13. Physical Quantities and UnitsLearning Objectives
The range of objects and phenomena studied in physics is immense. From the incredibly short lifetime of a nucleus to the age of the 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 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 Measurements of physical quantities are expressed in terms of There are two major systems of units used in the world: SI Units: Fundamental and Derived UnitsTable 1 gives the fundamental SI units that are used throughout this textbook. This text uses nonSI 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 nonSI units are discussed, they will be tied to SI units through conversions. Table 1: Fundamental SI Units
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 Units of Time, Length, and Mass: The Second, Meter, and KilogramThe SecondThe SI unit for time, the The MeterThe SI unit for length is the The KilogramThe SI unit for mass is the 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. Metric PrefixesSI units are part of the 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 The Quest for Microscopic Standards for Basic Units: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 smallscale 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 largescale currents and forces between wires. Table 2: Metric Prefixes for Powers of 10 and their Symbols
Known Ranges of Length, Mass, and TimeThe 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 3. Examination of this table will give you some feeling for the range of possible topics and numerical values. (See Figure 5 and Figure 6.) Unit Conversion and Dimensional AnalysisIt 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 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: $$80\overline{)\text{m}}\times \frac{\text{1 km}}{1000\overline{)\text{m}}}=0\text{.080 km.}$$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. Click here for a more complete list of conversion factors. Table 3: Approximate Values of Length, Mass, and Time
Example 1: Unit Conversions: A Short Drive HomeSuppose 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, $$\text{average speed =}\frac{\text{distance}}{\text{time}}\text{.}$$(2) Substitute the given values for distance and time. $$\text{average speed =}\frac{\text{10}\text{.}0\text{km}}{\text{20}\text{.}0\text{min}}=0\text{.}\text{500}\frac{\text{km}}{\text{min}}\text{.}$$(3) Convert km/min to km/h: multiply by the conversion factor that will cancel minutes and leave hours. That conversion factor is $\text{60 min/hr}$. Thus, $$\text{average speed =}0\text{.}\text{500}\frac{\text{km}}{\text{min}}\times \frac{\text{60}\text{min}}{1\text{h}}=\text{30}\text{.}0\frac{\text{km}}{\text{h}}\text{.}$$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: $$\frac{\text{km}}{\text{min}}\times \frac{1\text{hr}}{\text{60}\text{min}}=\frac{1}{\text{60}}\frac{\text{km}\cdot \text{hr}}{{\text{min}}^{2}}\text{,}$$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 $$\text{Average speed}=\text{30}\text{.}0\frac{\text{km}}{\text{h}}\times \frac{1\phantom{\rule{0ex}{0ex}}\text{h}}{\text{3,600 s}}\times \frac{\mathrm{1,}\text{000}\phantom{\rule{0ex}{0ex}}\text{m}}{\text{1 km}}\text{,}$$ $$\text{Average speed}=8\text{.}\text{33}\frac{\text{m}}{\text{s}}\text{.}$$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. Nonstandard Units:While there are numerous types of units that we are all familiar with, there are others that are much more obscure. For example, a firkin is a unit of volume that was once used to measure beer. One firkin equals about 34 liters. To learn more about nonstandard units, use a dictionary or encyclopedia to research different “weights and measures.” Take note of any unusual units, such as a barleycorn, that are not listed in the text. Think about how the unit is defined and state its relationship to SI units. Check Your UnderstandingSome hummingbirds beat their wings more than 50 times per second. A scientist is measuring the time it takes for a hummingbird to beat its wings once. Which fundamental unit should the scientist use to describe the measurement? Which factor of 10 is the scientist likely to use to describe the motion precisely? Identify the metric prefix that corresponds to this factor of 10. Show/Hide Solution SolutionThe scientist will measure the time between each movement using the fundamental unit of seconds. Because the wings beat so fast, the scientist will probably need to measure in milliseconds, or ${\text{10}}^{3}$ seconds. (50 beats per second corresponds to 20 milliseconds per beat.) Check Your UnderstandingOne cubic centimeter is equal to one milliliter. What does this tell you about the different units in the SI metric system? Show/Hide Solution SolutionThe fundamental unit of length (meter) is probably used to create the derived unit of volume (liter). The measure of a milliliter is dependent on the measure of a centimeter. Summary
Conceptual QuestionsExercise 1Identify some advantages of metric units. Problems & ExercisesExercise 1The speed limit on some interstate highways is roughly 100 km/h. (a) What is this in meters per second? (b) How many miles per hour is this? Show/Hide Solution Solution
Exercise 2A car is traveling at a speed of $\text{33 m/s}$. (a) What is its speed in kilometers per hour? (b) Is it exceeding the $\text{90 km/h}$ speed limit? Exercise 3Show that $1\text{.}\text{0 m/s}=3\text{.}\text{6 km/h}$. Hint: Show the explicit steps involved in converting $1\text{.}\text{0 m/s}=3\text{.}\text{6 km/h.}$ Show/Hide Solution Solution$\frac{\text{1.0 m}}{\mathrm{s}}=\frac{1\text{.}\text{0 m}}{\mathrm{s}}\times \frac{\text{3600 s}}{\text{1 hr}}\times \frac{\mathrm{1\; km}}{\text{1000 m}}$ $=3\text{.}\text{6 km/h}$. Exercise 4American football is played on a 100ydlong field, excluding the end zones. How long is the field in meters? (Assume that 1 meter equals 3.281 feet.) Exercise 5Soccer fields vary in size. A large soccer field is 115 m long and 85 m wide. What are its dimensions in feet and inches? (Assume that 1 meter equals 3.281 feet.) Show/Hide Solution Solutionlength: $\text{377 ft}$; $4\text{.}\text{53}\times {\text{10}}^{3}\text{in}\text{.}$ width: $\text{280ft}$; $3\text{.}3\times {\text{10}}^{3}\text{in}$. Exercise 6What is the height in meters of a person who is 6 ft 1.0 in. tall? (Assume that 1 meter equals 39.37 in.) Exercise 7Mount Everest, at 29,028 feet, is the tallest mountain on the Earth. What is its height in kilometers? (Assume that 1 kilometer equals 3,281 feet.) Show/Hide Solution Solution$8\text{.}\text{847 km}$ Exercise 8The speed of sound is measured to be $\text{342 m/s}$ on a certain day. What is this in km/h? Exercise 9Tectonic plates are large segments of the Earth’s crust that move slowly. Suppose that one such plate has an average speed of 4.0 cm/year. (a) What distance does it move in 1 s at this speed? (b) What is its speed in kilometers per million years? Show/Hide Solution Solution(a) $1\text{.}3\times {\text{10}}^{9}\text{m}$ (b) $\text{40 km/My}$ Exercise 10(a) Refer to Table 3 to determine the average distance between the Earth and the Sun. Then calculate the average speed of the Earth in its orbit in kilometers per second. (b) What is this in meters per second?
