Physics: An Algebra-Based Approach [1 ed.] 0176531866, 9780176531867

Citation preview

McFARLAND

HIRSCH

_O’MEARA

SI BASE UNITS There are seven SI base units, one for each of length, mass, time, electric current, temperature, amount of substance, and luminous inten-

sity. The following definitions of the base units were adopted by the Conférence générale des poids et mesures in the years indicated.

owl [Kitogram |ke

“The metre is the length of the path travelled by light in a vacuum during a time interval of 1/(299 792 458) of a second” (adopted in 1983).

“The kilogram is...the mass of the international prototype of the kilogram” (adopted in 1889 and 1901). “The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom” (adopted in 1967).

ampere

“The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed one metre apart in vacuum, would produce between these conductors a force equal to 2 X 10°’ N/m (newtons per metre of length)” (adopted in 1948).

“The kelvin...is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water” (adopted in 1967).

“The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kg of carbon-12” (adopted in 1967). candela

“The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 5.40 x 10'4 Hz and that has a radiant intensity in that direction of 1/683 watt per steradian” (adopted in 1979).

SOME Si DERIVED UNITS Derived units are formed from products and ratios of the base units. Some derived units are given special names, such as the newton or joule, to honour certain scientists.

Derived Unit

Quantity

Name

Symbol

area volume

speed

m/s or m-s_!

density

kg/m?

acceleration

Es

force

kg-m/s?

newton

energy

kg-m?/s? or N-m

joule

power

J/s

watt

frequency

I/s ors"!

hertz

pressure

N/m?

pascal coulomb

electric charge

electric potential electric resistance

V/A

ohm

electric capacitance

Cry

farad

magnetic field

N-A’!-m"!

tesla

(radio )activity

I/s ors"!

becquerel

PEEP ETP

NEL

S! PREFIXES Factor

| Prefix

10%

| Symbol

|

Origin

yotta

Italian ott(o) — eight (3 X 8 = 24)

zetta

Italian etta — seven (3 X 7 = 21)

peta

Greek peta — spread out

10° 10° 10°

giga mega kilo

Greek gigas — giant Greek mega — great Greek khilioi — thousand

10°

hecto

‘=

deca*

ex

tera

reek hekaton — hundred reek deka — ten

— =)

deci

Latin decimus — tenth

Om

centi

Latin centum — hundred

102

10% 10

milli

Latin mille — thousand

107}5

femto

Greek femten — fifteen

LO

atto

Danish atten — eighteen

105%

zepto

Greek micros — very small

107?

nano

Greek nanos — dwarf

Italian piccolo — small

Z

Greek (h)epto — seven (—3 X 7 = —21)

Greek octo — eight (~3 x 8 = ~24)

“Alternative spelling: deka.

Greek Alphabet Greek Letter | Name

| Equivalent |Sound When Spoken

Reon

eat

ee

[Sa

Bea

aya

a ae Ee

eal

Zeta

N IN 3 i 1m

9)

md

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zay-tah E

ay-tay

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Web Increased Engagement.

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Improved Outcomes.

A

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NELSON

PHYSICS

AN ALGEBRA-BASED APPROACH

Digitized by the Internet Archive in 2023 with funding from Kahle/Austin Foundation

https://archive.org/details/physicsalgebraba0000erni

PHYSICS AN ALGEBRA-BASED APPROACH

Ernie McFarland University of Guelph

Alan J. Hirsch

Joanne M. O’Meara University of Guelph

NELSON

NELSON Physics: An Algebra-Based Approach by Ernie McFarland, Alan J. Hirsch, and Joanne M. O'Meara Vice President, Editorial

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Library and Archives Canada Cataloguing in Publication

McFarland, Ernie, author Physics: an algebra-based approach / Ernie McFarland (University of Guelph), Alan J. Hirsch, Joanne M. O'Meara (University of Guelph). — First edition. Includes index. ISBN 978-0-17-653186-7 (bound)

1. Mathematical physics— Textbooks. 2. Algebraic logic— Textbooks. |. Hirsch, Alan J., author Il. O'Meara, Joanne M. QVoanne Michelle), author Ill. Title.

QC20.7.A4M34 2015 530.15 C2014-908371-8 ISBN-13: 978-0-17-653186-7 ISBN-10: 0-17-653186-6

| BRIEF TABLE TABLE OF OFCONTENTS CONTENTS |

Preface for Instructors About the Authors

xiv

xix

CHAPTER 1

Measurement and Types of Quantities

CHAPTER 2

One-Dimensional Kinematics

CHAPTER 3

Vectors and Trigonometry

CHAPTER 4

Two-Dimensional Kinematics

CHAPTER 5

Newton's Laws of Motion

105

CHAPTER 6

Applying Newton's Laws

139°

CHAPTER 7

Work, Energy, and Power

161

CHAPTER 8

Momentum and Collisions

CHAPTER 9

Gravitation

CHAPTER 1Q~

Rotational Motion

CHAPTER 11

Statics, Stability, and Elasticity

CHAPTERI12

Fluid Statics and Dynamics

CHAPTER13

Oscillations and Waves

CHAPTER 14

Soundand Music

CHAPTER 15

‘Reflection, Refraction, and Dispersion of Light

CHAPTER 16

Wave Optics

CHAPTER17

OpticalInstruments

CHAPTER 18

MHeatand Thermodynamics

CHAPTER 19

Electric Charge and Electric Field

CHAPTER 20

Electric Potential Energy, Electric Potential, and Current

CHAPTER 21

‘Electrical Resistance and Circuits

CHAPTER22

Magnetism

CHAPTER 23

Electromagnetic Induction

CHAPTER 24

Nuclear Physics: Theory and Medical Applications

Glossary

1

21

59 73

195

223 255 287

325

373

413 467

509 551 591 627 651

677

705 735 763

807

Appendix A: Answers to Exercises and End-of-Chapter Questions

Index NEL

823

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|CONTENTS | Preface for Instructors

xiv

2.5

Acceleration Due to Gravity

G

About

the Authors

CHAPTER

xix

Terminal Speed

1

CHAPTER REVIEW

Measurement in the Past

128

3

3.1 3

5

3.2

Conversions within the Metric System

8

oO

60

3.3 13

Sine Law

ScalarsandVectors

14

Cosine Law

16

LOOKING BACK...LOOKING AHEAD

3.4

65

Adding Vectors

Ee

66

Adding and Subtracting Vectors

66

Using Components

17

63

64

Trigonometric Identities

17

62

Vector Subtraction 62 Trigonometry 63 Sine, Cosine, and Tangent

14

Order-of-Magnitude Estimations

67

Subtracting Vectors

69

ee Vien One-Dimensional

a 3 Ba

LOOKING BACK...LOOKING AHEAD te

2.1

Distance andSpeed 22 Instantaneous Speed 24 Uniform Motion 24

ar

CHAPTER 4 Two-Dimensional Kinematics 73

2.2

Position, Displacement, and Velocity Average Velocity 26

Rinomaticcmel

=|

4.2 36

Solving Constant-Acceleration Problems 38 Equations of Kinematics for Constant Acceleration 38 Applying the Kinematics Equations 40

gs ae

74

Velocity in Two Dimensions

34

oO

: 5 Displacement and Velocity in

Instantaneous Velocity

Variable (Non-Constant) Acceleration

NEL

ce

é. 85

Displacement in Two Dimensions 29

70

70

Two Dimensions

32

Constant Acceleration

2.4

4.1

27

Graphing Non-Uniform Motion

Acceleration

25

CHAPTERREVIEW

27

Instantaneous Velocity Graphing Uniform Motion 2.3

60

Adding and Subtracting Vectors Using Scale Diagrams 61 Properties of Vector Addition

10

Calculations Based on Measurements

CHAPTER REVIEW

36 g S

60

Multiplying a Vector by a Scalar

Derived Units 6 Metric Prefixes 8

1.5

General Properties of Vectors

Vector Symbols 60 Directions of Vectors

6

Dimensional Analysis Significant Digits 12

5S

59

Vectors and Trigonometry

Metric Units, Prefixes, and Conversions Base Units in the Metric System 5

Round-off Error

ae

2

Measurement in Science and Physics

1.3. 1.4

52

CHAPTER 3

2

Scientific Notation

51

Kameel4u/Shutterstock

Measurement in Our Daily Lives

1.2

49

LOOKING BACK...LOOKING AHEAD

The Importance of

Measurement

44

Calculations Involving the Acceleration Due to Gravity 46

Measurement and Types of Quantities 1 1.1

44

Measuring the Acceleration Due to Gravity

74

76

77

Acceleration inTwo Dimensions

79

iistamaneous nccele/averier2e

4.3 4.4

Introduction to Projectile Motion 81 Solving Projectile Motion Problems 86 Maximum Horizontal Range 88

Vil

Viil

4.5

CONTENTS

Uniform Circular Motion

90

The Direction of Centripetal Acceleration

The Magnitude of Centripetal Acceleration 4.6

Velocity CHAPTER REVIEW

CHAPTER

91

7.4

Work Done By Friction

176

7.6

OtherTypes of Energy

178

Elastic Potential Energy in Sports

105

Energy Units

7.7

Gravity, Normal Force, Friction, andTension 106 Free-Body Diagrams 107

5.3

Newton’s Second Law of Motion

Schurt/Shutterstock Stefan

7.8

110

5.5

Solving Problems Using Newton’s First and Second Laws 116

CHAPTER REVIEW

114

CHAPTER

The Fundamental Forces

LOOKING BACK...LOOKING AHEAD

8.1

128

188

189

8

132

195

196

133

Average Force

140

198

Conservation of Momentum Dimension 201

Elastic and Inelastic Collisions

8.4

Conservation of Momentum Dimensions 213

LOOKING BACK...LOOKING AHEAD

6.3

Centrifugal “Force”

in One

8.3

Gunn/ Mitch Shutterstock.com

145

Forces in Circular Motion

SurangaSL/Shu

198

Applications of Conservation of Momentum

139

140

6.2

196

Contact Time during a Collision

8.2

Static and Kinetic Friction

Momentum

Momentum, Force, and Time

CHAPTER 6

146

CHAPTER REVIEW

204

207

in Two 217

217

151

Inertial and Noninertial Frames of Reference

Centrifuges and Centrifugal “Force”

152

Earth’s Surface as a Noninertial Frame of Reference 154

LOOKING BACK...LOOKING AHEAD

151

CHAPTER

Work, Energy, and Power

9.1

223

Law of Universal Gravitation

224

Determination of G 227

155

MarcelC/Think

Gravitational Force between Extended Objects 227

9.2

Gravity Due to Planets and Stars Variations ing on Earth

161 9.3

Gravitational Field

228

231

232

Calculating Gravitational Force from Gravitational Field 234

Work Done by a Constant Force 162

Chiyacat/Shutterstock

9.4

Using Vector Components to Determine Work 164 Work Done in Circular Motion

9

Gravitation

155

CHAPTER 7

7.1

186

Momentum and Collisions

134

Applying Newton's Laws

CHAPTER REVIEW

183

186

123

Newton’s Third Law of Motion

Controlling Friction

Efficiency of Energy Conversions

LOOKING BACK...LOOKING AHEAD

112

Newton’s First Law of Motion

Coefficients of Friction

182

Energy Sources and Conservation

5.4

6.1

181

Energy Return on Investment

More Problem-Solving

178

180

Kilowatt-Hour: An Energy Unit

108

Gravitational Acceleration and Force

CHAPTER REVIEW

Power

173

175

7.5

5

170

172

Law of Conservation of Energy Mechanical Energy

100

Adding Forces

5.8

Gravitational Potential Energy

99

5.2

5.7

7.3

Choosing the y = 0 Position

94

Newton's Laws of Motion

5.6

Kinetic Energy and the Work-Energy Theorem 167

Frames of Reference and Relative

LOOKING BACK...LOOKING AHEAD

5.1

7.2 90

Orbits, Kepler’s Laws, and Weightlessness Keplers Laws

166

Weightlessness

235

237 239 NEL

CONTENTS

9.5

Gravitational Potential Energy in General

CHAPTER

12

Fluid Statics and Dynamics 325

Escape Speed 245 Black Holes 247

Einstein’s Theory of Gravity

248

LOOKING BACK...LOOKING AHEAD CHAPTER REVIEW

242

12.1

249

Properties of Fluids Specific Gravity

250

Pressure

CHAPTER

10

326

Rotational Motion

255

328

Pressure in Liquids

Rotational Kinematics

10.2

Relations between Rotational and Linear Quantities 262 Tangential Acceleration

255

U-Tube Manometer TempSport/Corbis lundt/ Dimitri ©

12.2

263

265

Kinetic Energy of Rolling Objects

10.4

270

Angular Momentum

12.4

275

12.5

277

LOOKING BACK...LOOKING AHEAD

279

279

il

Centre of Mass

288

arindambanerjee/ Shutterstock.com

11.3

Stability 300 Stability and the Human Body 300 Other Examples of Stability 301

291

13.1

303

307

Shear and Bulk Stress and Strain;

309

Torsion of Cylinders 310 Bulk Stress and Bulk Strain

312

Stress and Strain in Architecture

313

314

LOOKING BACK...LOOKING AHEAD

NEL

362

363

Drag Force in General

355

316

367

368

13

373

Oscillations and Simple Harmonic Motion 374

StockSh ©

Period, Frequency, and Phase 304

359

365

Oscillations and Waves

Tensile Stress and Strain; Young’s Modulus

317

351

359

Sedimentation

CHAPTER

292

Stress and Strain; Elasticity

CHAPTER REVIEW

349

350

358

Turbulent Flow

CHAPTER REVIEW

Static Equilibrium

Spanning a Space

349

LOOKING BACK...LOOKING AHEAD

291

Elasticity and Plasticity

Viscosity

Poiseuille’s Law

11.2

Architecture

Fluid Dynamics

Newtonian and Non-Newtonian Fluids

Finding Centre of Mass Experimentally

11.5

344

Buoyancy and Archimedes’ Principle 346

Applications of Bernoulli’s Equation

12.6

Centre of Mass and Newton’s Laws of Motion 290

11.4

340

342

Bernoulli's Equation

280

Centre of Gravity

SurfaceTension

Equation of Continuity

Statics, Stability, and Elasticity 287 11.1

336

Streamline Flow and Ideal Fluids

Angular Momentum and Kinetic Energy

CHAPTER

333

Pascal’s Principle; Pressure in the Atmosphere 335

Capillary Action

Torque and Newton’s Second Law for Rotation 272

CHAPTER REVIEW

Measuring Blood Pressure

Measuring Atmospheric Pressure

12.3

Newton’s Second Law for Rotation

10.5

331

332

Pressure in the Atmosphere

Rotational Kinetic Energy and Moment of Inertia 267

329

329

Other Units of Pressure

10.1

10.3

endopac Think

327

Gauge Pressure and Absolute Pressure

Rolling Objects

IX

The Reference Circle

376

376

13.2

The Simple Pendulum

379

13.3

The Sinusoidal Nature of Simple Harmonic Motion 381

13.4

Energy in Simple Harmonic Motion; Damping; Resonance 386 Conservation of Mechanical Energy in SHM 386 Damped Harmonic Motion Resonance

390

389

X

CONTENTS

13.5

13.6

Energy Transfer with Waves 394 Transverse and Longitudinal Waves 394

CHAPTER 15 Reflection, Refraction, and

an Es3 2

Oceanic and Seismic Waves

Dispersion

E§5 F

PeriodicWaves

395

398

15.1

Reflection and Transmission of Waves

399

Diffraction of Waves

CHAPTER REVIEW

468

403 15.2

407

Measuring the Speed of Light

472

Mirrors and Their images

475

Rg ae

408

cele

S

14

14.2

=

413

9 $F

The Nature of Sound

414

Sound Transmission in Air 414 The Speed of Sound in Air 415 The Speed of Sound in G | 416 : eae he gn teal?

Hearing and the Intensity of Sound

ee A The Human Audible Range 423 Intensity and Loudness a Sound

14.4

The Doppler Effect and Subsonic and Supersonic Speeds in Sound 432

Snell’s Law of Refraction

Dispersion of Light Mirages 489

487

488

: Total Internal Reflection ET nanoee 1c

490

Diamonds and Other Jewels Fibre Optics 493

493

428

15.5

Lenses andTheir Images

432

Wena Aberrations

500

Subsonic and Supersonic Speeds

437

LOOKING BACK..LOOKING AHEAD

Diffraction, Interference, and Resonance : inSound 439

439

CHAPTER

Tonal Quality

14.8

Musical Instruments and Resonance in Stringed and Wind Instruments 447

442

er

Stringed Instruments

16.1

Acoustics

455

CHAPTER REVIEW

:

456

}

i

514

516

S17

Single-Slit Diffraction and Resolution A Qualitative Look at Single-Slit Diffraction

461

:E ZS

Interference Far from the

Measuring the Wavelengths of Visible Light

16.3

459

2

Double-Slit Interference of

Sources

457

462

;

=

509

Diffraction of Light and Huygens’ Principle 510

Two-Source

455

LOOKING BACK...LOOKING AHEAD

16

Light 511 Coherence of Light 514 Analyzing Two-Source Interference Patterns

447

Electrical and Electronic Instruments Virtual Music

16.2

452

Percussion Instruments The Human Voice

502

503

Wave Optics

441

14.6

Wind Instruments

CHAPTER REVIEW

a500

439

Resonance in Sound hc ee ~cales

495

Ray Diagrams for Lenses 496 The Thin-Lens Equation 498 Gradientindex Lenses

Diffraction of Sound

14.9

s=z

g Oe

484

486

436

Beats and Beat Frequency

re fee

Index of Refraction

Applications of the Doppler Effect

ye

’

Refraction and Dispersion of Light

eee

15.4

483

pac Apeicationsanaon

infrasonics, Ultrasonics, and Echgocation

14.5

15.3

423

14.3

Calculating the Observed Frequency

481

Applications of Curved Mirrors

coo

421

476

479

The Mirror Equation

oe

Sound and Music 14.1

eh

Images in Plane MUFOTS Curved Mirrors

CHAPTER

ceae

Light as Energy 468 Transmission of Light 470

406

LOOKING BACK...LOOKING AHEAD

Sources and Propagation of

Light

Interference of Waves 402 Interference in One Dimension 402 Standing Waves in One Dimension

13.8

467

a § 5 ra

The Universal Wave Equation

13.7

of Light

396

519 519

Quantitative Analysis of a Single-Slit Diffraction Pattern 521 NEL

CONTENTS. 16.4

Diffraction Gratings and Spectroscopy Spectroscopy

18.2

Heat, Specific Heat, and Latent Heat

18.3

Thermal Expansion

527

Latent Heats

Diffraction in Two and Three Dimensions

16.5

525

Thin-Film Interference Air-Wedge Interference

528

530

18.4

534

16.6

The Electromagnetic Spectrum

16.7

Polarization of Light

536

Convection

Polarization of Transverse Waves

Light Polarization by Selective Absorption

Light Polarization by Reflection Light Polarization by Scattering

CHAPTER REVIEW

543

First and Second Laws of Thermodynamics

18.5 544

Heat Engines and Entropy

18.6

CHAPTER 17

= 551

ge

552

“$

Vision Defects and Their Corrections

17.3

The Simple Magnifier

17.4

The Compound Light Microscope

17.5

Telescopes

Space Telescopes

573

Other Telescopes

17.6

564

556

19

19.1

Electric Charge and Atoms

628

19.2

Transfer of Electric Charge

628

19.3

Coulomb’s Law

19.5

Electric Field

631

576

638 639

Electric Field Due to More than One Point Charge

19.6

632

634

Electric Field Due to a Single Charge

580

Electric Field Lines

640

641

Discovery of the Elementary Charge—The Millikan Experiment 643

581 583

Motion of a Charged Particle in a Uniform Electric Field 644

586

LOOKING BACK...LOOKING AHEAD

587

646

647

18

Heat and Thermodynamics

591 &

592

Temperature and Kinetic Energy Internal Energy

CHAPTER 20

Electric Potential Energy, Electric Potential, and cowardlion/Shutterstock Current 651

Temperature and Internal Energy 592 Temperature

NEL

630

Conductors and Insulators

CHAPTER REVIEW

18.1

629

19.4 575 577

LOOKING BACK...LOOKING AHEAD

CHAPTER

Dave Massey/ Shutt

Charging by Contact (Conduction) and by Induction

5/74

Lasers and Holography

CHAPTER REVIEW

621

Static Electricity

571

Controlling Camera Variables Using Lasers

619

620

Electric Charge and Electric Field 627

570

Cameras and Photography

Holography

CHAPTER REVIEW

Electric Force

Structure and Operation of the Camera

17.7

Air Conditioners and Heat Pumps

567

Limiting Factors of a Telescope 573

618

561

567

Radio Telescopes

617

LOOKING BACK...LOOKING AHEAD

CHAPTER

17.2

Reflecting Telescopes

612

613

m

Distance from Sun to nearest star

4 x 10m

Average distance from Earth to Sun

1.5 X 10'"'m

Diameter of Earth

1.3 X 10’ m

Diameter of red blood cell

Sx

Diameter of typical bacteria (Figure 1-9)

mironov/

Figure 1-8 The Andromeda Galaxy, which consists of millions of stars, is similar in shape to our own Milky Way Galaxy. The distance from our solar system to the Andromeda Galaxy is 2 x 10° m. There are thousands of galaxies in the known universe.

MO ei

If SOP? rant

Diameter of hydrogen atom

XX

Diameter of proton

Ie S< 10” tay

Or

rea

Estimated age of universe

Age of solar system

53%

H Time since start of human existence

1X

Age of Egyptian pyramids

10% s

LALS< 10)!"s

One year (approximate value)

3.16 X 10's

Time interval between heartbeats

ls

: Duration of nerve impulse —

OP's

of typical radio waves

| Shortest pulse oflight in laboratory

1X

107s

eS

10nes

Lom

Lifetime of a W particle (a boson)

3

Library Photo Gaugler Gary

3.0 pm ————>

ks

Figure 1-9 This scanning electron microscope image shows rod-shaped Salmonella bacteria with a

ils

diameter of about 1 um and a length of about 3 um.

Masses Known universe (estimated)

aOr

ke:

Milky Way Galaxy

8 X 10" kg

Sun

2 X 10° kg

Earth

6 X 10% kg

| Small mountain

1 xX 10° kg

Ocean liner

8 X 10’kg

Car

1 X 10° kg

| Mosquito

wel

PX H0r ke

| Speck of dust

DO MO ONes

Bacterium

1 x 10°’ kg

_ |Uranium atom

4X 10 *kg

| |Electron

9X SS SSI APE

NEL

When manipulating numbers containing exponents, remember to apply the exponent laws where appropriate:

I

OEE ESE DTI EOE RSE EEO IIL EEE

10" kg LE AIOE APE LID IE Ses

se sje =

7% m+n

x™ alg x

=

Tee

(Ey

=

x™

(xy)" —

x y

Dr.

8

CHAPTER1

Measurement and Types of Quantities

Metric Prefixes Use your calculator to perform the calculation in Sample Problem 1-1 to be sure you are using the EE or EXP key correctly. Avoid the error made by some students who try to enter a number like 10” and get 10° instead because they enter 10 EE 24 instead of the correct entry, 1 EE 24.

Although all lengths could be stated using the base unit of the metre, it is sometimes more convenient to state lengths in larger or smaller units, such as the kilometre, the centimetre, or the very tiny nanometre. The metric prefixes, such as “centi,” “kilo,”

and “nano” are based on multiples of 10, a feature of the metric system that is a great advantage. The inside front cover of this book lists the 20 metric prefixes as well as their symbols, meanings, and origins.

Conversions within the Metric System Measurements to be added or subtracted must have a common unit. For example, you cannot add centimetres to metres without first converting to a common unit. Skill in converting one metric unit to another will assist you in analyzing physical situations and solving physics problems. In general, when converting from one metric unit to another, multiply by a conversion ratio such as (100 cm)/(1 m) in which the numerator and denominator are equivalent. Such a ratio equals, effectively, one; multiplying by one does not change the quantity. To determine the ratio either rely on your memory, which will improve with practice, or refer to the list of metric prefixes on the inside front cover.

SAMPLE PROBLEM | 1-2 | Convert 47.5 mm to metres. Solution

The conversion ratio is either (1 m)/(1000 mm) or (10°* m)/(1 mm). The “mm” must be in the denominator to cancel the “mm” in 47.5 mm. are) ion = 4

lm DS io) X< — 1000 mm

= 0.0475 m Thus, the length is 0.0475 m.

SAMPLE PROBLEM | 1-3 | Add 34 cm and 4.20 m. Solution

Before adding, the measurements must be expressed in the same units. In this case, we will use metres. DID YOU KNOW? Measurement errors can be very serious. In July, 1983, Air Canada Flight 143 from Montreal to Edmonton ran out of fuel before reaching Edmonton and was forced to crash land near

34 cm + 4.20 m = 0.34 m + 4.20 m = Al 4s iin

Thus, the sum is 4.54 m (or 454 cm).

Gimli, Manitoba, thus earning the nickname

“the Gimli Glider.” The crew responsible for fuel made an error in calculating the number of litres of fuel per kilogram required. In 1999, a space mission to Mars failed because the rocket manufacturer listed quantities in British Imperial units but the NASA scientists thought the quantities were in metric units.

SAMPLE PROBLEM |1-4| Convert 3.07 X 10" ps to megaseconds. Solution

From the inside front cover, pico =

10°" and mega =

10°. To avoid errors, the best

way to perform this conversion is to proceed in two steps, first converting picoseconds NE

1.2

to seconds, and then seconds to megaseconds. Both these steps can be carried out in one line:

OMA 3.07 < 10” ps = 3.07 X 10” ps X 1 ps

x

Metric Units, Prefixes, and Conversions

DID YOU KNOW? The world’s smallest lengths in an electron developed in Canada Council. Its divisions

TE 10°s

= 3.07 X 10°° Ms

9

ruler used for measuring microscope was for the National Research are only 18 atoms apart.

Thus, the time is 3.07 X 10° Ms. Notice how the ratios were arranged so that “ps” cancelled “ps,” and “s” cancelled “s,” leaving Ms.

SAMPLE PROBLEM |1-5| Convert the density measurement of 5.3 kg/cm’ to kilograms per cubic metre. Solution

The conversion relating centimetres and metres is 100 cm = | m. Since the denominator contains cm’, which represents cm-cm-cm, we must apply the factor three times, that is, we must cube it: (100 cm)’ = (1 m)’. Hence, we have

k Fes cm

k

100

cm:

= 5.3.

cm)" DID YOU KNOW?

lm

kg

For many people, their height and arm span are essentially the same value. Leonardo da Vinci and other Renaissance artists codified this observation.

le rs ee

Thus, the answer is 5.3 X 10° kg/m’.

©] rerrs Applying Estimation Skills Using your hand span (Figure 1-10), arm span, shoe length, and average walking pace is a skill that will help you estimate a great variety of measurements. Use a metre stick or metric tape to determine the following measurements in the unit stated: ¢ your hand span (in centimetres)

* your arm span (in metres) ¢ the sole of your shoe (in centimetres) ¢ the length of your natural pace (in metres)

Practise using the quantities you determined above to calculate, as accurately as possible, (a) the surface area of a desk or lab bench (b) the surface area of the floor of a room

(c) the distance you travel in 1.0 min while walking at a comfortable speed

EEO

+i

a

ae

te

Figure 1-10 Measuring your hand span.

EXERCISES 1-8

List reasons why the original definitions of the metre and second were unsatisfactory.

1-9

Use the values from Table 1-1 to calculate the approximate value of these ratios: (a) largest length to smallest length (b) longest time to shortest time

NEL

(c) greatest mass to least mass Which quantity has by far the greatest range of values?

1-10 If you could count one dollar each second, how long would it take to count to one billion (10°) dollars? Express your answer in seconds and years.

10

CHAPTER1

Measurement and Types of Quantities

1-11 Determine an estimate for the number of stars in our Milky Way Galaxy by assuming that the mass of the average star is the same as the Sun’s mass, and the masses of other bodies, such as planets,

are insignificant (Reference: Table 1-1).

1-12 The mass of a hydrogen atom is about 1.7 < 10” kg. If the Sun consists only of hydrogen atoms (actually, it is about 95% hydrogen), how many such atoms are in the Sun? (Use Table 1-1 as reference.)

(b) Each bristle of a carbon nanotube is 30 nm in diameter.

(c) At 1.23 X 10° km, the Hudson Bay shoreline is the longest bay shoreline in the world. (d) Each day, Voyageur | spacecraft travels another 10 *Tm away from Earth.

1-18 Perform the following unit conversions, showing Express the final answer in scientific notation. (a)

1-13 Simplify

1.486

x

your work.

20 ms to seconds

(b) 8.6 cm to micrometres

(a) (2.1 X 10°) (4.0 X 10”)

(c) 3.28 g to megagrams

() QS

(d)

OY Ox

IO VOX 10>)

105 MHz to kilohertz

(c) (6.4 X 10”) + (8.0 x 10”)

(e) 2.4 x 10 *MW/m*’ to milliwatts per square metre

(dd) G:88 x 107m) = 2.00'X 10°s)

(f) 9.8 m/s* to metres per square microsecond

1-14 State and give examples of the difference between a derived unit and a base unit. 1-15 Many derived units are named after famous scientists. Name four

(g) 4.7 g/cm* to kilograms per cubic metre

(h) 53 people/km’ to people per hectare (ha) (Note: | ha = 10* m*) 1-19 Simplify

of these and, in each case, state what the unit equals in terms of SI

(a) 280 mm

base units. (Reference: inside front cover.)

(b) 9850 mm

1-16 Describe what patterns can be seen in the list of metric prefixes on the inside front cover.

1-17 Write these measurements without using prefixes: (a) The longest cells in the human body are motor neurons at a length of 1.3 X 107' dam.

1.3

+ 37 cm —

1.68 m

1-20 An American tourist driving in Canada decides to convert the speed limit of 100 km/h to miles per hour, with which she is more familiar. She knows that there are 5280 ft in | mi, that | ft contains 12 in, and that | in = 2.54 cm. Using this information

and your knowledge of SI prefixes, determine the speed limit in miles per hour.

Dimensional Analysis

From earlier science courses, you are familiar with the definition of density as mass/ volume or, using symbols, p = m/V, where p is the Greek letter rho. Rearranging this equation to solve for mass, we have mass = density X volume or m = pV. If this equation is used to solve a problem, the unit resulting from the product of density and volume must equal the unit of mass. The equals sign dictates that the units as well as the numbers are equal on both sides of the equation. An example will illustrate this. Let us find the mass of 0.20 m’ of pure aluminum, which has a density of 2.7 X

10° kg/m’. m=

pV

kg

= 2.7 x 10° me= x 0.20 m

,

=5.4X 10°ke Thus, the mass is 5.4 X 10° kg. In this example, both sides of the equation have dimensions of mass. The symbols for the common

dimensions of length, mass, and time are written L, M, and T,

respectively. We use square brackets to indicate the dimensions of a quantity; for instance, [V] means the dimensions of volume V, and since volume is length X length X length, we have [V] = L’. The process of using dimensions to analyze a problem or an equation is called dimensional analysis _ the process of using dimensions, such as length, mass, and time, to analyze a problem or an equation

dimensional analysis. In this process, the quantities are expressed in terms of dimensions, such as length, mass, and time, and then the expressions are simplified algebraic-

ally. If you develop an equation during the solution of a physics problem, dimensional analysis is a useful tool to ensure that both the left-hand and right-hand sides of your

NEL

1.3

Dimensional Analysis

11

equation have the same dimensions. If they do not, you must have made an error in developing the equation. As an example, the dimensional analysis of the density equation, m = pV, involves determining the dimensions of both sides of the equation to ensure that they are equivalent: Dimensions of left-hand side (L.H.S.)

= [m] =M

M Dimensions of right-hand side (R.H.S.) =[p][V] = ia SEARS

Thus, the L.H.S. and R.H.S. of the equation m = pV have the same dimensions.

SAMPLE PROBLEM |1-6| A student reads a test question which asks for a distance d, given a time ¢t of 2.2 s and a constant acceleration a of 4.0 m/s’. Being unaware of the equation to use, the student decides to try dimensional analysis and comes up with the equation d = at’. Is the equation dimensionally correct? Solution The dimensions involved are L for distance, T for time, and, based on the units m/s’,

L/T’ for acceleration. Now, Dimensions of L.H.S = [d] = L

Dimensions of R.H.S = [aj[?] =

eee XT =L 2

Thus, the dimensions are equal on both sides of the equation, and the equation is dimensionally correct. Unfortunately for this student, the equation is not correct from the point of view of physics. The correct equation is d = 4at’, but since the “ye has no dimensions, it does not change the dimensions of the R.H.S. This illustrates a limitation of dimensional analysis; that is, it cannot detect errors involving multiplication or division by dimensionless constants.

EXERCISES 1-21 Write the dimensions of the following measurements: (a) a speed of 6.8 m/s

1-23 Use the information in the inside front cover to write the dimensions of these derived units:

(b) a speed of 100 km/h

(a) newton

(c) an acceleration of 9.8 m/s?

(b) watt

(d) adensity of 1.2 g/cm’

(c) hertz

1-22 Determine the final type of quantity produced (e.g., length, speed, etc.) in the following dimensional operations: @

IL KUT

(b) (LT) X TT

1-24 When a person is jogging at a constant speed, the distance travelled equals the product of speed and time, or d = vt. Show that the dimensions on both sides of this equation are equal.

1-25 Assume that for a certain type of motion, the distance travelled is related to time according to the equation d = kt’, where k is a constant. Determine the dimensions of k.

(CieMec bs GQ(L/Pieet

1-26. Prove that the equation d= v,t + Sat’ is dimensionally correct. (dis a distance, v, is initial velocity, tis time, and a is acceleration.) REN

EPSTEIN

ISSCESSIES DEESIDE ISISDDEABLS TO

IEEE DIE OLE SENDER I SELES ELPA USEIU ALLS BE ELE SEE EDIE

EISELE LEVIED YORI YEEDLEE ABIDE ELAM VASE

CERES LS PP LIEGE DEE PECL POLD SAGA BOIO IEISEN LESSEE SAE TIS

LRA

12

CHAPTER1

Measurement and Types of Quantities

1.4

Significant Digits

Quantities that are counted are exact, and there is no uncertainty about them. Two

examples are found in these statements: There are 27 bones in a human hand.

Ten dimes have the same value as one dollar.

in measurement, the range of values

Measured quantities are much different because they are never exact; every measurement has some uncertainty associated with it. Measurement uncertainty is the

in which the true value is expected to lie; also called peepMeane

range of values in which the true value is expected to lie; it is also called possible error. Uncertainty should accompany a measured value; in fact, the reporting of a measured

significant digits

value is not complete without stating the uncertainty of the measurement. The uncertainty may arise from the measurement device, environmental conditions, properties of the item being measured, or even the skill of thé person doing the measuring. As an example, suppose you are measuring the mass of a-friend in kilograms. You might try a bathroom scale and get 56 kg, then a scale in a physician’s office and get 55.8 kg, and finally a sensitive electronic scale in a scientific laboratory and get 55.778 kg. Not one of these measurements is exact; even the most precise measurement, the last one, has an uncertainty. For the bathroom scale, the reading of 56 kg could mean that the mass is closer to 56 kg than either 55 kg or 57 kg, and the measurement could be written as 56 kg + | kg. However, you may believe that you can read the scale to the closest 0.5 kg, in which case you would record the measurement as 56 kg + 0.5 kg. In many instruments, the uncertainty 1s stated either on the scale or in the owner’s manual. For example, the scale in the physician’s office could have an uncertainty of + 0.2 kg, so the measurement would be recorded as 55.8 kg + ().2 kg, and if the uncertainty of the electronic scale is + 0.001 kg, the measurement is recorded as 55.778 kg + 0.001 kg. In any measurement, the digits that are reliably known are called significant digits.

uncertainty

the digits in any measurement

that are reliably known

These include the digits known for certain and the single last digit that is estimated. Thus, the mass of 56 kg + | kg has two significant digits, and 55.778 kg + 0.001 kg has five significant digits. The number of significant digits can be determined by applying these rules: ¢

Zeroes placed before other digits are not significant: 0.089 kg has 2 significant digits.

¢

Zeroes placed between other digits are always significant: 4006 cm has 4 significant digits.

¢

Zeroes placed after other digits behind a decimal are significant: both 5.800 km and 703.0 N have 4 significant digits.

e

Zeroes at the end of a number are significant only if they are indicated to be so using scientific notation to place the zeroes after a decimal. For example, the distance 5 800 000 km may have anywhere from 2 to 7 significant digits. By using scientific notation, we can judge which digit is the estimated one, so we can determine the number of significant digits:

5.8 X 10° km has 2 significant digits

5.800 5.800 000 percent error

the difference between the measured

and accepted values of a measurement expressed as a percentage

10° km has 4 significant digits 10° km has 7 significant digits

If the accepted value of a measurement is known, the percent error of an experimental measurement can be found using

percent error =

|measured value — accepted value| accepted value

X 100%

NEL

1.4 Significant Digits

13

Calculations Based on Measurements Measurements made in experiments or given in problems are often used in calculations. For example, you might be asked to find the average speed of a cyclist given the measurement of the time it takes to travel a certain distance. The final answer of the problem should take into consideration the number of significant digits of each measurement, and it may have to be rounded off. When adding or subtracting measured quantities, the final answer should have no more than one estimated digit.

SAMPLE PROBLEM |1-7 | Add 123 cm + 12.4cm + 5.38 cm

tions were done first then the answer was rounded off. When calculated answers are

Solution

rounded off to the appropriate number of significant digits, the following rules apply:

123 cm (the “3” is estimated) 12.4 cm (the “4” is estimated)

+ 5.38 cm (the “8” is estimated) 140.78 cm (the “0,” the “7,” and the “8” are estimated)

Thus, the answer should be rounded off to one estimated digit, or 141 cm.

When multiplying or dividing measured quantities, the final answer should have the same number of significant digits as the original measurement with the least number of significant digits.

SAMPLE PROBLEM |1-8| A cyclist travels 4.00 X 10° m on a racetrack in 292.4 s. Calculate the average speed of the cyclist. Solution

Average speed (v) is distance (d) divided by time (f):

d_ 4,00 x 10°m

27

2904

0's

= 1.368.. X 10' m/s This answer should be rounded off to three significant digits. Hence, we obtain an average speed of 1.37 X 10' m/s or 13.7 m/s.

Most quantities written in this book have either two or three significant digits, although some quantities require more significant digits. The answers to the numerical problems have been written to the correct number of significant digits after rounding off following the rules given in this section. You will be able to practise the rules of significant digits with every numerical problem.

“In some specialized cases in statistical calculations, the digit preceding a 5 is not changed if it is even, but it is raised by 1 if it is odd; e.g., 1.265 becomes 1.26 and 1.275 becomes 1.28. This rule exists to

avoid the accumulated error that would occur if the 5 were always rounded up.

NEL

¢ If the first digit to be dropped is 4 or less, the preceding digit is not changed; e.g., to three significant digits 3.814 becomes 3.81. If the first digit to be dropped is more than 5, or a 5 followed by digits other than zeroes, or a 5 alone or followed by zeroes, the preceding digit is raised by 1; e.g., to three significant digits 5.476 becomes 5.48; 9.265 221 becomes 9.27;

and 1.265 becomes 1.27.

14

CHAPTER1

Measurement and Types of Quantities

Round-off Error Assume

you are helping a friend choose floor tiles for a room

with a length (/) of

5.17 dm and a width (w) of 3.41 dm. At a price of $2.49/dm?, how much will the tiles cost? You begin by finding the surface area (A) of the floor: A=IxXw

= 55117 Ginn < S42) Gham

= 1207

ahaa

If you follow the rules of rounding off answers, this area becomes 17.6 dm?. Next you calculate the cost: Cost =A X rate =

17/6 cian?

Mo

pay

~

dm:

= $43.824, or $43.80 to three significant digits But if you had simply left the area calculation in your calculator before multiplying by the rate, the calculated cost would have been: Cost —"A ~ rate

$2.49 = {7207 cline S< °: 2

= $43.89795, or $43.90 to three significant digits You can see that, to obtain a final answer valid to three significant digits, it is necessary to keep more than three significant digits in intermediate answers. Rounding off too early introduces error, which is referred to as round-off error. Hence, when doing calculations, remember to keep all the digits in your calculator until the final answer is determined, and then round off this answer to the correct number of significant digits. If it is necessary to write down an intermediate answer, use the correct number of significant digits, but keep the intermediate answer with all its digits in your calculator.

Order-of-Magnitude Estimations

order-of-magnitude estimation

a calculation

based on reasonable assumptions to obtain a value expressed to a power of 10; also called a Fermi question after Enrico Fermi

When solving numerical problems in physics, as well as in everyday experiences, it is good to be able to estimate the value of a quantity. This skill applies when checking that a calculated answer makes sense, but it also can be applied when you are trying to decide if you can believe some advertising or reports in the media. An order-of-magnitude estimation is a calculation based on reasonable assumptions to obtain a value expressed to a power of 10. Often there is only one significant digit in the calculated value. Consider, for example, an estimation of the volume of air in litres inhaled by a person in one year. We begin with two assumptions. First, a person would inhale about one litre (1 L) of air with each breath. (You might find a more

accurate value by

devising a water-displacement experiment.) Second, the number of breaths a person would take per minute would likely be between 10 and 20, let’s say 15. Now we are ready for the calculation. Notice the cancellation of units in this calculation.

volume of air =

8, breath

x

= 8 X 10°L/

15 breaths min

x

60min h

x

24h_ day

x

365d _=—year

year (one significant digit)

Thus the volume of air inhaled by a person is approximately 8 < 10° L/year. Orderof-magnitude estimations are often rounded off to the nearest power of ten. In this example, we would then obtain 10’ L/year.

NEL

1.4

Significant Digits

15

This type of question will be asked from time to time in this text. Such a problem is called a Fermi question, named after Enrico Fermi, a famous physicist and professor who often asked his students to estimate quantities impossible to measure directly.

(2) PROFILES IN PHYSICS Enrico Fermi (1901-1954): Combining Education with Experimental and Theoretical Physics Enrico Fermi was a brilliant physicist who was at the forefront of 20th century science (Figure 1-11). He is celebrated as one of the leaders in understanding nuclear reactions and learning how to control them. He is remembered as a great teacher who used estimation questions, along with many other methods, to inspire his students to “think physics.” He was honoured by having an element named after him (Fermium, atomic number 100) as well as a class of particles called “fermions,” which have a particular characteristic spin. Born in Rome in 1901, Fermi entered higher education at an institute in Pisa, Italy, at age 17. By age 20 he was already publishing articles in scientific journals. He studied at various European universities and became a physics professor at the University of Rome when he was 24. When Fermi began concentrating less on theoretical physics and more on experimental physics, he discovered how to cause neutrons to slow down during a nuclear reaction. This important discovery later led to the

development of controlled nuclear fission reactions and nuclear fission bombs. For this discovery as well as other contributions, Fermi was awarded the Nobel Prize in Physics in 1938. Shortly after receiving this award in Sweden, he and his family moved to the U.S.A. where he later became a U.S. citizen. Once in the U.S.A., Fermi joined other eminent physicists working on the secret “Manhattan Project” in Chicago, where they created the world’s first chain-reaction “atomic pile.” In July 1945, while witnessing a major nuclear explosion test, he performed the type of calculation he often asked his students to do: he estimated the power of the nuclear blast. Not surprisingly, his estimation was “in the ballpark.” Enrico Fermi combined experimental and theoretical physics in a way that no other leading physicist did. He was highly respected and well-liked, and he preferred easier explanations to harder ones. He died at age 53 of cancer, possibly related USA/Alamy Pictures Keystone © to his close work with radioactive nuclear materials.

EXERCISES 1-27 Three different scales are for sale at three different prices. A potential buyer measures the mass of a box on the scales and obtains these values:

42.40 kg = 0.005 kg 42.4kg + 0.05 kg

424kg + 100g Which scale do you think would be the most expensive? the least expensive? Why? 1-28) State the number of significant digits in each measurement:

(a) 0.04 Tm (b) 400.20 pm (c) 8.10 X 10°kg (d) 0.008 200 ws

1-29 Round off each quantity to three significant digits, and write the answer 1n scientific notation. (a) 38510 Gm (b) 0.000 940 488 MW (c)

55.055 dam

(d) 876.50 kL (e) 0.076 550 pg 1-30 A student performs measurements to determine the density of pure water at 4°C, and obtains an average result of 1.08 X 10° kg/m’. The accepted value is 1.00 < 10° kg/m*. What is the percent error in the experimental measurement? 1-31 A rectangular mirror measures 1.18 m by 0.378 m. Find the perimeter and the surface area of the mirror, expressing your answers to the appropriate number of significant digits.

16

CHAPTER1

Measurement and Types of Quantities

(a) Assume that the mass of an average cell in your body is 1 x 10“ kg. Calculate the number of cells in your body.

1-32 Use the data on the inside back cover to determine the difference in masses between

(b) What is the mass in kilograms of all the hamburger patties consumed in North America in one year?

(a) aneutron and a proton (b) a proton and an electron

(c) Assume that a Ferris wheel at an amusement park were to fall off its support and become a gigantic rolling wheel. How many rotations, travelling in a straight line, would it have to make to travel from Calgary to Winnipeg?

1-33 Assume that the Earth-Sun distance has a constant value of 1.495 988 = 10'' m and the Earth-Moon distance is constant at 3.844 x 10° m. Also assume that all three lie in the same plane. Determine the greatest and least Moon—Sun distances, taking into consideration significant digits.

(d) Canada’s coastline, including along its islands, is the longest in the world at about 2 X 10° km. If Canada’s entire population were lined up side-by-side with outstretched arms just touching, how many total populations would equal the coastline length?

1-34 Find the time it takes light, travelling at 3.00 x 10° m/s, to travel from the Sun to Earth. (Use the distance given in Question 1-33, and remember significant digits.) 1-35 Fermi Question: Determine a reasonable order-of-magnitude estimation in each case. Show your reasoning.

1.5

Scalars and Vectors

A tourist, trying to find a museum scalar quantity

a quantity with magnitude (or

size) but no direction (also called a scalar)

vector quantity

a quantity with both magnitude

and direction (also called a vector)

DID YOU KNOW? Scalar is derived from the Latin word scala,

which means “ladder” or “steps” and implies magnitude. Vector is a copy of the Latin word vector, which means “carrier” and implies something being carried from one place to another in a certain direction. In biology, a vector can be an organism that carries disease or an agent that transfers genetic material.

in an unfamiliar city, asks how to get there and is

told, “All you have to do is walk one-and-a-half kilometres from here.” This information is obviously not very useful without a direction. A measurement of 1.5 km is an example of a scalar quantity, one with magnitude but no direction. (Magnitude means size.) A measurement such as 1.5 km west is an example of a vector quantity, one with both magnitude and direction. Some common scalar quantities and typical examples are Length or distance

A race track is 100 m long.

Mass

The mass of a newborn baby is approximately 3 kg. The time interval between heartbeats is about 1.0 s.

Vector quantities are common study of motion and forces are Displacement

in physics. The ones used most frequently in the

A crane is used to lift a steel beam 40 m upward.

Velocity

A military jet was travelling at 1600 km/h east.

Acceleration

The acceleration due to gravity on the Moon is 1.6 m/s’ downward (toward the centre of the Moon).

A person falling about 50 cm onto a hard surface and landing on a heel experiences a force of about 2 X 10* N upward, enough to fracture a bone.

Scalar quantities, also called scalars, are easy to manipulate algebraically. The ordinary rules of addition, subtraction, multiplication, and division apply. Vector quantities, or vectors, are more complex. They require special symbols and rules of addition, subtraction, and so on. The general nature of scalars and vectors will be considered as you proceed through Chapter 2. However, details regarding vectors are presented in Chapter 3, which prepares you for vector analysis in two-dimensional motion in Chapter 4.

EXERCISES 1-36 List three scalar quantities other than those already given, and give an example of each.

1-37 For motion that you have experienced today, write two specific examples of displacement and two of velocity.

Chapter Review

|LOOKING

WVé

BACK...LOOKING AHEAD AHEAD

This chapter has emphasized the importance of measurement and basic mathematical skills in physics. Details of the SI units were presented; this system will be used throughout the text. The topics of scientific notation, uncertainties, significant digits, and order-of-magnitude estimations were featured. Scalar and vector quantities were introduced and compared.

In the remainder of the text, you will apply and extend your knowledge of measurement and skills in mathematical operations. The next chapter deals with motion in one dimension. Then Chapter 3 describes mathematical details of vectors and trigonometry required for the study of motion in two dimensions in Chapter 4. Displacement, velocity, and acceleration vectors are important there.

| CONCEPTS AND AND SKILLS. SKILLS Having completed this chapter, you should now be able to do the following: ° State the SI base units of length, mass, and time and give examples of derived units. ¢ Convert from one metric unit to another. ¢ Write numbers using scientific notation. e Apply the exponent laws, especially for multiplication and division involving powers.

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e ¢ ¢

Verify that the dimensions of an equation are correct by using dimensional analysis. Write measurements to the appropriate number of significant digits. Calculate the percent error of a measured value knowing the accepted value.

¢

Perform calculations (+,—,x, and +) involving measured quantities

¢

and round off the answer to the appropriate number of significant digits. Develop skill in estimating quantities and calculate order-ofmagnitude estimations.

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You should now be able to define or explain each of the following words or phrases:

SI

significant digits

scientific notation

percent error

qualitative description

nanotechnology

derived unit

order-of-magnitude estimations

quantitative description

standard unit

dimensional analysis

scalar quantity (or scalar)

measurement

base (or fundamental) unit

uncertainty

vector quantity (or vector)

Chapter Review MULTIPLE-CHOICE QUESTIONS 1-38

The mass closest to

of a heavyweight

(a) 10° (b) 10° (c) 10° 1-39

wrestler,

in grams,

is

(d) 10° (e) 10’

(a) DEBNESA (@) IBC. ID)AN

(a) 7

(d) 4

(b) 6

Crs

+

How many significant digits are there in the product 8.005 00 x 0.005 380 1?

(a) 7 (b) 6

(d) 10° nm (e) 10" pm

(d) B,D, A,C (e) none of these

How many significant digits are there in the sum 460.299 390.0008 + 6.123 + 5.07?

(@) 3S

1-42

Order the following four masses in descending size: A: 10 *Mg; BatOukeC210/° Ye; Di 107 Ge. (b) A, C, B, D

NEL

male

Which length is the smallest? (a) 10°cm (b) 10° mm (c) 10° mm

1-40

1-41

@)

1-43

(d) 4 (e) 3

5

An airline passenger of mass 78 kg has a 20.7 kg checked bag and a 4.19 kg carry-on bag. The total mass, to the correct number of significant digits, is

(a) 1.0289 X 10° kg (b) 1.029 X 10° kg (c)

1.03 X 10° kg

(d) 1.0 X 10? kg (e)

1 X 10°kg

18

CHAPTER

1

Measurement and Types of Quantities

Review Questions and Problems 1-44

List reasons why measurement is important for use in (a) society

1-53

and (b) physics.

1-45

List the SI base units of length, time, and mass.

1-46

Why is it not necessary to have a base unit for area in the SI?

1-47

A high-rise apartment building is 30 storeys high. Estimate the height of the building in metres and decametres, showing your reasoning.

1-48

Convert each measurement to the units indicated in parentheses. State the answers in scientific notation.

|Force (F) |Energy (E)

|(F)= Mur | l= ML"

|Power (P)

|[P] = [EVT

Pressure (p)

[p] = [FVL?

(a) What do the symbols M, L, and T mean?

(a) Mt. Everestis 8.85 km high. (metres, decimetres, centimetres)

(b) Make a similar list for these quantities: speed (v), accelera-

(b) The biggest animal known was a blue whale with a mass of 1.90 X 10° Mg. (kilograms, grams, centigrams)

tion (a), and area (A).

(c) Express the dimensions of power and pressure in terms of M,

(c) A “cosmic year” is the time our solar system takes to complete one revolution around the centre of the Milky Way Galaxy. It is

L, and T.

;

(d) How can the dimensions of energy be expressed in terms of mass and speed?

about 2.2 X 10° years. (seconds, microseconds, and exaseconds)

1-49

The list below shows the symbols and dimensions of some quantities used in physics.

(e)

Determine the ratio of each of the following masses to your own mass:

How can the dimensions of power be expressed in terms of mass, speed, and acceleration?

(a) The world’s largest erratic boulder (1.e., one moved by a glacier), located in Alberta, has a mass of 16 Mg.

1-54

If two measurements have different dimensions, can they be added? multiplied? Give an example to illustrate each answer.

(b) The lightest human birth on record is 0.26 kg.

1-55

Before a wooden metre stick is imprinted with millimetre and centimetre markings, it 1s important that the wood be cured (..e., properly dried). Explain why.

1-56

An experiment is used to determine the speed of light in a transparent material. The measured value is 1.82 < 10° m/s and the accepted value is 1.86 X 10° m/s. What is the percent error in the measurement?

1-57

A customer buys three 120 cm pieces of rope from a new roll that is 500 m long. Assume both measurements have three significant digits. What length of rope is left on the roll? (Don’t forget significant digits.)

1-58

The official diameter of the discus for women is 182 mm and for men it is 22] mm. If each discus is rolled along the ground like

(c) The heaviest mass D2. 10 kg:

supported

(d) A prize winning pumpkin (Figure 1-12).

by a person’s

had a mass

shoulders

of 6.7 X

is

10° dg

(e) A recipe calls for 28 g of sea salt.

a wheel for 25.0 revolutions, how much farther does the men’s

discus travel? swisshippo/Fotolia.com ©

1-59

A soccer goal is 7.32 m wide and 2.44 m high. Calculate the area of the goal opening.

1-60

The equation for the volume of a cylinder is V = rh, where r is the radius and / is the height. Find the volume in cubic centimetres of a solid cylinder of gold that is 8.4 cm in diameter and 22.8 cm in height. Write your answer in scientific notation with the appropriate number of significant digits.

1-61

Figure 1-12 A typical prize-winning pumpkin (Question 1-49(d)).

1-50

The larva of a certain moth consumes 86 000 times its own mass at birth within its first 48 h. How much would a human with a mass at birth of 3.0 kg have to consume in the first 48 h to compete with this phenomenal eater? Express your answer in kilograms, then in a unit that requires a number between | and 1000.

1-51

Simplify

Fermi Question: Determine an order-of-magnitude estimation for each quantity. (Show your reasoning.)

1-52

(a) 2.00 X 10° km — 3.0 X 10°m

(a) the number of your normal paces it would take to walk 1.6 km

(b) (4.4 X 10? m) + (2.0 X 10° s’)

(b) the number of kernels of corn in a container that holds 1.0 L

(c) (4.4 X 10’ m) + (2.0 x 10° s)’

(Hint: It may help you to know that a cube 1.0 dm (= 10 cm) on a side has a capacity of 1.0 L.)

The equation for the area of a circle is A = mr’, and the equation for the area of a right-angled triangle is A = bh/2. (a) What do the symbols r, b, and h represent?

(b) Determine the dimensions of the right side of each equation. What do you conclude?

(c) volume of water (in litres) in a typical above-ground backyard swimming pool (See the hint in (b) above.) (d wa the number of times your heart has beat in your lifetime to date

(e) the number of people in the world who are sleeping at the time you are answering this question

NEL

Applying Your Knowledge

1-62

Classify each of the following as either scalar or vector.

(d) the reading on a car’s speedometer

(a) the force exerted by your biceps on your forearm as you hold a weight in your hand

(e) the gravitational force of the Moon on Earth

19

(f) the age of the universe

(b) the number of cars in a parking lot (c) the reading on a car’s odometer

Applying Your Knowledge 1-63

Do the prefixes kilo, mega, and giga when used in connection with computers (e,g., kilobyte) have the same meaning as in SI? Explain your answer.

1-64

Fermi Question: Determine an estimate for the number of cells in your index finger. Assume that the cells are spherical and have

a diameter of about 10 um.

1-65

Fermi Question: The heart of an average adult pumps about 5 L of blood per minute. Estimate the volume of blood your heart will pump from now until the end of your own life expectancy.

1-66

Determine the approximate time it takes a fingernail to grow 1 nm. State your assumptions and show your calculations.

DID YOU KNOW? The answer to Question 1-66 indicates how quickly layers of atoms are assembled in the process of protein synthesis. It took many years to grow the longest fingernails in the world, a world record of 8.65 m in total for all ten fingernails.

falconhy/Fotolia.c ©

Figure 1-13 A peregrine falcon in flight (Question 1-69(b)).

1-70

Recent

measurements

have caused

scientists to state that the

length of a mean solar day increases by | ms each century. (a) How much longer would a day be 3000 years from now?

1-67

one microcentury. Determine how close his estimate was.

1-68

(b) How long will it take from now for the day to be one minute

At one time, Enrico Fermi stated that his 50 min lecture lasted If volume were a base dimension (V) in the SI, what would be the

longer?

1-71

Show mathematically why the stars in the photograph appear to move at the rate of 15° every hour.

dimensions of length? of area?

1-69

NEL

(a) In Canada, the speed limit on many highways is 100 km/h. Convert this measurement to metres per second. (b) The fastest measured speed of any animal is 97 m/s, recorded as a peregrine falcon was diving (Figure 1-13). Convert the measurement to kilometres per hour. (c) Suggest a convenient way of changing metres per second to kilometres per hour and vice versa.

Refer to Figure 1-3(b) on page 3.

Determine approximately how long the time-exposure photograph lasted.

1-72

Fermi Question: Imagine that you are able to cover a typical football field with $10 bills laid out flat. Determine an estimate of the height to which those bills would reach if stacked tightly flat on top of each other.

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