# Unit Iv :semiconductorsapplied Physics

### Learning Objectives

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

• Perform unit conversions both in the SI and English units.
• Explain the most common prefixes in the SI units and be able to write them in scientific notation.

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)

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 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.

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. Oct-19 unit-4 (qp) OCT-19 UNIT-5 (QP) OCT-19 UNIT-6 (QP) OCT-19 UNIT-4 (ms) OCT-19 UNIT-5 (MS). Jan-15 UNIT-4 (MS) Jan-15 UNIT-5 (MS). Unit 1: Introduction to Physics Our first step in this course is to gain a basic understanding of the language and analytical techniques that are specific to physics. This unit will include a brief outline of physics and the scientific method, measurement units and scientific notation, and the concepts of significant figures, order-of-magnitude.

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.

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.

Table 1. Fundamental SI Units
LengthMassTimeElectric Current
meter (m)kilogram (kg)second (s)ampere (A)

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 current. (Note that electric current will not be introduced until much later in this text.) All other physical quantities, such as force and electric charge, 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.

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

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 3.) 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 3. 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 4.) 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.

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.

Figure 4. 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.

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 non-metric 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, 101, 102, 103, 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 × 102, and the number 450 can be written as 4.5 × 102. Thus, the numbers 800 and 450 are of the same order of magnitude: 102. 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 10-9m while the diameter of the Sun is on the order of 109m.

### The Quest for Microscopic Standards for Basic Units

Download perfect uninstaller torrent. 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.

Table 2. Metric Prefixes for Powers of 10 and their Symbols
PrefixSymbolValueExample (some are approximate)
exaE[latex]{text{10}}^{text{18}}[/latex]exameterEm[latex]{text{10}}^{text{18}}text{ m}[/latex]distance light travels in a century
petaP[latex]{text{10}}^{text{15}}[/latex]petasecondPs[latex]{text{10}}^{text{15}}text{ s}[/latex]30 million years
teraT[latex]{text{10}}^{text{12}}[/latex]terawattTW[latex]{text{10}}^{text{12}}text{ W}[/latex]powerful laser output
gigaG[latex]{text{10}}^{9}[/latex]gigahertzGHz[latex]{text{10}}^{9}text{ Hz}[/latex]a microwave frequency
hectoh[latex]{text{10}}^{2}[/latex]hectoliterhL[latex]{text{10}}^{2}text{ L}[/latex]26 gallons
[latex]{text{10}}^{0}[/latex] (=1)
decid[latex]{text{10}}^{-1}[/latex]deciliterdL[latex]{text{10}}^{-1}text{ L}[/latex]less than half a soda
centic[latex]{text{10}}^{-2}[/latex]centimetercm[latex]{text{10}}^{-2}text{ m}[/latex]fingertip thickness
millim[latex]{text{10}}^{-3}[/latex]millimetermm[latex]{text{10}}^{-3}text{ m}[/latex]flea at its shoulders
microµ[latex]{text{10}}^{-6}[/latex]micrometerµm[latex]{text{10}}^{-6}text{ m}[/latex]detail in microscope
nanon[latex]{text{10}}^{-9}[/latex]nanogramng[latex]{text{10}}^{-9}text{ g}[/latex]small speck of dust
femtof[latex]{text{10}}^{-text{15}}[/latex]femtometerfm[latex]{text{10}}^{-text{15}}text{ m}[/latex]size of a proton
attoa[latex]{text{10}}^{-text{18}}[/latex]attosecondas[latex]{text{10}}^{-text{18}}text{ s}[/latex]time light crosses an atom

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 5 and Figure 6.)

Figure 5. 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 Collections)

Figure 6. 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.)

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:

[latex]80overline{)text{m}}times frac{text{1 km}}{1000overline{)text{m}}}=0text{.080 km.}[/latex]

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.

Table 3. Approximate Values of Length, Mass, and Time
Lengths in metersMasses in kilograms (more precise values in parentheses)Times in seconds (more precise values in parentheses)
[latex]{text{10}}^{-text{18}}[/latex]Present experimental limit to smallest observable detail[latex]{text{10}}^{-text{30}}[/latex]Mass of an electron [latex]left(9text{.}text{11}times {text{10}}^{-text{31}}text{ kg}right)[/latex][latex]{text{10}}^{-text{23}}[/latex]Time for light to cross a proton
[latex]{text{10}}^{-text{15}}[/latex]Diameter of a proton[latex]{text{10}}^{-text{27}}[/latex]Mass of a hydrogen atom [latex]left(1text{.}text{67}times {text{10}}^{-text{27}}text{ kg}right)[/latex][latex]{text{10}}^{-text{22}}[/latex]Mean life of an extremely unstable nucleus
[latex]{text{10}}^{-text{14}}[/latex]Diameter of a uranium nucleus[latex]{text{10}}^{-text{15}}[/latex]Mass of a bacterium[latex]{text{10}}^{-text{15}}[/latex]Time for one oscillation of visible light
[latex]{text{10}}^{-text{10}}[/latex]Diameter of a hydrogen atom[latex]{text{10}}^{-5}[/latex]Mass of a mosquito[latex]{text{10}}^{-text{13}}[/latex]Time for one vibration of an atom in a solid
[latex]{text{10}}^{-8}[/latex]Thickness of membranes in cells of living organisms[latex]{text{10}}^{-2}[/latex]Mass of a hummingbird[latex]{text{10}}^{-8}[/latex]Time for one oscillation of an FM radio wave
[latex]{text{10}}^{-6}[/latex]Wavelength of visible light[latex]text{1}[/latex]Mass of a liter of water (about a quart)[latex]{text{10}}^{-3}[/latex]Duration of a nerve impulse
[latex]{text{10}}^{-3}[/latex]Size of a grain of sand[latex]{text{10}}^{2}[/latex]Mass of a person[latex]text{1}[/latex]Time for one heartbeat
[latex]text{1}[/latex]Height of a 4-year-old child[latex]{text{10}}^{3}[/latex]Mass of a car[latex]{text{10}}^{5}[/latex]One day [latex]left(8text{.}text{64}times {text{10}}^{4}text{s}right)[/latex]
[latex]{text{10}}^{2}[/latex]Length of a football field[latex]{text{10}}^{8}[/latex]Mass of a large ship[latex]{text{10}}^{7}[/latex]One year (y) [latex]left(3text{.}text{16}times {text{10}}^{7}text{s}right)[/latex]
[latex]{text{10}}^{4}[/latex]Greatest ocean depth[latex]{text{10}}^{text{12}}[/latex]Mass of a large iceberg[latex]{text{10}}^{9}[/latex]About half the life expectancy of a human
[latex]{text{10}}^{7}[/latex]Diameter of the Earth[latex]{text{10}}^{text{15}}[/latex]Mass of the nucleus of a comet[latex]{text{10}}^{text{11}}[/latex]Recorded history
[latex]{text{10}}^{text{11}}[/latex]Distance from the Earth to the Sun[latex]{text{10}}^{text{23}}[/latex]Mass of the Moon [latex]left(7text{.}text{35}times {text{10}}^{text{22}}text{ kg}right)[/latex][latex]{text{10}}^{text{17}}[/latex]Age of the Earth
[latex]{text{10}}^{text{16}}[/latex]Distance traveled by light in 1 year (a light year)[latex]{text{10}}^{text{25}}[/latex]Mass of the Earth [latex]left(5text{.}text{97}times {text{10}}^{text{24}}text{ kg}right)[/latex][latex]{text{10}}^{text{18}}[/latex]Age of the universe
[latex]{text{10}}^{text{21}}[/latex]Diameter of the Milky Way galaxy[latex]{text{10}}^{text{30}}[/latex]Mass of the Sun [latex]left(1text{.}text{99}times {text{10}}^{text{30}}text{ kg}right)[/latex]
[latex]{text{10}}^{text{22}}[/latex]Distance from the Earth to the nearest large galaxy (Andromeda)[latex]{text{10}}^{text{42}}[/latex]Mass of the Milky Way galaxy (current upper limit)
[latex]{text{10}}^{text{26}}[/latex]Distance from the Earth to the edges of the known universe[latex]{text{10}}^{text{53}}[/latex]Mass of the known universe (current upper limit)

### Example 1. Unit Conversions – A Short Drive Home

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,

[latex]text{average speed =}frac{text{distance}}{text{time}}[/latex].

(2) Substitute the given values for distance and time.

[latex]text{average speed =}frac{text{10}text{.}0text{ km}}{text{20}text{.}0text{ min}}=0text{.}text{500}frac{text{ km}}{text{ min}}[/latex].

(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,

[latex]text{average speed =}0text{.}text{500}frac{text{ km}}{text{ min}}times frac{text{60}text{ min}}{1text{ h}}=text{30}text{.}0frac{text{ km}}{text{ h}}[/latex].

#### Discussion for (a)

(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:

[latex]frac{text{ km}}{text{min}}times frac{1text{ hr}}{text{60}text{ min}}=frac{1}{text{60}}frac{text{ km}cdot text{hr}}{{text{ min}}^{2}}[/latex],

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

[latex]text{Average speed}=text{30}text{.}0frac{text{km}}{text{h}}times frac{1text{h}}{text{3,600 s}}times frac{1,text{000}text{m}}{text{1 km}}[/latex],
[latex]text{Average speed}=8text{.}text{33}frac{text{m}}{text{s}}[/latex].

#### 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.

1. Some 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.

2. One cubic centimeter is equal to one milliliter. What does this tell you about the different units in the SI metric system?

1. The 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 10-3 seconds. (50 beats per second corresponds to 20 milliseconds per beat.)

2. The 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.

• Physical quantities are a characteristic or property of an object that can be measured or calculated from other measurements.
• Units are standards for expressing and comparing the measurement of physical quantities. All units can be expressed as combinations of four fundamental units.
• The four fundamental units we will use in this text are the meter (for length), the kilogram (for mass), the second (for time), and the ampere (for electric current). These units are part of the metric system, which uses powers of 10 to relate quantities over the vast ranges encountered in nature.
• The four fundamental units are abbreviated as follows: meter, m; kilogram, kg; second, s; and ampere, A. The metric system also uses a standard set of prefixes to denote each order of magnitude greater than or lesser than the fundamental unit itself.
• Unit conversions involve changing a value expressed in one type of unit to another type of unit. This is done by using conversion factors, which are ratios relating equal quantities of different units.

### Problems & Exercises

1. The 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?

2. A car is traveling at a speed of 33 m/s. (a) What is its speed in kilometers per hour? (b) Is it exceeding the 90 km/h speed limit?

3. Show that 1.0 m/s = 3.6 km/h. Hint: Show the explicit steps involved in converting 1.0 m/s = 3.6 km/h.

4. American football is played on a 100-yd-long field, excluding the end zones. How long is the field in meters? (Assume that 1 meter equals 3.281 feet.)

5. Soccer 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.)

6. What is the height in meters of a person who is 6 ft 1.0 in. tall? (Assume that 1 meter equals 39.37 in.)

7. Mount 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.)

8. The speed of sound is measured to be 342 m/s on a certain day. What is this in km/h?

9. Tectonic 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?

10. (a) Refer to Table 2: Metric Prefixes for Powers of 10 and their Symbols 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?

## Glossary

physical quantity:
a characteristic or property of an object that can be measured or calculated from other measurements
units:
a standard used for expressing and comparing measurements
SI units:
the international system of units that scientists in most countries have agreed to use; includes units such as meters, liters, and grams
English units:
system of measurement used in the United States; includes units of measurement such as feet, gallons, and pounds
fundamental units:
units that can only be expressed relative to the procedure used to measure them
derived units:
units that can be calculated using algebraic combinations of the fundamental units
second:
the SI unit for time, abbreviated (s)
meter:
the SI unit for length, abbreviated (m)
kilogram:
the SI unit for mass, abbreviated (kg)
metric system:
a system in which values can be calculated in factors of 10
order of magnitude:
refers to the size of a quantity as it relates to a power of 10
conversion factor:
a ratio expressing how many of one unit are equal to another unit

### Selected Solutions to Problems & Exercises

1. (a) 27.8 m/s (b) 62.1 mph

3. [latex]frac{text{1.0 m}}{s}=frac{1text{.}text{0 m}}{s}times frac{text{3600 s}}{text{1 hr}}times frac{1 km}{text{1000 m}}[/latex]

5. length: 377 ft; 4.53 × 103 in. width: 280 ft; 3.3 × 103 in.

7. 8.847 kn

9. (a) 1.3 × 109 m (b) 40 km/My

1. See Appendix A for a discussion of powers of 10. ↵

## Discussion

### definitions

Informally, viscosity is the quantity that describes a fluid's resistance to flow. Fluids resist the relative motion of immersed objects through them as well as to the motion of layers with differing velocities within them.

Formally, viscosity (represented by the symbol η 'eta') is the ratio of the shearing stress (F/A) to the velocity gradient (vx/∆z or dvx/dz) in a fluid.

 η = F/A ∆vx/∆z

or

 η = F/A dvx/dz

The more usual form of this relationship, called Newton's equation, states that the resulting shear of a fluid is directly proportional to the force applied and inversely proportional to its viscosity. The similarity to Newton's second law of motion (F = ma) should be apparent.

 F = η ∆vx A ∆z
 F = m ∆v ∆t

Or if you prefer calculus symbols (and who doesn't)…

 F = η dvx A dz
 F = m dv dt

The SI unit of viscosity is the pascal second [Pa s], which has no special name. Despite its self-proclaimed title as an international system, the International System of Units has had little international impact on viscosity. The pascal second is rarely used in scientific and technical writing today. The most common unit of viscosity is the dyne second per square centimeter [dyne s/cm2], which is given the name poise [P] after the French physiologist Jean Poiseuille (1799–1869). Ten poise equal one pascal second [Pa s] making the centipoise [cP] and millipascal second [mPa s] identical.

 1 Pa s = 10 P 1000 mPa s = 10 P 1 mPa s = 0.01 P 1 mPa s = 1 cP

There are actually two quantities that are called viscosity. The quantity defined above is sometimes called dynamic viscosity, absolute viscosity, or simple viscosity to distinguish it from the other quantity, but is usually just called viscosity. The other quantity called kinematic viscosity (represented by the Greek letter ν 'nu') is the ratio of the viscosity of a fluid to its density.

 ν = η ρ

Kinematic viscosity is a measure of the resistive flow of a fluid under the influence of gravity. It is frequently measured using a device called a capillary viscometer — basically a graduated can with a narrow tube at the bottom. When two fluids of equal volume are placed in identical capillary viscometers and allowed to flow under the influence of gravity, the more viscous fluid takes longer than the less viscous fluid to flow through the tube. Capillary viscometers will be discussed in more detail later in this section.

The SI unit of kinematic viscosity is the square meter per second [m2/s], which has no special name. This unit is so large that it is rarely used. A more common unit of kinematic viscosity is the square centimeter per second [cm2/s], which is given the name stokes [St] after the Irish mathematician and physicist George Stokes (1819–1903). One square meter per second is equal to ten thousand stokes.

 1 cm2/s = 1 St 1 m2/s = 10,000 cm2/s 1 m2/s = 10,000 St

Even this unit is a bit too large, so the most common unit is probably the square millimeter per second [mm2/s] or the centistokes [cSt]. One square meter per second is equal to one million centistokes.

 1 mm2/s = 1 cSt 1 m2/s = 1,000,000 mm2/s 1 m2/s = 1,000,000 cSt

The stokes is a rare example of a word in the English language where the singular and plural forms are identical. Fish is the most immediate example of a aword thatbehaves like this. 1 fish, 2 fish, red fish, blue fish; 1 stokes, 2 stokes, some stokes, few stokes.

### factors affecting viscosity

Viscosity is first and foremost a function of material. The viscosity of water at 20 °C is 1.0020 millipascal seconds (which is conveniently close to one by coincidence alone). Most ordinary liquids have viscosities on the order of 1 to 1,000 mPa s, while gases have viscosities on the order of 1 to 10 μPa s. Pastes, gels, emulsions, and other complex liquids are harder to summarize. Some fats like butter or margarine are so viscous that they seem more like soft solids than like flowing liquids. Molten glass is extremely viscous and approaches infinite viscosity as it solidifies. Since the process is not as well defined as true freezing, some believe (incorrectly) that glass may still flow even after it has completely cooled, but this is not the case. At ordinary temperatures, glasses are as solid as true solids.

From everyday experience, it should be common knowledge that viscosity varies with temperature. Honey and syrups can be made to flow more readily when heated. Engine oil and hydraulic fluids thicken appreciably on cold days and significantly affect the performance of cars and other machinery during the winter months. In general, the viscosity of a simple liquid decreases with increasing temperature. As temperature increases, the average speed of the molecules in a liquid increases and the amount of time they spend 'in contact' with their nearest neighbors decreases. Thus, as temperature increases, the average intermolecular forces decrease. The actual manner in which the two quantities vary is nonlinear and changes abruptly when the liquid changes phase.

Viscosity is normally independent of pressure, but liquids under extreme pressure often experience an increase in viscosity. Since liquids are normally incompressible, an increase in pressure doesn't really bring the molecules significantly closer together. Simple models of molecular interactions won't work to explain this behavior and, to my knowledge, there is no generally accepted more complex model that does. The liquid phase is probably the least well understood of all the phases of matter.

While liquids get runnier as they get hotter, gases get thicker. (If one can imagine a 'thick' gas.) The viscosity of gases increases as temperature increases and is approximately proportional to the square root of temperature. This is due to the increase in the frequency of intermolecular collisions at higher temperatures. Since most of the time the molecules in a gas are flying freely through the void, anything that increases the number of times one molecule is in contact with another will decrease the ability of the molecules as a whole to engage in the coordinated movement. The more these molecules collide with one another, the more disorganized their motion becomes. Physical models, advanced beyond the scope of this book, have been around for nearly a century that adequately explain the temperature dependence of viscosity in gases. Newer models do a better job than the older models. They also agree with the observation that the viscosity of gases is roughly independent of pressure and density. The gaseous phase is probably the best understood of all the phases of matter.

Since viscosity is so dependent on temperature, it shouldn't never be stated without it.

Viscosities of selected materials (note the different unit prefixes)
simple liquidsT (°C)η (mPa s)gasesT (°C)η (μPa s)
alcohol, ethyl (grain)201.1air1517.9
alcohol, isopropyl202.4hydrogen08.42
alcohol, methyl (wood) 200.59helium (gas)018.6
blood373–4nitrogen016.7
ethylene glycol2516.1oxygen018.1
ethylene glycol1001.98complex materialsT (°C)η (Pa s)
freon 11 (propellant)−250.74caulk201000
freon 11 (propellant)00.54glass201018–1021
freon 11 (propellant)+250.42glass, strain pt.5041015.2
freon 12 (refrigerant)-15?glass, annealing pt.5461012.5
freon 12 (refrigerant)0?glass, softening pt.724106.6
freon 12 (refrigerant)+150.20glass, working pt.103
glycerin201420glass, melting pt.101
glycerin40280honey2010
helium (liquid)4 K0.00333ketchup2050
mercury151.55lard201000
milk253molasses205
oil, vegetable, canola2557mustard2570
oil, vegetable, canola4033peanut butter20150–250
oil, vegetable, corn 2065sour cream25100
oil, vegetable, corn 4031syrup, chocolate2010–25
oil, vegetable, olive2084syrup, corn252–3
oil, vegetable, olive40?syrup, maple202–3
oil, vegetable, soybean2069tar2030,000
oil, vegetable, soybean 4026vegetable shortening201200
oil, machine, light20102
oil, machine, heavy20233
oil, motor, SAE 2020125
oil, motor, SAE 3020200
oil, motor, SAE 4020319
propylene glycol2540.4
propylene glycol1002.75
water01.79
water201.00
water400.65
water1000.28

### motor oil

Motor oil is like every other fluid in that its viscosity varies with temperature and pressure. Since the conditions under which most automobiles will be operated can be anticipated, the behavior of motor oil can be specified in advance. In the United States, the organization that sets the standards for the performance of motor oils is the Society of Automotive Engineers (SAE). The SAE numbering scheme describes the behavior of motor oils under low and high temperature conditions — conditions that correspond to starting and operating temperatures. The first number, which is always followed by the letter W for winter, describes the low temperature behavior of the oil at start up while the second number describes the high temperature behavior of the oil after the engine has been running for some time. Lower SAE numbers describe oils that are meant to be used under lower temperatures. Oils with low SAE numbers are generally runnier (less viscous) than oils with high SAE numbers, which tend to be thicker (more viscous).

For example, 10W-40 oil would have a viscosity no greater than 7,000 mPa s in a cold engine crankcase even if its temperature should drop to −25 °C on a cold winter night and a viscosity no less than 2.9 mPa s in the high pressure parts of an engine near the point of overheating (150 °C).

Viscosity characteristics of motor oil gradesSource: Society of Automotive Engineers (SAE)
* 0W-40, 5W-40, 10W-40 15W-40, 20W-40, 25W-40
low temperature specifications
sae
prefix
dynamic viscosity
cranking maximum
dynamic viscosity
pumping maximum
00W06,200 mPa s(−35 °C)60,000 mPa s(−40 °C)
05W06,600 mPa s(−30 °C)60,000 mPa s(−35 °C)
10W07,000 mPa s(−25 °C)60,000 mPa s(−30 °C)
15W07,000 mPa s(−20 °C)60,000 mPa s(−25 °C)
20W09,500 mPa s(−15 °C)60,000 mPa s(−20 °C)
25W13,000 mPa s(−10 °C)60,000 mPa s(−15 °C)
high temperature specifications
sae
suffix
kinematic viscosity
low shear rate
dynamic viscosity
high shear rate
0804.0–6.10 mm2/s(100 °C)>1.7 mPa s(150 °C)
1205.0–7.10 mm2/s(100 °C)>2.0 mPa s(150 °C)
1606.1–8.20 mm2/s(100 °C)>2.3 mPa s(150 °C)
2005.6–9.30 mm2/s(100 °C)>2.6 mPa s(150 °C)
3009.3–12.5 mm2/s(100 °C)>2.9 mPa s(150 °C)
*40*12.5–16.3 mm2/s(100 °C)>2.9 mPa s(150 °C)
4012.5–16.3 mm2/s(100 °C)>3.7 mPa s(150 °C)
5016.3–21.9 mm2/s(100 °C)>3.7 mPa s(150 °C)
6021.9–26.1 mm2/s(100 °C)>3.7 mPa s(150 °C)

### capillary viscometer

The the mathematical expression describing the flow of fluids in circular tubes was determined by the French physician and physiologist Jean Poiseuille (1799–1869). Since it was also discovered independently by the German hydraulic engineer Gotthilf Hagen (1797–1884), it should be properly known as the Hagen-Poiseuille equation, but it is usually just called Poiseuille's equation. I will not derive it here. (Please don't ask me to.) For non-turbulent, non-pulsatile fluid flow through a uniform straight pipe, the volume flow rate (qm) is…

• directly proportional to the pressure difference (P) between the ends of the tube
• inversely proportional to the length () of the tube
• inversely proportional to the viscosity (η) of the fluid
• proportional to the fourth power of the radius (r4) of the tube
 qm = π∆Pr4 8ηℓ

Solve for viscosity if that's what you want to know.

 η = π∆Pr4 8qmℓ

capillary viscometer… keep writing…

### falling sphere

The mathematical expression describing the viscous drag force on a sphere was determined by the 19th century British physicist George Stokes. I will not derive it here. (Once again, don't ask.)

R = 6πηrv

The formula for the buoyant force on a sphere is accredited to the Ancient Greek engineer Archimedes of Syracuse, but equations weren't invented back then.

B = ρfluidgVdisplaced

The formula for weight had to be invented by someone, but I don't know who.

### Semiconductor Physics And Devices

W = mg = ρobjectgVobject

Let's combine all these things together for a sphere falling in a fluid. Weight goes down, buoyancy goes up, drag goes up. After a while, the sphere will fall with constant velocity. When it does, all these forces cancel. When a sphere is falling through a fluid it is completely submerged, so there is only one volume to talk about — the volume of a sphere. Let's work through this.

 B + R = W ρfluidgV + 6πηrv = ρobjectgV 6πηrv = (ρobject − ρfluid)gV 6πηrv = ∆ρg43πr3

And here we are.

 η = 2∆ρgr2 9v

Drop a sphere into a liquid. If you know the size and density of the sphere and the density of the liquid, you can determine the viscosity of the liquid. If you don't know the density of the liquid you can still determine the kinematic viscosity. If you don't know the density of the sphere, but you know its mass and radius, well then you do know its density. Why are you talking to me? Go back several chapters and get yourself some education.

Should I write more?

### non-newtonian fluids

Newton's equation relates shear stress and velocity gradient by means of a quantity called viscosity. A newtonian fluid is one in which the viscosity is just a number. A non-newtonian fluid is one in which the viscosity is a function of some mechanical variable like shear stress or time. (Non-newtonian fluids that change over time are said to have a memory.)

Some gels and pastes behave like a fluid when worked or agitated and then settle into a nearly solid state when at rest. Such materials are examples of shear-thinning fluids. House paint is a shear-thinning fluid and it's a good thing, too. Brushing, rolling, or spraying are means of temporarily applying shear stress. This reduces the paint's viscosity to the point where it can now flow out of the applicator and onto the wall or ceiling. Once this shear stress is removed the paint returns to its resting viscosity, which is so large that an appropriately thin layer behaves more like a solid than a liquid and the paint does not run or drip. Think about what it would be like to paint with water or honey for comparison. The former is always too runny and the latter is always too sticky.

Toothpaste is another example of a material whose viscosity decreases under stress. Toothpaste behaves like a solid while it sits at rest inside the tube. It will not flow out spontaneously when the cap is removed, but it will flow out when you put the squeeze on it. Now it ceases to behave like a solid and starts to act like a thick liquid. when it lands on your toothbrush, the stress is released and the toothpaste returns to a nearly solid state. You don't have to worry about it flowing off the brush as you raise it to your mouth.

Shear-thinning fluids can be classified into one of three general groups. A material that has a viscosity that decreases under shear stress but stays constant over time is said to be pseudoplastic. A material that has a viscosity that decreases under shear stress and then continues to decrease with time is said to be thixotropic. If the transition from high viscosity (nearly semisolid) to low viscosity (essentially liquid) takes place only after the shear stress exceeds some minimum value, the material is said to be a bingham plastic.

Materials that thicken when worked or agitated are called shear-thickening fluids. An example that is often shown in science classrooms is a paste made of cornstarch and water (mixed in the correct proportions). The resulting bizarre goo behaves like a liquid when squeezed slowly and an elastic solid when squeezed rapidly. Ambitious science demonstrators have filled tanks with the stuff and then run across it. As long as they move quickly the surface acts like a block of solid rubber, but the instant they stop moving the paste behaves like a liquid and the demonstrator winds up taking a cornstarch bath. The shear-thickening behavior makes it a difficult bath to get out of. The harder you work to get out, the harder the material pulls you back in. The only way to escape it is to move slowly.

Materials that turn nearly solid under stress are more than just a curiosity. They're ideal candidates for body armor and protective sports padding. A bulletproof vest or a kneepad made of of shear-thickening material would be supple and yielding to the mild stresses of ordinary body motions, but would turn rock hard in response to the traumatic stress imposed by a weapon or a fall to the ground.

Shear-thickening fluids are are also divided into two groups: those with a time-dependent viscosity (memory materials) and those with a time-independent viscosity (non-memory materials). If the increase in viscosity increases over time, the material is said to be rheopectic. If the increase is roughly directly proportional to the shear stress and does not change over time, the material is said to be dilatant.

Classes of nonlinear fluids with examples and applications
shear-thinningshear-thickening
time-dependent
(memory materials)
thixotropic
ketchup, honey, quicksand, snake venom, polymeric thick film inks
rheopectic
cream being whipped
time-independent
(non-memory materials)
pseudoplastic
paint, styling gel, whipped cream, cake batter, applesauce, ballpoint pen ink, ceramic-metal inks
dilatant
starch pastes, silly putty, synovial fluid, chocolate syrup, viscous coupling fluids, liquid armor
with a yield stressbingham plastic
toothpaste, drilling mud, blood, cocoa butter, mayonnaise, yoghurt, tomato puree, nail polish, sewage sludge
n/a

With a bit of adjustment, Newton's equation can be written as a power law that handles the pseudoplastics and the dilantants — the Ostwald-de Waele equation

 F = k ⎛⎜⎝ dvx ⎞n⎟⎠ A dz

where η the viscosity is replaced with k the flow consistency index [Pa sn] and the velocity gradient is raised to some power n called the flow behavior index [dimensionless]. The latter number varies with the class of fluid.

 n < 1 n = 1 n > 1 pseudoplastic newtonian dilatant

A different modification to Newton's equation is needed to handle Bingham plastics — the Bingham equation

 F = σy + ηpl dvx A dz

where σy is the yield stress [Pa] and ηpl is the plastic viscosity [Pa s]. The former number separates Bingham plastics from newtonian fluids.

 σy < 0 σy = 0 σy > 0 impossible newtonian bingham plastic

Combining the Ostwald-de Waele power law with the Bingham yield stress gives us the more general Herschel-Bulkley equation

 F = σy + k ⎛⎜⎝ dvx ⎞n⎟⎠ A dz

where again, σy is the yield stress [Pa], k is the flow consistency index [Pa sn], and n is the flow behavior index [dimensionless].

### viscoelasticity

When a force (F) is applied to an object, one of four things can happen.

1. It could accelerate as a whole, in which case Newton's second law of motion would apply…

F = ma

This term is not interesting to us right now. We've already discussed this kind of behavior in earlier chapters. Mass (m) is resistance to acceleration (a), which is the second derivative of position (x). Let's move on to something new.

2. It could flow like a fluid, which could be described by this relationship…

F = −bv

This is the simplified model where drag is directly proportional to speed (v), the first derivative of position (x). We used this in terminal velocity problems just because it gave differential equations that were easy to solve. We also used it in the damped harmonic oscillator, again because it gave differential equations that were easy to solve (relatively easy, anyway). The proportionality constant (b) is often called the damping factor.

3. It could deform like a solid according to Hooke's law…

F = −kx

The proportionality constant (k) is the spring constant. Position (x) is not the part of any derivative nor is it raised to any power.

4. It could get stuck

F = −f

That symbol f makes it look like we're discussing static friction. In fluids (non-newtonian fluids, to be specific) a term like this is associated with yield stress. Position (x) is not involved in any way.

Put everything together and state acceleration and velocity as derivatives of position.

 F = m d2x − b dx − kx − f dt2 dt

This differential equation summarizes the possible behaviors of an object. The interesting thing is that it mixes up the behaviors of fluids and solids. The more interesting thing is that there are occasions when both behaviors will be present in one thing. Materials that both flow like fluids and deform like solids are said to be viscoelastic — an obvious mash-up of viscosity and elasticity. The study of materials with fluid and solid properties is called rheology, which comes from the Greek verb ρέω (reo), to flow.

What old book gave me this idea? What should I write next?