to determine a conversion factor relating meters to kilometers. A conversion factor is a ratio expressing how many of one unit are equal to another unit. For example, there are 12 inches in 1 foot, 100 centimeters in 1 meter, 60 seconds in 1 minute, and so on. In this case, we know that there are 1,000 meters in 1 kilometer.
Now we can set up our unit conversion. We will write the units that we have and then multiply them by the conversion factor so that the units cancel out, as shown:
Note that the unwanted m unit cancels, leaving only the desired km unit. You can use this method to convert between any types of unit.
Mass of an electron
\((9.11\times 10^{-31}kg)\)
Mass of a hydrogen atom
\((1.67\times 10^{-27}kg)\)
One day
\((8.64\times 10^{4} s)\)
One year (y)
\((3.16\times 10^{7} s)\)
Mass of the Moon
\((7.35\times 10^{22} kg)\)
Mass of the Earth
\((5.97\times 10^{24} kg)\)
Mass of the Sun
\((1.99\times 10^{30} kg)\)
Suppose that you drive the 10.0 km from your university to home in 20.0 min. Calculate your average speed (a) in kilometers per hour (km/h) and (b) in meters per second (m/s). (Note: Average speed is distance traveled divided by time of travel.)
Strategy
First we calculate the average speed using the given units. Then we can get the average speed into the desired units by picking the correct conversion factor and multiplying by it. The correct conversion factor is the one that cancels the unwanted unit and leaves the desired unit in its place.
Solution for (a)
(1) Calculate average speed. Average speed is distance traveled divided by time of travel. (Take this definition as a given for now—average speed and other motion concepts will be covered in a later module.) In equation form,
(2) Substitute the given values for distance and time.
(3) Convert km/min to km/h: multiply by the conversion factor that will cancel minutes and leave hours. That conversion factor is \(60\;min/hr\). Thus,
Discussion for (a)
To check your answer, consider the following:
(1) Be sure that you have properly cancelled the units in the unit conversion. If you have written the unit conversion factor upside down, the units will not cancel properly in the equation. If you accidentally get the ratio upside down, then the units will not cancel; rather, they will give you the wrong units as follows:
which are obviously not the desired units of km/h.
(2) Check that the units of the final answer are the desired units. The problem asked us to solve for average speed in units of km/h and we have indeed obtained these units.
(3) Check the significant figures. Because each of the values given in the problem has three significant figures, the answer should also have three significant figures. The answer 30.0 km/hr does indeed have three significant figures, so this is appropriate. Note that the significant figures in the conversion factor are not relevant because an hour is defined to be 60 minutes, so the precision of the conversion factor is perfect.
(4) Next, check whether the answer is reasonable. Let us consider some information from the problem—if you travel 10 km in a third of an hour (20 min), you would travel three times that far in an hour. The answer does seem reasonable.
Solution for (b)
There are several ways to convert the average speed into meters per second.
(1) Start with the answer to (a) and convert km/h to m/s. Two conversion factors are needed—one to convert hours to seconds, and another to convert kilometers to meters.
(2) Multiplying by these yields
Discussion for (b)
If we had started with 0.500 km/min, we would have needed different conversion factors, but the answer would have been the same: 8.33 m/s.
You may have noted that the answers in the worked example just covered were given to three digits. Why? When do you need to be concerned about the number of digits in something you calculate? Why not write down all the digits your calculator produces? The module Accuracy, Precision, and Significant Figures will help you answer these questions.
Which of the following units belongs to the International System? Choose 3 correct options.
The speed of sound is measured to be \(v = 340\;{m \over s}\) on a certain day. What is this in \(km \over h\)?
Tectonic plates are large segments of the Earth's crust that move slowly. Suppose that one such plate has an average speed of \(v = 9\;{cm \over year}\).
What distance, in meters, does it move in 4 s at this speed?
Assume that 1 year has 3.15E7 seconds.
What is its speed in kilometers per million years?
Convert 396 \(\;km \over h\) to meters per second.
Tectonic plates are large segments of the Earth's crust that move slowly. Suppose that one such plate has an average speed of \(v = 8\;{cm \over year}\). What distance, in meters, does it move in 4 s at this speed?
What is the area, in cm2, of a circle 86.42 mm in diameter?
The Moon's diameter is about 3,474 km. What is its diameter in meters?
To calculate the average speed of a planet in its orbit, you may use the formula \(v = {d \over t} = {2 \pi r \over t}\), where \(r\) is the average distance between the planet and the Sun during 1 year.
If \(r = 10^{7}\;km\), what is the average speed in kilometers per second?
What is the average speed in meter per second?
A soccer player has an average record of 1375 yards over the last two years. What is the equivalent in meters?
Remember that 1 yard is equivalent to 0.914 meters.
What is the area, in cm2, of a circle 46.03 mm in diameter?
The speed of sound is measured to be \(v = 341\;{m \over s}\) on a certain day. What is this in \(km \over h\)?
Convert 33 \(km \over h\) to \(m\over s\).
How many micrometers are there in 5 millimeters?
Soccer fields vary in size. A large soccer field is 100 m long and 81 m wide. What is its length in feet?
Consider 1 ft = 0.3048 m
What is its length in inches?
Consider 1 ft = 12 in
What is its width in feet?
What is its width in inches?
Mount Everest, at 29028 feet, is the tallest mountain on Earth. What is its height in kilometers? (Assume that 1 kilometer equals 3,281 feet.)
What is the area, in cm2, of a circle 93.02 mm in diameter?
What is the height in meters of a person who is 6 ft and 6 in tall? (Assume that 1 meter equals 39.37 in and 1 ft equals 12 in)
If \(r = 10^{8}\;km\), what is the average speed in kilometers per second?
What is the average speed in meters per second?
A certain sport is played on a 115-yd-long field, excluding the end zones. How long is the field in meters? (Assume that 1 meter equals 1.09361 yds.)
A soccer player has an average record of 1739 yards over the last two years. What is the equivalent in meters?
Certain sport is played on a 110-yd-long field, excluding the end zones. How long is the field in meters? (Assume that 1 meter equals 1.09361 yds.)
How many atmospheres are there in 4480 Pa?
Remember that \(101,325 \space Pa = 1 \space atm\)
Tectonic plates are large segments of the Earth's crust that move slowly. Suppose that one such plate has an average speed of \(v = 4\;{cm \over year}\).
What distance does it move in 4 s at this speed? Express your answer in meters.
Which is equivalent to the units of the force, Newton?
The speed of sound is measured to be \(v = 345\;{m \over s}\) on a certain day. What is this in \(km \over h\)?
What is the height in meters of a person who is 5 ft and 5 in tall? (Assume that 1 meter equals 39.37 in and 1 ft equals 12 in)
If \(r = 10^{9}\;km\), what is the average speed in kilometers per second?
What is the height in meters of a person who is 6 ft and 2 in tall? (Assume that 1 meter equals 39.37 in and 1 ft equals 12 in)
Certain sport is played on a 112-yd-long field, excluding the end zones. How long is the field in meters? (Assume that 1 meter equals 1.09361 yds.)
This is a large modal.
