L ENGTH AND L ENGTH -R ELATED P ARAMETERS

7

CHAPTER

L ^{ENGTH AND} L ^ENGTH -R ^ELATED

When you become a practicing engineer, you will find out that you don’t stop learn- ing new things even after obtaining your engineering degree. For example, you may work on a project in which the noise of a machine is a concern, and you may be asked to come up with ways to reduce the level of noise. It may be the case that during the four or five years of your engineering education, you did not take a class in noise con- trol. Considering your lack of understanding and background in noise reduction, you may first try to find someone who specializes in noise control who could solve the problem for you. But your supervisor may tell you that because of budget constraints and because this is a one-time project, you must come up with a reasonable solution yourself. Therefore, you will have to learn something new and learn fast. If you have a good grasp of underlying engineering concepts and fundamentals, the learning process could be fun and quick. The point of this story is that during the next four years you need to make sure that you learn the fundamentals well.

TABLE 6.7 Fundamental Dimensions and How They Are Used in Defining Variables that Are Used in Engineering Analysis and Design Fundamental

Dimension Related Engineering Variables

Length (L) Radian , Area (L²) Volume (L³) Area moment

[Chapter 7] Strain of inertia (L⁴)

Time (t) Angular speed ,

[Chapter 8] Angular acceleration Volume flow rate

Linear speed , Linear acceleration

Mass (M) Mass flow rate , Density ,

[Chapter 9] Momentum Specific volume

Kinetic energy

Force (F) Moment (LF), Pressure ,

[Chapter 10] Work, energy (FL), Stress ,

1

2 1

^Lt³

2

1

^LL

1

^LL

2 2

Linear impulse (Ft), Modulus of Specific weight ,

Power elasticity ,

Modulus of rigidity

Temperature (T) Linear thermal expansion Volume thermal

[Chapter 11] expansion

Specific heat

Electric Current (I) Charge (It) Current density

[Chapter 12]

1

L^I²

2

1

MT^FL

2 1

L^L³³T

2

1

LT^L

2 ,

1

L^F²

2 1

L^F²

2

1

^FLt

2 1

L^F³

2

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As an engineering student, you also need to develop a keen awareness of your surroundings. In this chapter, we will investigate the role of length, area, and vol- ume along with other length-related variables in engineering applications. You will learn how these physical variables affect engineering design decisions. The topics introduced in this chapter are fundamental in content, so developing a good grasp of them will make you a better engineer. You may see some of the concepts and ideas introduced here in some other form in the engineering classes that you will take later. The main purpose of introducing these concepts here is to help you become aware of their importance and learn to look for their relation to other engineering parameters in your future classes when you study a speciﬁc topic in detail.

Table 6.7 is repeated here to show the relationship between the content of this chapter and other fundamental dimensions discussed in other chapters.

77..11 LLe en ng gtth h a ass a a FFu un nd da am me en ntta all D Diim me en nssiio on n

When you walk down a hallway, can you estimate how tall the ceiling is? Or how wide a door is? Or how long the hallway is? You should develop this ability because having a “feel” for dimensions will help you become a better engineer. If you decide to become a design engineer, you will ﬁnd out that size and cost are important design parameters. Having developed a “feel” for the size of objects in your surroundings enables you to have a good idea about the acceptable range of values when you design something.

As you know, every physical object has a size. Some things are bigger than others. Some things are wider or taller than others. These are some common ways of expressing the relative size of objects. As discussed in Chapter 6, through their observation of nature, people recognized the need for a physical quantity or a physical dimension (which today we calllength) so that they could describe their surroundings better. They also realized that having a common deﬁnition for a physical quantity, such as length, makes communication easier. Earlier humans may have used a ﬁnger, arm, stride, stick, or rope to measure the size of an object. Chapter 6 also emphasized the need for having scales or divisions for the dimension length so that numbers could be kept simple and manageable. Today, we call these divisions or scalessystems of units. In this chapter, we will focus our attention on length and such length-related derived quantities as area and volume.

Length is one of the seven fundamental or base dimensions that we use to properly express what we know of our natural world. In today’s globally driven economy, where products are made in one place and assembled somewhere else, there exists an even greater need for a uni- form and consistent way of communicating information about the fundamental dimension length and other related length variables so that parts manufactured in one place can easily be combined on an assembly line with parts made in other places. An automobile is a good example of this concept. It has literally thousands of parts that are manufactured by various companies in different parts of the world.

7.1 Length as a Fundamental Dimension

161

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^Chapter7 Length and Length-Related Par ameters

As explained in Chapter 6, we have learned from our surroundings and formulated our observations into laws and principles. We use these laws and physical principles to design, develop, and test products and services. Are you observing your surroundings carefully? Are you learning from your everyday observations? Here are some questions to consider: Have you thought about the size of a soda can? What are its dimensions? What do you think are the important design factors? Most of you drink a soda every day, so you know that it fits in your hand. You also know that the soda can is made from aluminum, so it is lightweight. What do you think are some of its other design factors? What are important considerations when designing signs for a highway? How wide should a hallway be? When designing a supermarket, how wide should an aisle be? Most of you have been going to class for at least 12 years, but have you thought about classroom seating arrangements? For example, how far apart are the desks? Or how far above the floor is the presentation board? For those of you who may take a bus to school, how wide are the seats in a bus? How wide is a highway lane? What do you think are the important factors when determining the size of a car seat? You can also look around your home to think about the dimension length. Start with your bed: What are its dimensions, how far above the floor is it? What is a typical standard height for steps in a stairway? When you tell someone that you own a 32-inch TV, to what dimension are you referring? How high off the floor are a doorknob, showerhead, sink, light switch, and so on?

You are beginning to see that length is a very important fundamental dimension, and it is thus commonly used in engineering products. Coordinate systems are examples of another application where length plays an important role. Coordinate systems are used to locate things with respect to a known origin. In fact, you use coordinate systems every day, even though you don’t think about it. When you go from your home to school or a grocery store or to meet a friend for lunch, you use coordinate systems. The use of coordinate systems is almost second nature to you. Let’s say you live downtown, and your school is located on the northeast side of town. You know the exact location of the school with respect to your home. You know which streets to take for what distance and in which directions to move to get to school. You have been using a coordinate system to locate places and things most of your life, even if you did not know it. You also know the speciﬁc location of objects at home relative to other objects or to yourself. You know where your TV is located relative to your sofa or bedroom.

There are different types of coordinate systems such as rectangular, cylindrical, spherical, and so on, as shown in Figure 7.1. Based on the nature of a particular problem, we may use one or another. The most common coordinate system is the rectangular, or Cartesian, coordinate system (Figure 7.1). When you are going to school from home or meeting a friend for lunch you use the rectangular coordinate system. But you may not call it that; you may use the directions north, east, west, or south to get where you are going. You can think of the axes of a rectangular coordinate system as aligning with, for example, the east and north direction. People who are blind are expert users of rectangular coordinate systems. Because they cannot rely on their visual perception, people with a vision disability know how many steps to take and in which direction to move to go from one location to another. So to better understand coordinate systems, perform the following experiment. While at home, close your eyes for a few min- utes and try to go from your bedroom to the bathroom. Note the number of steps you had to take and in which directions you had to move. Think about it!

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7.1 Length as a Fundamental Dimension

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All of us, at one time or another, have experienced not knowing where we are going. In other words, we were lost! The smarter ones among us use a map or stop and ask someone for directions and distances (xandycoordinates) to the desired place. An example of using a map to locate a place is shown in Figure 7.2. Coordinate systems are also integrated into software that drives computer numerically controlled (CNC) machines, such as a milling machine or a lathe that cuts materials into speciﬁc shapes.

Now that you understand the importance of the length dimension, let us look at its divisions or units. There are several systems of units in use in engineering today. We will focus on two of these systems: the International System of Units (SI) and the United States Customary units.

The unit of length in SI is the meter (m). We can use the multiples and fractions of this unit according to Table 6.2. Common multipliers of the meter are micrometer (␮m), millimeter (mm), centimeter (cm), and kilometer (km). Recall from our discussion of units and multiplication preﬁxes in Chapter 6 that we use these multiplication preﬁxes to keep the numbers manageable. The International System of Units is used almost universally, except in the United States.

The unit of length in the U.S. Customary system is foot (ft). The relation between foot and meter units is given by 1 ft⫽ 0.3048 m. Table 7.1 shows other commonly used units and their

x y

y A

Cartesian system (x, y, z)

z r

y A

Cylindrical system (r, ␪, z)

␪

y A

Spherical system (r, ␪, ␾)

␪

␾ 0

■

^{Figure 7.1}

Examples of coordinate systems.

To locate an object at pointA, with respect to the origin (point 0) of the Cartesian system, you move along thexaxis byxamount (or steps) and then you move along the dashed line parallel to theyaxis byyamount. Finally, you move along the dashed line parallel to the zdirection byzamount. How would you get to pointAusing the Cylindrical or Spherical system? Explain.

164

^Chapter7 Length and Length-Related Par ameters

equivalent values in an increasing order and includes both SI and U.S. Customary units to give you a sense of their relative magnitude.

Some interesting dimensions in the natural world are

The highest mountain peak (Everest): 8848 m (29,028 feet)

Paciﬁc Ocean: Average depth: 4028 m (13,215 feet)

Greatest known depth: 11,033 m (36,198 feet) y

Home x

Bank

■

^{Figure 7.2}

An example demonstrating the use of a coordinate system.

TABLE 7.1 Units of Length and their Equivalent Values

Units of Length in Increasing Order Equivalent Value

1 angstrom 1 ⫻10^⫺10meter (m)

1 micrometer or 1 micron 1 ⫻10^⫺6meter (m)

1 mil 1/1000 inch ⬇2.54 ⫻10^⫺5meter (m)

1 point (printer’s) 3.514598 ⫻10^⫺4(m)

1 millimeter 1/1000 meter (m)

1 pica (printer’s) 4.217 millimeter (mm)

1 centimeter 1/100 meter (m)

1 inch 2.54 centimeter (cm)

1 foot 12 inches (in.)

1 yard 3 feet (ft)

1 meter 1.0936 yard ⬇1.1 yard (yd)

1 kilometer 1000 meter (m)

1 mile 1.6093 kilometer (km)⫽5280 feet (ft)

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7.2 Measurement of Length

165 77..22 M Me ea assu urre em me en ntt o off LLe en ng gtth h

Early humans may have used ﬁnger length, arm span, stride length (step length), a stick, rope, chains, and so on to measure the size or displacement of an object. Today, depending on how accurate the measurement needs to be and the size of the object being measured, we use other measuring devices, such as a ruler, a yardstick, and a steel tape. All of us have used a ruler or tape measure to measure a distance or the size of an object. These devices are based on internationally deﬁned and accepted units such as millimeters, centimeters, or meters or inches, feet, or yards. For more accurate measurements of small objects, we have developed measurement tools such as the micrometer or the Vernier caliper, which allow us to measure dimensions within 1/1000 of an inch. In fact, machinists use micrometers and Vernier calipers every day.

On a larger scale, you have seen the milepost markers along interstate highways. Some people actually use the mileposts to check the accuracy of their car’s odometer. By measuring the time between two markers, you can also check the accuracy of the car’s speedometer. In the last few decades, electronic distance measuring instruments (EDMI) have been developed that allow us to measure distances from a few feet to many miles with reasonable accuracy. These electronic distance measuring devices are used quite commonly for surveying purposes in civil engineering applications. The instrument sends out a light beam that is reflected by a system of reflectors located at the unknown distance. The instrument and the reflector system are sit- uated such that the reflected light beam is intercepted by the instrument. The instrument then interprets the information to determine the distance between the instrument and the reflector.

The Global Positioning System (GPS) is another example of recent advances in locating objects on the surface of the earth with good accuracy. As of the year 2006, radio signals were sent from approximately 30 artiﬁcial satellites orbiting the earth. Tracking stations are located around the world to receive and interpret the signals sent from the satellites. Although originally the GPS

A Vernier caliper and a micrometer.

Source:Saeed Moaveni

166

^Chapter7 Length and Length-Related Par ameters

was funded and controlled by the U.S. Department of Defense for military applications, it now has millions of users. GPS navigation receivers are now common in airplanes, automobiles, buses, cellphones, and hand-held receivers used by hikers.

Sometimes, distances or dimensions are determined indirectly using trigonometric principles. For example, let us say that we want to determine the height of a building similar to the one shown in Figure 7.3. but do not have access to accurate measuring devices. With a drink- ing straw, a protractor, and a steel tape we can determine a reasonable value for the height of the building by measuring the angle ␣and the dimensionsdandh₁. The analysis is shown in Figure 7.3. Note thath₁represents the distance from the ground to the eye of a person look- ing through the straw. The angle ␣(alpha) is the angle between the straw, which is focused on the edge of the roof, and a horizontal line. The protractor is used to measure the angle that the straw makes with the horizontal line.

It is very likely that you have used trigonometric tools to analyze some problems in the past.

It is also possible that you may have not used them recently. If that is the case, then the tools have been sitting in your mental toolbox for a while and quite possibly have collected some rust in your head! Or you may have forgotten altogether how to properly use the tools. Because of their importance, let us review some of these basic relationships and deﬁnitions. For a right triangle, the Pythagorean relation may be expressed by

a²⫹b²⫽c² An example of an electronic

distance measuring

instrument used in surveying.

Source:Courtesy Pentax Corporation

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7.2 Measurement of Length

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In theright triangle shown, the angle facing sidea is denoted by a(alpha), and the angle facing sidebby b(beta). The sine, cosine, and the tangent of an angle are deﬁned by

The sine and the cosine law (rule) for anarbitrarily shaped triangleis

cos u⫽ a²⫹b²⫺c² 2ba cos b⫽ a²⫹c²⫺b²

2ac cos a⫽ b²⫹c²⫺a²

2bc

c²⫽a²⫹b²⫺2ba1cos u2 b²⫽a²⫹c²⫺2ac1cos b2 The cosine rule: a²⫽b²⫹c²⫺2bc1cos a2

The sine rule: a

sin a⫽ b sin b⫽ c

sin u

tan b⫽ sin b

cos b⫽ opposite adjacent⫽ b cos b⫽ adjacent a

hypotenuse ⫽a sin b⫽ opposite c

hypotenuse⫽ b c

tan a⫽ sin a

cos a⫽ opposite adjacent ⫽ a cos a⫽ adjacent b

hypotenuse ⫽ b sin a⫽ opposite c

hypotenuse⫽ a c H

h₂

h₁ d ␣

tan ␣⁼ ^h_d²

H = h₁⫹ h₂ = h₁⫹ d tan ␣

90 80

100 100 110 70 80

120 60

130 50

140 40

15030

16020

17010

70 110 60 120 50

130 40

140 30

150 20

160 170 10

Straw Protractor

String Mass

␣

■

^{Figure 7.3}

Measuring the height of a building indirectly.

Hypotenuse

a ␤ c

␣

a c

␤

␣

␪

168

^Chapter7 Length and Length-Related Par ameters

Some other useful trigonometry identities are

As you will learn later in your physics, statics, and dynamics classes, the trigonometry relations are quite useful.

77..33 N No om miin na all S Siizze ess vve errssu uss A Ac cttu ua all S Siizze ess

You have all seen or used a 2 ⫻4 piece of lumber. If you were to measure the dimensions of the cross section of a 2 ⫻4, you would find that the actual width is less than 2 inches (approximately 1.5 inches) and the height is less than 4 inches (approximately 3.5 inches). Then why is it referred to as a “2 by 4”? Manufacturers of engineering parts (See Figure 7.4) use round numbers so that it is easier for people to remember the size and thus more easily refer to a specific part. The 2 ⫻4 is called thenominal sizeof the lumber. If you were to investigate other structural members, such as I-beams, you would also note that the nominal size given by the manufacturer is different from the actual size. You will find a similar situation for pipes, tubes, screws, wires, and many other engineering parts. Agreed-upon standards are followed by manufacturers when providing information about the size of the parts that they make. But manufacturers do provide actual sizes of parts in addi- tion to nominal sizes. This fact is important because, as you will learn in your future engineering classes, you need the actual size of parts for various engineering calculations. Examples of nominal sizes versus actual sizes of some engineering parts are given in Table 7.2.

cos1a⫺b2⫽cos a cos b⫹sin a cos b cos1a⫹b2⫽cos a cos b⫺sin a sin b sin1a⫺b2⫽sin a cos b⫺sin b cos a sin1a⫹b2⫽sin a cos b⫹sin b cos a cos1⫺a2⫽cos a

sin1⫺a2 ⫽ ⫺sin a

cos 2a⫽cos² a⫺sin² a⫽2 cos² a⫺1⫽1⫺2 sin² a sin 2a⫽2 sin a cos a

sin² a⫹cos² a⫽1

■

^{Figure 7.4} Manufacturers provide actual and nominal sizes for many parts and structural members.

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