Vật lý đại cương Principles of Physic Chương I

Cuốn sách có nội dung phong phú và hình ảnh đẹp, được tác giả Serway và JewettChương I mô tả về các đại lượng Vật lý và các phép toán Vecto được sử dụng trong vật lý. Cuốn sách mô tả đầy đủ về các đại lượng vật lý, và có phần bài tập phù hợp. Cuốn sách phù hợp cho giáo viên dạy hsg và dạy các trường THPT Chuyên hoặc dạy học vật lý bằng tiếng Anh thì đây là tài liệu hết sức hữu dụng.

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Principles of Physics: A Calculus-Based Text,

Fifth Edition, International Edition Raymond A Serway

John W Jewett, Jr

Publisher, Physical Sciences: Mary Finch Publisher, Astronomy and Physics: Charles Hartford Development Editor: Ed Dodd

Assistant Editor: Brandi Kirksey Editorial Assistant: Brendan Killion Media Editor: Rebecca Berardy Schwartz Marketing Manager: Jack Cooney Marketing Communications Manager:

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Printed in China

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We dedicate this book to our wives Elizabeth and Lisa and all our children and grandchildren for their loving understanding when we spent time on writing instead of being with them.

1.6 Coordinate Systems 1.7 Vectors and Scalars 1.8 Some Properties of Vectors 1.9 Components of a Vector and Unit Vectors

1.10 Modeling, Alternative Representations, and Problem-Solving Strategy SUMMARY

The goal of physics is to provide a quantitative understanding of cer-tain basic phenomena that occur in our Universe Physics is a science based on experimental observations and mathematical analyses The main objectives behind such experiments and analyses are to develop theories that explain the phenomenon being studied and to relate those theories to other established theories Fortunately, it is possible to explain the behavior of various physical systems using relatively few fundamental laws Analytical procedures require the expression of those laws in the language of mathematics, the tool that provides a bridge between theory and experiment In this chapter, we shall discuss a few math-ematical concepts and techniques that will be used throughout the text In addition,

we will outline an effective problem-solving strategy that should be adopted and used in your problem-solving activities throughout the text

1.1 | Standards of Length, Mass, and Time

To describe natural phenomena, we must make measurements associated with physical quantities, such as the length of an object The laws of physics can be expressed as mathematical relationships among physical quantities that will be

A signpost in Saint Petersburg, Florida, shows the distance and direction to several cities Quantities that are defi ned by both a magnitude and a direction are called vector quantities.

Interactive content from this and other chapters may

be assigned online in Enhanced WebAssign.

1.1 | Standards of Length, Mass, and Time 5

introduced and discussed throughout the book In mechanics, the three tal quantities are length, mass, and time All other quantities in mechanics can be expressed in terms of these three

fundamen-If we measure a certain quantity and wish to describe it to someone, a unit for the quantity must be specifi ed and defi ned For example, it would be meaningless for

a visitor from another planet to talk to us about a length of 8.0 “glitches” if we did not know the meaning of the unit glitch On the other hand, if someone familiar with our system of measurement reports that a wall is 2.0 meters high and our unit

of length is defi ned to be 1.0 meter, we then know that the height of the wall is twice our fundamental unit of length An international committee has agreed on a system

of defi nitions and standards to describe fundamental physical quantities It is called

the SI system (Système International) of units Its units of length, mass, and time are

the meter, kilogram, and second, respectively

Length

In a.d 1120, King Henry I of England decreed that the standard of length in his country would be the yard and that the yard would be precisely equal to the distance from the tip of his nose to the end of his outstretched arm Similarly, the original standard for the foot adopted by the French was the length of the royal foot of King Louis XIV This standard prevailed until 1799, when the legal standard of length

in France became the meter, defi ned as one ten-millionth of the distance from the

equator to the North Pole

Many other systems have been developed in addition to those just discussed, but the advantages of the French system have caused it to prevail in most countries and

in scientifi c circles everywhere Until 1960, the length of the meter was defi ned as the distance between two lines on a specifi c bar of platinum – iridium alloy stored under controlled conditions This standard was abandoned for several reasons, a principal one being that the limited accuracy with which the separation be tween the lines can

be determined does not meet the current requirements of science and technology

The defi nition of the meter was modifi ed to be equal to 1 650 763.73 wavelengths of orange – red light emitted from a krypton-86 lamp In October 1983, the meter was

redefi ned to be the distance traveled by light in a vacuum during a time interval of 1y299 792 458 second. This value arises from the establishment of the speed of light in

a vacuum as exactly 299 792 458 meters per second We will use the standard scientifi c notation for numbers with more than three digits in which groups of three digits are separated by spaces rather than commas Therefore, 1 650 763.73 and 299 792 458

in this paragraph are the same as the more popular American cultural notations of 1,650,763.73 and 299,792,458 Similarly,  5 3.14159265 is written as 3.141 592 65.

Mass

Mass represents a measure of the resistance of an object to changes in its motion The

SI unit of mass, the kilogram, is defi ned as the mass of a specifi c platinum – iridium alloy cylinder kept at the International Bureau of Weights and Measures at Sèvres, France. At this point, we should add a word of caution Many beginning students of

physics tend to confuse the physical quantities called weight and mass For the

pres-ent we shall not discuss the distinction between them; they will be clearly defi ned in later chapters For now you should note that they are distinctly different quantities

Time

Before 1967, the standard of time was defi ned in terms of the average length of a mean

solar day (A solar day is the time interval between successive appearances of the Sun

at the highest point it reaches in the sky each day.) The basic unit of time, the ond, was defi ned to be (1/60)(1/60)(1/24) 5 1/86 400 of a mean solar day In 1967, the second was redefi ned to take advantage of the great precision obtainable with a device known as an atomic clock (Fig 1.1), which uses the characteristic frequency of the

sec-c Defi nition of the meter

c Defi nition of the kilogram

Figure 1.1 A cesium fountain atomic clock The clock will neither gain nor lose a second in 20 million years.

cesium-133 atom as the “reference clock.” The second is now defi ned as 9 192 631 770 times the period of oscillation of radiation from the cesium atom. It is possible today

to purchase clocks and watches that receive radio signals from an atomic clock in Colorado, which the clock or watch uses to continuously reset itself to the correct time

Approximate Values for Length, Mass, and Time

Approximate values of various lengths, masses, and time intervals are presented in Tables 1.1, 1.2, and 1.3, respectively Note the wide range of values for these quantities.1

You should study the tables and begin to generate an intuition for what is meant by a mass of 100 kilograms, for example, or by a time interval of 3.2 3 107 seconds

Systems of units commonly used in science, commerce, manufacturing, and

everyday life are (1) the SI system, in which the units of length, mass, and time are the meter (m), kilogram (kg), and second (s), respectively; and (2) the U.S customary

system, in which the units of length, mass, and time are the foot (ft), slug, and second, respectively Throughout most of this text we shall use SI units because they are almost universally accepted in science and industry We will make limited use of U.S

customary units in the study of classical mechanics

Some of the most frequently used prefi xes for the powers of ten and their abbreviations are listed in Table 1.4 For example, 1023 m is equivalent to 1 millimeter (mm), and 103 m

is 1 kilometer (km) Likewise, 1 kg is 103 grams (g), and 1 megavolt (MV) is 106 volts (V)

The variables length, time, and mass are examples of fundamental quantities A much larger list of variables contains derived quantities, or quantities that can be expressed as

a mathematical combination of fundamental quantities Common examples are area, which is a product of two lengths, and speed, which is a ratio of a length to a time interval.

Another example of a derived quantity is density The density  (Greek letter rho;

a table of the letters in the Greek alphabet is provided at the back of the book) of any

substance is defi ned as its mass per unit volume:

 ; m

c Defi nition of density

Pitfall Prevention | 1.1 Reasonable Values

Generating intuition about typical values of quantities when solving problems is important because you must think about your end result and determine if it seems reasonable For example, if you are calculating the mass of a housefl y and arrive

at a value of 100 kg, this answer is

unreasonable and there is an error somewhere.

1 If you are unfamiliar with the use of powers of ten (scientifi c notation), you should review Appendix B.1.

TABLE 1.1 |Approximate Values of Some Measured Lengths

Length (m)

Distance from the Earth to the nearest large galaxy (M 31, the Andromeda galaxy) 2 3 10 22

1.2 | Dimensional Analysis 7

which is a ratio of mass to a product of three lengths For example, aluminum has a density of 2.70 3 103 kg/m3, and lead has a density of 11.3 3 103 kg/m3 An extreme difference in density can be imagined by thinking about holding a 10-centimeter (cm) cube of Styrofoam in one hand and a 10-cm cube of lead in the other

1.2 | Dimensional Analysis

In physics, the word dimension denotes the physical nature of a quantity The

dis-tance between two points, for example, can be measured in feet, meters, or furlongs, which are all different ways of expressing the dimension of length

The symbols used in this book to specify the dimensions2 of length, mass, and time are L, M, and T, respectively We shall often use square brackets [ ] to denote the dimensions of a physical quantity For example, in this notation the dimensions

of speed v are written [v] 5 L/T, and the dimensions of area A are [A] 5 L2 The mensions of area, volume, speed, and acceleration are listed in Table 1.5, along with their units in the two common systems The dimensions of other quantities, such as force and energy, will be described as they are introduced in the text

di-In many situations, you may be faced with having to derive or check a specifi c equation Although you may have forgotten the details of the derivation, a useful and

powerful procedure called dimensional analysis can be used as a consistency check, to

assist in the derivation, or to check your fi nal expression Dimensional analysis makes use of the fact that dimensions can be treated as algebraic quantities For example, quantities can be added or subtracted only if they have the same dimensions Fur-thermore, the terms on both sides of an equation must have the same dimensions

By following these simple rules, you can use dimensional analysis to help determine

TABLE 1.4 |

Some Prefi xes for Powers of Ten Power Prefi x Abbreviation

One day (time interval for one revolution of the Earth about its axis)

Pitfall Prevention | 1.2 Symbols for Quantities

Some quantities have a small number of symbols that represent them For example, the symbol

for time is almost always t Other

quantities might have various symbols depending on the usage Length may be described with symbols such

as x, y, and z (for position); r (for radius); a, b, and c (for the legs of a right triangle); ℓ (for the length of

an object); d (for a distance); h (for a

height); and so forth.

2 The dimensions of a variable will be symbolized by a capitalized, nonitalic letter, such as, in the case of length, L The

symbol for the variable itself will be italicized, such as L for the length of an object or t for time.

TABLE 1.5 |Dimensions and Units of Four Derived Quantities

whether an expression has the correct form because the relationship can be correct only if the dimensions on the two sides of the equation are the same.

To illustrate this procedure, suppose you wish to derive an expression for the

posi-tion x of a car at a time t if the car starts from rest at t 5 0 and moves with constant acceleration a In Chapter 2, we shall fi nd that the correct expression for this special case is x 512at2 Let us check the validity of this expression from a dimensional analy-sis approach

The quantity x on the left side has the dimension of length For the equation to be

dimensionally correct, the quantity on the right side must also have the dimension of length We can perform a dimensional check by substituting the basic dimensions for acceleration, L/T2 (Table 1.5), and time, T, into the equation x 512at2 That is, the

dimensional form of the equation x 512at2 can be written as

[x] 5 L

T2 T25LThe dimensions of time cancel as shown, leaving the dimension of length, which is

the correct dimension for the position x Notice that the number 12 in the equation has no units, so it does not enter into the dimensional analysis

Q U I C K Q U I Z 1.1 True or False: Dimensional analysis can give you the numerical

value of constants of proportionality that may appear in an algebraic expression

1.3 | Conversion of Units

Sometimes it is necessary to convert units from one system to another or to convert within a system, for example, from kilometers to meters Equalities between SI and U.S customary units of length are as follows:

1 mile (mi) 5 1 609 m 5 1.609 km 1 ft 5 0.304 8 m 5 30.48 cm

1 m 5 39.37 in 5 3.281 ft 1 inch (in.) 5 0.025 4 m 5 2.54 cm

A more complete list of equalities can be found in Appendix A

Units can be treated as algebraic quantities that can cancel each other To

per-form a conversion, a quantity can be multiplied by a conversion factor, which is a

frac-tion equal to 1, with numerator and denominator having different units, to provide the desired units in the fi nal result For example, suppose we wish to convert 15.0 in

to centimeters Because 1 in 5 2.54 cm, we multiply by a conversion factor that is the appropriate ratio of these equal quantities and fi nd that

15.0 in 5 (15.0 in.) 12.54 cm

1 in 2538.1 cm

Pitfall Prevention | 1.3 Always Include Units

When performing calculations, make

it a habit to include the units with every quantity and carry the units through the entire calculation Avoid the temptation to drop the units during the calculation steps and then apply the expected unit to the number that results for an answer By including the units in every step, you can detect errors if the units for the answer are incorrect.

Exa m p l e 1.1 | Analysis of an Equation

Show that the expression v 5 at, where v represents speed, a acceleration, and t an instant of time, is dimensionally

Therefore, v 5 at is dimensionally correct because we have the same dimensions on both sides (If the expression were given

as v 5 at 2, it would be dimensionally incorrect Try it and see!)

1.4 | Order-of-Magnitude Calculations 9

1.4 | Order-of-Magnitude Calculations

Suppose someone asks you the number of bits of data on a typical musical compact disc In response, it is not generally expected that you would provide the exact num-ber but rather an estimate, which may be expressed in scientifi c notation The esti-

mate may be made even more approximate by expressing it as an order of magnitude,

which is a power of ten determined as follows:

1 Express the number in scientifi c notation, with the multiplier of the power of ten between 1 and 10 and a unit

2 If the multiplier is less than 3.162 (the square root of ten), the order of tude of the number is the power of ten in the scientifi c notation If the multi-plier is greater than 3.162, the order of magnitude is one larger than the power

magni-of ten in the scientifi c notation

We use the symbol , for “is on the order of.” Use the procedure above to verify the orders of magnitude for the following lengths:

0.008 6 m , 1022m 0.002 1 m , 1023m 720 m , 103mUsually, when an order-of-magnitude estimate is made, the results are reliable to within about a factor of ten If a quantity increases in value by three orders of magni-tude, its value increases by a factor of about 103 5 1 000

where the ratio in parentheses is equal to 1 Notice that we express 1 as 2.54 cm/1 in

(rather than 1 in./2.54 cm) so that the inch cancels with the unit in the original quantity The remaining unit is the centimeter, which is our desired result

QUICK QUIZ 1.2 The distance between two cities is 100 mi What is the number

of kilometers between the two cities? (a) smaller than 100 (b) larger than 100 (c) equal to 100

Convert seconds to hours: (2.36 3 1022 mi/s)1 60 s

1 min2 160 min

1 h 2585.0 mi/hThe driver is indeed exceeding the speed limit and should slow down

What If? What if the driver were from outside the United States and is familiar with speeds measured in kilometers per hour? What is the speed of the car in km/h?

Answer We can convert our fi nal answer to the appropriate units:

(85.0 mi/h)11.609 km

1 mi 25137 km/hFigure 1.2 shows an automobile speedometer displaying speeds in both mi/h and km/h Can you check the conversion we just performed using this photograph?

Figure 1.2 (Example 1.2) The speedometer

of a vehicle that shows speeds in both miles per hour and kilometers per hour.

Exa m p l e 1.3 | The Number of Atoms in a Solid

Estimate the number of atoms in 1 cm3 of a solid

Exa m p l e 1.4 | Breaths in a Lifetime

Estimate the number of breaths taken during an average human lifetime

SOLUTION

We start by guessing that the typical human lifetime is about 70 years Think about the average number of breaths that a person takes in 1 min This number varies depending on whether the person is exercising, sleeping, angry, serene, and so forth To the nearest order of magnitude, we shall choose 10 breaths per minute as our estimate (This estimate is certainly closer to the true average value than an estimate of 1 breath per minute or 100 breaths per minute.)

Find the approximate number of minutes in a year: 1 yr1400 days1 yr 2125 h

1 day2160 min

1 h 256 3 105 minFind the approximate number of minutes in a 70-year number of minutes 5 (70 yr)(6 3 105 min/yr)

Find the approximate number of breaths in a lifetime: number of breaths 5 (10 breaths/min)(4 3 107 min)

5 4 3 108 breathsTherefore, a person takes on the order of 109 breaths in a lifetime Notice how much simpler it is in the fi rst calculation above to multiply 400 3 25 than it is to work with the more accurate 365 3 24

What If? What if the average lifetime were estimated as 80 years instead of 70? Would that change our fi nal estimate?

Answer We could claim that (80 yr)(6 3 105 min/yr) 5 5 3 107 min, so our fi nal estimate should be 5 3 108 breaths

This answer is still on the order of 109 breaths, so an order-of-magnitude estimate would be unchanged

1.5 | Signifi cant Figures

When certain quantities are measured, the measured values are known only to within the limits of the experimental uncertainty The value of this uncertainty can depend

on various factors, such as the quality of the apparatus, the skill of the experimenter,

and the number of measurements performed The number of signifi cant fi gures in a

measurement can be used to express something about the uncertainty The number

of signifi cant fi gures is related to the number of numerical digits used to express the measurement, as we discuss below

As an example of signifi cant fi gures, suppose we are asked to measure the radius

of a compact disc using a meterstick as a measuring instrument Let us assume the accuracy to which we can measure the radius of the disc is 60.1 cm Because of the uncertainty of 60.1 cm, if the radius is measured to be 6.0 cm, we can claim only that its radius lies somewhere between 5.9 cm and 6.1 cm In this case, we say that the

1.5 | Signifi cant Figures 11

measured value of 6.0 cm has two signifi cant fi gures Note that the signifi cant fi gures

include the fi rst estimated digit. Therefore, we could write the radius as (6.0 6 0.1) cm

Zeros may or may not be signifi cant fi gures Those used to position the decimal point in such numbers as 0.03 and 0.007 5 are not signifi cant Therefore, there are one and two signifi cant fi gures, respectively, in these two values When the zeros come after other digits, however, there is the possibility of misinterpretation For example, suppose the mass of an object is given as 1 500 g This value is ambiguous because we do not know whether the last two zeros are being used to locate the deci-mal point or whether they represent signifi cant fi gures in the measurement To re-move this ambiguity, it is common to use scientifi c notation to indicate the number

of signifi cant fi gures In this case, we would express the mass as 1.5 3 103g if there are two signifi cant fi gures in the measured value, 1.50 3 103g if there are three sig-nifi cant fi gures, and 1.500 3 103g if there are four The same rule holds for numbers less than 1, so 2.3 3 1024has two signifi cant fi gures (and therefore could be written 0.000 23) and 2.30 3 1024has three signifi cant fi gures (also written as 0.000 230)

In problem solving, we often combine quantities mathematically through tiplication, division, addition, subtraction, and so forth When doing so, you must make sure that the result has the appropriate number of signifi cant fi gures A good rule of thumb to use in determining the number of signifi cant fi gures that can be claimed in a multiplication or a division is as follows:

mul-When multiplying several quantities, the number of signifi cant fi gures in the fi nal answer is the same as the number of signifi cant fi gures in the quantity having the smallest number of signifi cant fi gures The same rule applies to division

Let’s apply this rule to fi nd the area of the compact disc whose radius we sured above Using the equation for the area of a circle,

mea-A 5 r2 5 (6.0 cm)2 5 1.1 3 102 cm2

If you perform this calculation on your calculator, you will likely see 113.097 335 5

It should be clear that you don’t want to keep all of these digits, but you might be tempted to report the result as 113 cm2 This result is not justifi ed because it has three signifi cant fi gures, whereas the radius only has two Therefore, we must report the result with only two signifi cant fi gures as shown above

For addition and subtraction, you must consider the number of decimal places when you are determining how many signifi cant fi gures to report:

When numbers are added or subtracted, the number of decimal places in the sult should equal the smallest number of decimal places of any term in the sum

re-or difference

As an example of this rule, consider the sum

23.2 1 5.174 5 28.4Notice that we do not report the answer as 28.374 because the lowest number of deci-mal places is one, for 23.2 Therefore, our answer must have only one decimal place

The rules for addition and subtraction can often result in answers that have a different number of signifi cant fi gures than the quantities with which you start For example, consider these operations that satisfy the rule:

1.000 1 1 0.000 3 5 1.000 41.002 2 0.998 5 0.004

In the fi rst example, the result has fi ve signifi cant fi gures even though one of the terms, 0.000 3, has only one signifi cant fi gure Similarly, in the second calculation, the result has only one signifi cant fi gure even though the numbers being subtracted have four and three, respectively

Pitfall Prevention | 1.4 Read Carefully

Notice that the rule for addition and subtraction is different from that for multiplication and division

For addition and subtraction, the important consideration is the

number of decimal places, not the number of signifi cant fi gures.

In this book, most of the numerical examples and end-of-chapter problems will yield answers having three signifi cant fi gures When carrying out estima-tion calculations, we shall typically work with a single signifi cant fi gure.

If the number of signifi cant fi gures in the result of a calculation must be reduced, there is a general rule for rounding numbers: the last digit retained is increased by 1

if the last digit dropped is greater than 5 (For example, 1.346 becomes 1.35.) If the last digit dropped is less than 5, the last digit retained remains as it is (For example, 1.343 becomes 1.34.) If the last digit dropped is equal to 5, the remaining digit should be rounded to the nearest even number (This rule helps avoid accumulation

of errors in long arithmetic processes.)

A technique for avoiding error accumulation is to delay the rounding of numbers

in a long calculation until you have the fi nal result Wait until you are ready to copy the

fi nal answer from your calculator before rounding to the correct number of signifi cant

fi gures In this book, we display numerical values rounded off to two or three signifi cant fi gures This occasionally makes some mathematical manipulations look odd or incorrect For instance, looking ahead to Example 1.8 on page 21, you will see the op-eration 217.7 km 1 34.6 km 5 17.0 km This looks like an incorrect subtraction, but that is only because we have rounded the numbers 17.7 km and 34.6 km for display

-If all digits in these two intermediate numbers are retained and the rounding is only performed on the fi nal number, the correct three-digit result of 17.0 km is obtained

c Signifi cant fi gure guidelines used in this book

Pitfall Prevention | 1.5 Symbolic Solutions

When solving problems, it is very useful to perform the solution completely in algebraic form and wait until the very end to enter numerical values into the fi nal symbolic expression This method will save many calculator keystrokes, especially

if some quantities cancel so that you never have to enter their values into your calculator! In addition, you will only need to round once, on the

fi nal result.

1.6 | Coordinate Systems

Many aspects of physics deal in some way or another with locations in space For ample, the mathematical description of the motion of an object requires a method for specifying the object’s position Therefore, we fi rst discuss how to describe the position of a point in space by means of coordinates in a graphical representation

ex-A point on a line can be located with one coordinate, a point in a plane is located with two coordinates, and three coordinates are required to locate a point in space

A coordinate system used to specify locations in space consists of

• A fi xed reference point O, called the origin

• A set of specifi ed axes or directions with an appropriate scale and labels on the axes

• Instructions that tell us how to label a point in space relative to the origin and axes

One convenient coordinate system that we will use frequently is the Cartesian

co-ordinate system, sometimes called the rectangular coordinate system Such a system in two

dimensions is illustrated in Figure 1.3 An arbitrary point in this system is labeled with

the coordinates (x, y) Positive x is taken to the right of the origin, and positive y is upward from the origin Negative x is to the left of the origin, and negative y is down- ward from the origin For example, the point P, which has coordinates (5, 3), may be

reached by going fi rst 5 m to the right of the origin and then 3 m above the origin

Exa m p l e 1.5 | Installing a Carpet

A carpet is to be installed in a rectangular room whose length is measured to be 12.71 m and whose width is measured to

be 3.46 m Find the area of the room

SOLUTION

If you multiply 12.71 m by 3.46 m on your calculator, you will see an answer of 43.976 6 m2 How many of these numbers should you claim? Our rule of thumb for multiplication tells us that you can claim only the number of signifi cant fi gures in your answer as are present in the measured quantity having the lowest number of signifi cant fi gures In this example, the lowest number of signifi cant fi gures is three in 3.46 m, so we should express our fi nal answer as 44.0 m2

Figure 1.3 Designation of points

in a Cartesian coordinate system

Each square in the xy plane is 1 m

on a side Every point is labeled with

O

1.7 | Vectors and Scalars 13

(or by going 3 m above the origin and then 5 m to the right) Similarly, the point Q

has coordinates (23, 4), which correspond to going 3 m to the left of the origin and

4 m above the origin

Sometimes it is more convenient to represent a point in a plane by its plane polar

coordinates (r, ), as in Active Figure 1.4a In this coordinate system, r is the length

of the line from the origin to the point, and  is the angle between that line and a

fi xed axis, usually the positive x axis, with  measured counterclockwise From the

right triangle in Active Figure 1.4b, we fi nd that sin  5 y/r and cos  5 x/r (A

review of trigonometric functions is given in Appendix B.4.) Therefore, starting with plane polar coordinates, one can obtain the Cartesian coordinates through the equations

coor-an coor-angle measured counterclockwise from the positive x axis Other choices are made in

navigation and astronomy If the reference axis for the polar angle  is chosen to be

other than the positive x axis or if the sense of increasing  is chosen differently, the

corresponding expressions relating the two sets of coordinates will change

1.7 | Vectors and Scalars

Each of the physical quantities that we shall encounter in this text can be placed in

one of two categories, either a scalar or a vector A scalar is a quantity that is

com-pletely specifi ed by a positive or negative number with appropriate units On the

other hand, a vector is a physical quantity that must be specifi ed by both magnitude

represented by the distance r and

the angle , where  is measured in a

counterclockwise direction from the

positive x axis (b) The right triangle used to relate (x, y) to (r, ).

O

(x, y) y

x r

x

sin = y rcos = xr tan = x y

u u

u

a

b

Figure 1.5 (a) The number of grapes in this bunch is one example of a scalar quantity

Can you think of other examples? (b) This helpful person pointing in the correct direction tells us to travel fi ve blocks north to reach the courthouse A vector is a physical quantity that is specifi ed by both magnitude and direction.

Mack Henley/Visuals Unlimited, Inc © Cengage Learning/George Semple

the information; no specifi cation of direction is required Other examples of scalars are temperature, volume, mass, and time intervals The rules of ordi-nary arithmetic are used to manipulate scalar quantities; they can be freely added and subtracted (assuming that they have the same units!), multiplied and divided.

Force is an example of a vector quantity To describe the force on an object pletely, we must specify both the direction of the applied force and the magnitude of the force

com-Another simple example of a vector quantity is the displacement of a particle,

de-fi ned as its change in position The person in Figure 1.5b is pointing out the direction

of your desired displacement vector if you would like to reach a destination such as the courthouse She will also tell you the magnitude of the displacement along with the direction, for example, “5 blocks north.”

Suppose a particle moves from some point 𝖠 to a point 𝖡 along a straight path, as in Figure 1.6 This displacement can be represented by drawing an arrow from 𝖠 to 𝖡, where the arrowhead represents the direction of the dis-placement and the length of the arrow represents the magnitude of the displace-ment If the particle travels along some other path from 𝖠 to 𝖡, such as the broken line in Figure 1.6, its displacement is still the vector from 𝖠 to 𝖡 The vector displacement along any indirect path from 𝖠 to 𝖡 is defined as being equivalent to the displacement represented by the direct path from 𝖠 to 𝖡 The magnitude of the displacement is the shortest distance between the end points

Therefore, the displacement of a particle is completely known if its initial and final coordinates are known. The path need not be specified In other words,

the displacement is independent of the path if the end points of the path

are fixed

Note that the distance traveled by a particle is distinctly different from its

displace-ment The distance traveled (a scalar quantity) is the length of the path, which in general can be much greater than the magnitude of the displacement In Figure 1.6, the length of the curved broken path is much larger than the magnitude of the solid black displacement vector

If the particle moves along the x axis from position x i to position x f, as in

Fig-ure 1.7, its displacement is given by x f 2 x i (The indices i and f refer to the initial and fi nal values.) We use the Greek letter delta (D) to denote the change in a quan-

tity Therefore, we defi ne the change in the position of the particle (the ment) as

From this defi nition we see that Dx is positive if x f is greater than x i and negative if

x f is less than x i For example, if a particle changes its position from x i 5 25 m to

x f 5 3 m, its displacement is Dx 5 1 8 m

Many physical quantities in addition to displacement are vectors They include velocity, acceleration, force, and momentum, all of which will be defi ned in later chapters In this text, we will use boldface letters with an arrow over the letter, such as

A

:

, to represent vectors Another common notation for vectors with which you should

be familiar is a simple boldface character: A

The magnitude of the vector A: is written with an italic letter A or, alternatively,

u A:u The magnitude of a vector is always positive and carries the units of the quantity that the vector represents, such as meters for displacement or meters per second for velocity Vectors combine according to special rules, which will be discussed in Sections 1.8 and 1.9

QUICK QUIZ 1.3 Which of the following are vector quantities and which are

scalar quantities? (a) your age (b) acceleration (c) velocity (d) speed (e) mass

c Displacement

c Distance

Figure 1.6 After a particle moves from 𝖠 to 𝖡 along an arbitrary path represented by the broken line, its displacement is a vector quantity shown by the arrow drawn from 𝖠 to 훾.

훽

훾

Figure 1.7 A particle moving along

the x axis from x i to x f undergoes a

1.8 | Some Properties of Vectors 15

1.8 | Some Properties of Vectors

Equality of Two Vectors

Two vectors A: and B: are defi ned to be equal if they have the same units, the same

magnitude, and the same direction That is, A: 5 B: only if A 5 B and A: and B: point

in the same direction For example, all the vectors in Figure 1.8 are equal even though they have different starting points This property allows us to translate a vec-tor parallel to itself in a diagram without affecting the vector

Addition

The rules for vector sums are conveniently described using a graphical method

To add vector B: to vector A:, fi rst draw a diagram of vector A:on graph paper, with its

mag nitude represented by a convenient scale, and then draw vector B: to the same scale

with its tail starting from the tip of A:, as in Active Figure 1.9a The resultant vector

Reasoning Unless you have a very unusual commute, the distance traveled must

be larger than the magnitude of the displacement vector The distance includes the results of all the twists and turns you make in following the roads from home

to work or school On the other hand, the magnitude of the displacement vector

is the length of a straight line from your home to work or school This length is often described informally as “the distance as the crow fl ies.” The only way that the distance could be the same as the magnitude of the displacement vector is

if your commute is a perfect straight line, which is highly unlikely! The distance

could never be less than the magnitude of the displacement vector because the

shortest distance between two points is a straight line b

‹ ‡

A

R A B

S

B

S

b

Active Figure 1.9 (a) When vector B:is added to vector A:, the resultant R:is the

vector that runs from the tail of A:to the tip of B: (b) This construction shows that

A 1 B:5 B:1 A:; vector addition is commutative.

Pitfall Prevention | 1.6 Vector Addition Versus Scalar Addition

Keep in mind that A:1 B:5 C: is very

different from A 1 B 5 C The fi rst

equation is a vector sum, which must

be handled carefully, such as with the graphical method described in Active Figure 1.9 The second equation is a simple algebraic addition of numbers that is handled with the normal rules

of arithmetic.

If three or more vectors are added, their sum is independent of the way in which they are grouped A geometric demonstration of this property for three vectors is

given in Figure 1.10 This property is called the associative law of addition:

A

:

1(B:1 C:) 5 (A:1 B:) 1 C: 1.8bGeometric constructions can also be used to add more than three vectors,

as shown in Figure 1.11 for the case of four vectors The resultant vector R: 5 A: 1

B

:

1 C: 1 D: is the vector that closes the polygon formed by the vectors being added

In other words, R: is the vector drawn from the tail of the fi rst vector to the tip of the last vector Again, the order of the summation is unimportant

In summary, a vector quantity has both magnitude and direction and also obeys the laws of vector addition as described in Active Figure 1.9 and Figures 1.10 and 1.11 When two or more vectors are added together, they must all have the same units and they must all be the same type of quantity It would be meaningless to add

a velocity vector (for example, 60 km/h to the east) to a displacement vector (for example, 200 km to the north) because these vectors represent different physical quantities The same rule also applies to scalars For example, it would be meaning-less to add time intervals to temperatures

Negative of a Vector

The negative of the vector A: is defi ned as the vector that, when added to A:, gives

zero for the vector sum That is, A: 1 (2A:) 5 0 The vectors A: and 2A: have the same magnitude but point in opposite directions

Subtraction of Vectors

The operation of vector subtraction makes use of the defi nition of the negative of a

vector We defi ne the operation A:2 B: as vector 2B: added to vector A::

A

:

The geometric construction for subtracting two vectors is illustrated in Figure 1.12

Multiplication of a Vector by a Scalar

If a vector A: is multiplied by a positive scalar quantity s, the product sA: is a vector

that has the same direction as A: and magnitude sA If s is a negative scalar quantity, the

vector s A: is directed opposite to A: For example, the vector 5A: is fi ve times longer

than A: and has the same direction as A:; the vector 213A: has one-third the

magni-tude of A: and points in the direction opposite A:

Multiplication of Two Vectors

Two vectors A: and B: can be multiplied in two different ways to produce either a

sca-lar or a vector quantity The scasca-lar product (or dot product) A: ? B:is a scalar quantity

equal to AB cos , where  is the angle between A: and B: The vector product (or cross product) A: 3 B: is a vector quantity whose magnitude is equal to AB sin  We shall

discuss these products more fully in Chapters 6 and 10, where they are fi rst used

QUICK QUIZ 1.4 The magnitudes of two vectors A: and B: are A 5 12 units and

B 5 8 units Which pair of numbers represents the largest and smallest possible values

for the magnitude of the resultant vector R: 5 A: 1 B:? (a) 14.4 units, 4 units (b) 12 units, 8 units (c) 20 units, 4 units (d) none of these answers

QUICK QUIZ 1.5 If vector B: is added to vector A:, under what condition does the

resultant vector A: 1 B:have magnitude A 1 B ? (a) A: and B: are parallel and in the

same direction (b) A: and B: are parallel and in opposite directions (c) A: and B: are perpendicular

Figure 1.10 Geometric constructions for verifying the associative law of addition.

Add and ; then add the result to .A

A A A

B

B B

Figure 1.11 Geometric construction for summing four vectors The

resultant vector R:closes the polygon and points from the tail of the fi rst vector to the tip of the fi nal vector.

A

B C

DS

A B

B

ˆ

A B

We would draw here if we were adding it to .

Adding ˆ to

is equivalent to

subtracting from

B

A A B

1.9 | Components of a Vector and Unit Vectors 171.9 | Components of a Vector and Unit Vectors

The graphical method of adding vectors is not recommended whenever high accuracy is required or in three-dimensional problems In this section, we de-scribe a method of adding vectors that makes use of the projections of vectors

along coordinate axes These projections are called the components of the tor or its rectangular components Any vector can be completely described by its

vec-components

Consider a vector A: lying in the xy plane and making an arbitrary angle 

with the positive x axis as shown in Figure 1.13a This vector can be expressed as

the sum of two other component vectors A:x , which is parallel to the x axis, and A:y, which

is parallel to the y axis From Figure 1.13b, we see that the three vectors form a right

triangle and that A:5 A:x1 A:y We shall often refer to the “components of a vector

A

:

,” written A x and A y (without the boldface notation) The component A x

rep-resents the projection of A: along the x axis, and the component A y represents

the projection of A: along the y axis These components can be positive or negative

The component A x is positive if the component vector A:x points in the positive x

direction and is negative if A:x points in the negative x direction A similar statement

is made for the component A y.From Figure 1.13b and the defi nition of the sine and cosine of an angle, we see that cos  5 A x /A and sin  5 A y /A Hence, the components of A: are given by

A x 5 A cos  and A y 5 A sin  1.10bThe magnitudes of these components are the lengths of the two sides of a right tri-

angle with a hypotenuse of length A Therefore, the magnitude and direction of A:

are related to its components through the expressions

To solve for , we can write  5 tan21 (A y /A x), which is read “ equals the angle whose

tangent is the ratio A y /A x ” Note that the signs of the components A x and A y depend on the angle  For example, if  5 1208, A x is negative and A y is positive If  5 2258, both A x and A y are negative Figure 1.14 summarizes the signs of the components when

A

:

lies in the various quadrants

If you choose reference axes or an angle other than those shown in Figure 1.13, the components of the vector must be modifi ed accordingly In many applications, it

is more convenient to express the components of a vector in a coordinate system ing axes that are not horizontal and vertical but are still perpendicular to each other

hav-c Magnitude of :

c Direction of :

Figure 1.14 The signs of the

components of a vector A:depend on the quadrant in which the vector is located.

Equation 1.10 associates the cosine

of the angle with the x component

and the sine of the angle with the

y component This association is

true only because we measured the

angle  with respect to the x axis, so

do not memorize these equations

If  is measured with respect to the

y axis (as in some problems), these equations will be incorrect Think about which side of the triangle containing the components is adjacent to the angle and which side

is opposite and then assign the cosine and sine accordingly.

Figure 1.13  (a) A vector A:lying in the xy plane can be represented by

its component vectors A:x and A:y (b) The y component vector A:y can be

moved to the right so that it adds to A:x The vector sum of the component

vectors is A: These three vectors form a right triangle.

y

x O

Suppose a vector B:makes an angle 9 with the x9 axis defi ned in Figure 1.15 The

com-ponents of B:along these axes are given by B x9 5 B cos 9 and B y9 5 B sin 9, as in

Equa-tion 1.10 The magnitude and direcEqua-tion of B:are obtained from expressions equivalent

to Equations 1.11 and 1.12 Therefore, we can express the components of a vector in

any coordinate system that is convenient for a particular situation

Q U I C K Q U I Z 1.6 Choose the correct response to make the sentence true: A

component of a vector is (a) always, (b) never, or (c) sometimes larger than the

magnitude of the vector

Unit Vectors

Vector quantities often are expressed in terms of unit vectors A unit vector is a

dimensionless vector having a magnitude of exactly 1 Unit vectors are used to specify

a given direction and have no other physical signifi cance We shall use the symbols iˆ,

jˆ , and kˆ to represent unit vectors pointing in the x, y, and z directions, respectively

The “hat” over the letters is a common notation for a unit vector; for example, iˆ is called “i-hat.” The unit vectors iˆ, jˆ, and k ˆ form a set of mutually perpendicular vectors

as shown in Active Figure 1.16a, where the magnitude of each unit vector equals 1;

that is, u iˆ u 5 u jˆ u 5 u k ˆ u 5 1.

Consider a vector A:lying in the xy plane, as in Active Figure 1.16b The product of the component A x and the unit vector iˆ is the component vector A:x5A xiˆ, which lies

on the x axis and has magnitude A x Likewise, A y jˆ is a component vector of magnitude

A y lying on the y axis Therefore, the unit- vector notation for the vector A:is

A

:

Now suppose we wish to add vector B:to vector A:, where B:has components B x and

B y The procedure for performing this sum is simply to add the x and y com ponents

separately The resultant vector R: 5 A:1 B:is therefore

R

:

5(A x1B x ) iˆ 1 (A y1B y) jˆ 1.14bFrom this equation, the components of the resultant vector are given by

R x5A x1B x

1.15b

R y5A y1B y

Therefore, we see that in the component method of adding vectors, we add all the

x components to fi nd the x component of the resultant vector and use the same

Active Figure 1.16 (a) The unit

vectors iˆ, jˆ, and kˆ are directed along

the x, y, and z axes, respectively

(b) A vector A: lying in the xy plane has component vectors A xiˆ and A y jˆ,

where A x and A y are the components

z

ˆj ˆi ˆk

A

Figure 1.15 The component vectors

of vector B:in a coordinate system that is tilted.

B

S