A students pocket companion fundamentals of physics

ment, velocity, acceleration, and force are vector quanti-ties;mass, speed,charge,and temperatureare scalarquan- Displace-tities.. Ifthe particle has position vector rj at timeti and pos

A STUDENT'S POCKET COMPANION

ClevelandStateUniversity

New York Chichester Brisbane Toronto Singapore

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AStudent'sPocketCompanionioFundamentalsofPhysics,fifth

ineachsectionofthetext,withafewsentencesabouteach,andgives thebasic equations Itservesthreepurposes First,itcan betakentoclassas

a substitute forthetext. You might wantto checkoffthe topics covered

and makeshortnotestoremindyourselfofimportantpoints A wideleft

this purpose Second, it can beused as ahandyreferencefor ideasand

equations whileworking problemassignments Third, itcan be used to

reviewtext material before an exam orwhen you need to recallanidea

fromapreviouschapter

read-ing thetext. Some derivationsand applicationsareoutlinedin^Pocket

Student's Companion but they are necessarilyshortened The text

preferablybeforeclass,then use>4Student'sPocketCompaniontoremind

yourselfofthematerialyou havestudied If it fails tojogyour memory,

restudytheappropriate portion ofthetext. Ashortvocabularylist isvided at thebeginning ofeach chapter Inorderto understand thema-

phrases Somedefinitionsaregivenin^Student'sPocket Companion-,for

otherdefinitionsyoushouldrefertothetext.

Fullunderstanding ofthe ideas outlinedin^Student'sPocket

solv-ing techniques are not explicitlycovered Forhelp insolving problems

refertotheSample Problemsofthetext,thefullsizeStudent's

Compan-ion, and the Solutions Manual Also read the Problem Solving Tacticssectionsofthetext.

Acknowledgements Many goodpeopleatJohnWiley& Sonshelped puttogethery4 Student'sPocketCompanion Amongthem, CliffMills,Joan

Kalkut, Erica Liu, and Rita Kerriganwere instrumentalin the

cur-rent Physics Editor, has supported the latest edition Monica Stipanov

andJennifer Bruer have each contributed in a great manyways I am

gratefulto them all Iam also grateful toKaren Christman,who

Christman,whosesupportand encouragement seemtoknow no bound.

U.S CoastGuard Academy

TABLE OF CONTENTS

Chapter! Motion AlongaStraightLine 5

Contents

Chapter25 Electric Potential 179

andAlternating Current 245

Chapter 1

MEASUREMENT

Physicsis anexperimental scienceand reliesstrongly onaccurate

ei-ther directorindirect,withstandards Thismeansthat foreveryquantity

you mustnot onlyhavea qualitativeunderstanding ofwhatthe quantityrepresents butalso an understanding of howit is measured A length

measurement is a familiar example You should know that the length

ofan objectrepresents itsextentin spaceand alsothatlengthmight be

understandingbothaspectsofeach newquantity asit isintroduced

Important Concepts

n unit n conversionfactor

n standard D meter

n basequantity D second

(baseunit,base standard) D kilogram

n InternationalSystemofUnits D atomic massunit

n Aunitisa well-definedquantitywithwhichotherquantities

are compared in a measurement Examples: the unit oflengthisthemeter,the unitoftimeisthesecond,the unit

ofmassisthekilogram

n Some units are defined interms ofothers Forexample,

unitsandaredefinedinterms of standards Ideally,a

stan-dardshouldbeaccessibleandinvariable

n Asystem ofunits consistsofa unit foreachphysical

quan-tity,organized sothatallcanbederivedfromasmall

num-ber ofindependentbaseunits

1-2 TheInternationalSystemofUnits

n This systemiscalled the SIsystem (previously,themetric

mass: kilogram(abbreviation: kg)

D SI prefixes areused to represent powersoften The

fol-lowing are usedthemost:

Prefix PowerofTbn Svmbol

P Memorizethem Whenevaluatinganalgebraic expression,substitute the value using the appropriate power of ten.That is, for example, if a length is given as 25/im, sub-

the kilogram. Thus,a massof 25 kgissubstituteddirectly,

whileamassof25 gissubstituted as25 x 10"^kg

quantity Forexample,lengthcanbe measuredinmeters,

feet,yards, miles,light years,and otherunits

n Aquantitygiveninoneunitisconvertedtoanotherby

n Carefully study Section 1-3 to see how a quantity given

in one unit is converted to another Cultivate the good

habitofsaying thewordsassociatedwitha conversion

you that 1ft is equivalent to 0.3048m Say "Since 1ft is

equivalent to 0.3048m, then 50ft must be equivalent to

(50ft) X (0.3048m/ft) = 15m".

n The SI standard for the meteris the distance traveled by

n Thismakesthespeedoflightexactly 299, 792,458m/s

n T^ble 1-3givessomelengths Note thewiderange of

val-ues

1-5 Time

n TheSI standard for the second is the time taken for

platinum-reau ofWeights and MeasuresnearParis,France

anddefined sothemassofacarbon-12atomisexactly 12u.

1u = 1.6605402 X lO'^^kg

NOTES:

Chapter 2

MOTION ALONG A STRAIGHT LINE

This chapter introducesyoutosomeoftheconceptsusedtodescribe

particularattention to their definitionsandtothe relationshipsbetween

them

Important Concepts

n particle n (instantaneous)speed

n coordinateaxis D averageacceleration

n origin n (instantaneous) acceleration

n coordinate n motionwith constant

D displacement acceleration

n averagevelocity n free-fallacceleration

n averagespeed n free-fallmotion

n (instantaneous)velocity

2-1 Motion

n In this sectionofthe text, objects are treated as particles

Aparticlehasnoextentinspaceandhasnointernal parts

properties,suchasmass

initmovealongparallellines Itcannotrotateanditnotdeform Ifan extendedobjectcan betreated asa par-ticle, we maypick one point on the object and followits

can-motion The position ofa crate, for example, means thepositionofthepointonthecratewe have chosento follow,

2-2 Positionand Displacement

bygivingitscoordinate x asa functionof timet. Drawa

coordinateaxisalongthelineofmotionoftheparticleand

fromtheorigintotheparticleisthemagnitudeof the dinate Thecoordinateispositiveif it isonthe sideoftheorigin designated positiveand negative if it is onthe side

coor-designatednegative

n You must carefullydistinguishbetween aninstantoftime

andhasnoextension Thus,tmightbeexactly12min,2.43safternoon ona certain day At anyothertime, nomatter

intervalextendsfrom someinitialtime tosomefinaltime:

twoinstantsoftimearerequiredtodescribeit. Notethata

value ofthetimemay bepositiveornegative,depending on

as^ = 0.

n Similarly, avalue ofthecoordinatex specifies apoint on

theXaxis Ithasnoextensioninspace

n Adisplacementisadifferenceintwocoordinates Ifa ticle goes from xi to X2 during someinterval oftime, its

par-displacement duringthatintervalisAx = X2 -xi. Notice

thattheinitialcoordinateissubtractedfromthefinaldinate Thisdefinitionisvalidno matterwhat the signsof

coor-XI andX2'

displacement no matterwhat their motions The tude ofthedisplacement duringatimeintervalmay bedif-ferentfromthe distance traveled duringtheinterval Thedifferenceispronouncedif,forexample,theparticlemoves

magni-back and forth several timesintheinterval

2-3 AverageVelocityand Average Speed

n Ifa particle goes from xj at time ^i to X2 at time ^2» its

averagevelocity vinthe interval from<i to<2isgivenby

Ifyouare given the functionx(^) andareaskedforthe

av-eragevelocityinsomeinterval fromti to/2» firstevaluatethe function for< = <i to find xj, then evaluate the func-tion fort = t2to find X2 and finallysubstitute thevalues

intothe defining equation

from^1to^2isgivenbytheslopeofthelinefrom^i,xj to

averagevelocity in this interval is the slope of thedotted

line.

A downward slopingline(fromlefttoright)hasanegativeslopeandindicatesanegativeaveragevelocity An upwardslopinglinehasa positiveslopeandindicatesa positive ve-locity.

Carefully distinguishbetweenaveragevelocityandaverage

speed TheaveragespeedoveratimeintervalAtisdefined

where dis distance traveled in theinterval This may be

quite differentfromthedisplacementiftheparticlemoves

back and forthduringtheinterval

2-4 InstantaneousVelocityand Speed

n The instantaneousvelocityis the velocity at an instantof

value oftheaveragevelocity in an interval as the interval

shrinks tozero The term"velocity" meansinstantaneous

velocity Theadjective"instantaneous"isimplied

n Apositivevelocitymeansthattheparticleistravelinginthepositive X direction Similarly, a negative velocity meansthattheparticleistravelinginthe negative xdirection

n Ifthe functionx(t)isknown, the velocityatany timeti is

evaluating theresult for/ = ti. Thevelocityatany timet

isgivenby

ofthestraight line thatistangenttothecurveatthe point

correspondingto^ = ^i. Onthegraph belowtheslope ofthe slanteddottedlinegivesthe velocityattime = 7.0s.

2 4 6 8 10

t(s)

n Theinstantaneous speed (orjustplainspeed) ofa particle

forexample, thenthespeed is+5.0m/s

2-5 Acceleration

D Ifthe velocityofa particlechangesfromvi attime^i to V2

interval fromti tot2 isgivenby

t2-ti " a7'

whereAt; = t;2- viand At = ii-^i- Noticethatthe

velocity atthe end Alsonotice thatinstantaneousvelocities

n Ifthefunctionx(/)isgiven,firstdifferentiateitwithrespect

theresult fortimesti and^2 to findvaluesforv^ and

t;2-Substitutethevaluesintothedefiningequation forthe

av-erageacceleration

in-tervalfrom tiioti'^theslopeofthelinefromt\yVi ioti,

V2

n The instantaneous acceleration gives the acceleration of

inter-val It isthe limitingvalue oftheaverageacceleration as theintervalbecomes zero "Acceleration"and "instantaneous

acceleration"meanthesamething

n Ifthe function v{i) is known, the instantaneous

function x{t) is known, the instantaneous acceleration is

Chapter2: Motion Along aStraightLine 9

n Onagraphofvvs t,theinstantaneousaccelerationatanytimeti isthe slopeofthestraight line thatistangenttothe

curve at/ = ti.

n Notethata positive accelerationdoes notnecessarilymean

does notnecessarilymeantheparticlespeed isdecreasing

thesamesignanddecreasesiftheyhave oppositesigns,no

matterwhatthesigns

n At the instant aparticleismomentarilyat rest its ation is not necessarily 0 If, at an instant, the velocity is

acceler-zerobutthe accelerationisnot,theninthenext instant thevelocityisnot zeroand theparticleismoving

accel-erationa, itscoordinateandvelocityaregivenasfunctions

oftimetby

wherexq is itscoordinateat/ = and vq is itsvelocityat

t = 0. Noticethatthesecond equationisthe derivativeof

thefirst

n Eq.2-16ofthetextisalsoextremelyuseful It is

V = Vq -{- 2a{x—Xq).

Itgivesthevelocityasafunction of the coordinate

workedbysolvingtheseequationsfortheunknowns allytwoevents aredescribedintheproblemstatement Se-

Usu-lectthetimetobezeroatoneofthe events, xqisthedinateoftheparticleand vq is itsvelocity then Theotherevent occursatsometimet. xisthecoordinateandvisthevelocity then Theotherquantity that enters is the accel-

coor-erationa. Ofthesixquantities thatoccurinthe equations,

fourare usually given orimpliedand twoare unknown.

10 Chapter2: MotionAlongaStraightLine

2-7 Another LookatConstantAcceleration

n Integrationcan beusedtoderivethe equationsformotion

with constantacceleration Ifaisthe acceleration,thenthevelocityisgivenbyf(^) = J adt + C = at+ Cy where Cisaconstant ofintegration Itsphysicalmeaningisthe velocity

at/ = 0. lbseethis,justset/ = intheequation above.Thus, C = VQ andv(t) = vq -k- at.

n Thecoordinateisgivenbyx(t) = /v(t) d< + C = f(vQ +

integration It isthecoordinatewhen^ = 0. Thus,C' = xq

and x(t) = Xq + 1^0^+ jat^

2-8 Free-FailAcceleration

bythegravitationalforceoftheEarth,has thesameeration, calledthefree-fallaccelerationand denoted byg.

fromplacetoplaceonthe Earth

n Intheabsence ofairresistance,aball(oranyotherobject)

dur-ingitsupwardflight,duringitsdownward fall,and even at

the verytopofitstrajectory

n Theconstantaccelerationequations arerevisedsomewhat

theupward directionbeingpositiveandtheequations are

written

-J9^^y '^(O = ^0-9iy

and v'^ = v^-2g{y- y^).

They can beobtained fromtheequations givenpreviously

fora.

Chapter2: Motion AlongaStraightLine 11

n Problems dealing with free fall are solved in exactly thesame way as otherproblems dealingwith constant accel-

eration

2-9 TheParticles ofPhysics

made upofquanta ofmatter, calledparticles Other

quan-tities,such asenergy, are also "lumpy" (quantized) at the

electrical forces

n Anatomic nucleusiscomposedofelectricallyneutral ticles, called neutrons, and electrically charged particles,

par-called protons The protons repel eachotherelectrically,

butallneutronsandprotonsinanucleusattracteach other

via a strong nuclearforce The neutrons provide thegluethatholdsanucleustogether

n Protons,neutrons,and manyotherparticles(butnot

elec-trons)arenotfundamentalbutratherarecomposedofticlesknownasquarks

NOTES:

12 Chapter2: MotionAlongaStraightLine

Chapter 3

VECTORS

Youwill dealwithvectorquantitiesthroughout thecourse Inthis

chap-ter,youwill learn about theirproperties and howtheyaremanipulated

mathematically Asolidunderstanding ofthismaterialwillpay handsome

dividendslater.

Important Concepts

n vector n vectorsubtraction

n scalar D multiplicationofavector

n componentofavector bya scalar

n unitvector D scalarproduct oftwovectors

n vector addition D vectorproduct oftwovectors

n negativeofavector

5-1 VectorsandScalars

n Avectorquantityhasa direction as well as amagnitude and

obeys the rules of vector addition (discussed later in this

chapter) Incontrast, ascalarquantity has only a

magni-tudeandobeysthe rulesof ordinaryarithmetic ment, velocity, acceleration, and force are vector quanti-ties;mass, speed,charge,and temperatureare scalarquan-

Displace-tities.

n Avectoris representedgraphicallyby an arrowin the

di-rectionofthe vector,with lengthproportionaltothe

mag-nitude of the vector (accordingtosomescale) As anbraicsymbol, a vectoriswrittenin boldface(a, forexam-

of a iswrittena,initalics (ornot boldandwithoutan

ar-row) oras |a| (or \a\). Besureyoufollowthis convention

It helps you distinguishvectors from scalars and

compo-nents of vectors It helps you communicate properlywithyourinstructorsand examgraders Donotwritea+6when

you mean a + 6,forexample Theyhave entirely different

meanings!

thischapter Adisplacementvectorisavectorfromthe

po-sitionof aparticle at thebeginningof atimeinterval toitspositionattheendoftheinterval Notethatadisplacementvectortellsusnothingaboutthepathof theobject,only therelationship betweenthe initial and final positions When

ofvectors,think ofdisplacementvectors

n Ifdi is the displacementvector from point .4 to point B

anddiisthedisplacementvectorfrompointB topointC,then thesum (writtend^ -I- d2) is the displacementvector

thehead of thefirst, thendraw the vectorfromthe tail ofthefirsttotheheadof thesecond Theorderinwhich you

draw the vectors is not important: a-)-b = b + a. Itis

important that the tail ofone be atthe head of the other

and thatthe resultant vectorbefromthe "free" tail tothe

"free" head,asin thediagramabove

re-sultant vectorisnot thesumofthe magnitudes ofthe

Remember you canreposition avectoraslongasyou donot

youwishtoaddgraphicallydonothappentobeplacedwiththe tail ofone at the head ofthe other, simply move one

into theproperposition

The idea ofthe negative ofavectoris used todefine tor subtraction Thenegative ofavectorisavectorthatis

D Vectorsubtractionisdefinedbya-b = a+(-b) Thatis,

n Noticethatvectorsubtractionisdefined sothatifa+b = 0,

then a = -b andifc = a+b,then a = c-b. Just subtract

3-3 Vectorsand Their Components

n lbfind the x component ofa vector, drawlines from the

TheXcomponentofthevectoristhe projectionofthe tor on the axis and is indicated by the separation ofthepointswherethelinesmeettheaxis. Similarly,to findtheycomponent drawlinesfrom thehead andtailtotheyaxis.

y

a

n

they can be either positiveor negative Thevector in the

negative ycomponent.

withtheXaxis,youcan findthecomponentsusing

Forthese expressions tobevalid,6must be measured

coun-terclockwisefromthe positivexdirection:

D

negative and the y component is positive; if^ is between

180° and270°,boththex andycomponentsare negative;

and theycomponentisnegative

You mustalso beable tofind themagnitude and tionofavectorwhen youare givenitscomponents. Sup-

orienta-poseavectoraliesin thexy planeanditscomponents a^

andQy are given Interms ofthecomponents,themagni

solutions totheequation for9, theone givenby your

cal-culatorandthatangleplus 180° You mustlookatthe

ori-entation of the vector tosee which makes physical sense

Forexample, ifOy/ox = 0.50, then 6 = 26.6° or206.6°

Inthefirstcase,bothcomponentsarepositive,whileinthesecond,botharenegative

3-4 UnitVectors

n Theunit vectorsi, j,and kareusedwhenavectoriswritten

interms ofitscomponents Thesevectorshave magnitude

1andareinthe positivex,t/,andzdirections respectively

n Oxi isavectorparallel tothe xaxiswith x componenta^,

Oyj is avectorparallel totheyaxiswith y componentay,

anda^kisavectorparalleltothez axiswithzcomponent

Gz Thevectoraisgivenby

a = Oxi +flyj + flzk

where the rulesof vector additionapply

n Thesymbolst, j,and kareusedtohandwrite the unit

vec-tors

n Units are associated with the components Ox, Oy, and a^

ofa vectorbut not withthe unit vectorsi, j, and k. Thus,thesameunitvectors can be usedto writeanyvector, no

matterwhatitsunits

3-5 AddingVectorsby Components

In terms ofthe components of a and b: Cx = a^ + 6^,

Cy = ay '\- by,2^(1 cz - az ^^ hz.

Cy = ay —byy2iT[(lCz — az— hz'

3-6 VectorsandtheLawsofPhysics

n Manyofthe laws ofphysicsare writteninvectornotation

For example,Newton's second law ofmotion tells us that

a single forceFactingon anobjectofmass mproducesan

accelerationa accordingtoF = ma. Noticethatboth therightand lefthandsides ofthe equation arevectors The

advantageto writingthelawinvectorformisthatthe

nocoordinatesystemneed bespecifiedwhentheequation

iswritten

n Thevectorequation a = bstands for a^ = hx, ay = by,and az = bzy no matterwhat the orientationofthe coor-

dinate system used to find the components For different

problems, different orientations ofthecoordinate systemare convenient The diagram shows two possible systems

compo-nents

2/'

y

The x' and y'components are givenbya^ -acosQ'and

o!y = asin0'. Theangles and(9'are relatedby6> = 6>'+<^,

where (^ is theangle by whichtheprimed coordinate

sys-temisrotatedfromtheunprimed. It ispositive for a

coun-terclockwise rotation (Forthe coordinatesystems inthe

3-7 Multiplying Vectors

n Vectorscan bemultipliedbyscalars.Letabeavectorands

ascalar Then,saisa vector If sispositive,itsdirectionis

thedirectionof 5ais oppositethatof a andits magnitude

multipli-n The scalarproduct (ordot product) oftwovectorsis

de-finedinterms ofthemagnitudesofthetwovectorsandtheanglebetween them whentheyaredrawnwiththeirtailsat

thesamepoint: a•b = a6cos<^. The geometryisshownin

b

n Rememberthatthe scalarproduct oftwovectorsisascalar

andhasnodirectionassociatedwithit.

n a•bispositiveif (^ isbetween and90°; it isnegativeif <j)

n In terms ofcomponents,a•b = axhx + ayhy -»- azhz

n Youshouldnotice thata•b can beinterpreted asthe

mag-nitude of a timesthecomponentofb along thedirectionof

a orasthemagnitudeofb timesthemagnitudeof a alongthedirectionofb. Thiswillbea useful interpretationwhen youstudyworkinChapter7.

n Thevectorproduct(or crossproduct) oftwovectors,

writ-tenaxb,isa vector Itsmagnitudeisgivenby

|axb| = absiTKf)

where istheanglebetweenaandbwhenthey aredrawn

with theirtailsatthesamepoint <j) isalways inthe range

alwayspositive

n Thedirectionofthe vectorproductisalways perpendicular

thevectorswith their tails at the samepointand pretend

thereisahingeatthatpoint Curl thefingersofyourright

handso theyrotate a intob. Your thumbwillthenpointin

thedirectionofaxb Notethatbxa = -axb.

n Thevector product oftwovectorswith given magnitudes

iszeroifthe vectors areparallel orantiparallel((/> = or

180°); ithas itsmaximumvalue ifthey are perpendicular

toeachother((?!. = 90°)

NOTES:

Chapter 4

MOTION IN TWO AND THREE DIMENSIONS

Theideasofposition,velocity,andacceleration thatwereintroducedlierinconnection withone-dimensionalmotionarenow extended You

ear-should paycloseattentiontothe definitionsandrelationships discussed

cir-cularmotion,andrelativemotion

Important Concepts

n positionvector D averageacceleration

n displacement vector D (instantaneous)acceleration

D averagevelocity Q projectilemotion

n (instantaneous)velocity D uniformcircularmotion

4-2 Positionand Displacement

particle movingin two or three dimensions is its positionvector Thetailofthisvectorisalwaysattheoriginandatanyinstantthehead isatthe particle The cartesiancom-

andsoisafunction oftime

D Adisplacement vector is used to describe a change in a

position vector Ifthe particle has position vector rj at

timeti and position vectorr2 at alater time <2» then the

displacement vector for this interval is Ar = r2 - ri- If

theparticle has coordinates xi,yi,zi attimeti and

coor-dinates X2yy2,z2 ^t time /2> then the components of the

displacement vector are givenby (Ar)^ = Ax = X2 - x^

and

(Ar)y = Ay = t/2- Vh and(Ar)^ = Az = Z2-zi- When

using these equations, payclose attention to the order ofthe terms: a displacement vectoris a positionvectorat a

4-3 VelocityandAverageVelocity

n Interms ofthedisplacement vectorAr,theaverage

-n lbusethe definition tocalculatethe averagevelocityover

vectorforthebeginningand endof theinterval Just as for

rep-resent pointsonacoordinateaxis. Theyarenotnecessarilyrelatedinanywaytothe distance traveledbytheparticle

n The instantaneous velocity v at any time t is the limitoftheaveragevelocity overa timeintervalthatincludest,as

the duration oftheintervalbecomesvanishinglysmall In

terms of theposition vector,it isgivenbythe derivative

dr

" =

57-Interms oftheparticlecoordinates,itscomponents are

The term'Velocity"meansthesameas"instantaneous

ve-locity"

n Youshould be aware that theinstantaneousvelocity,

un-liketheaveragevelocity,isassociatedwith asingle instant

oftime At anyotherinstant,nomatterhowclose, the

in-stantaneousvelocitymight bedifferent

n lbusethe definition to calculatetheinstantaneousvelocity,

you must know the positionvectoras a function oftime

Thisisidenticaltoknowingthecoordinatesasfunctionsof

time Youshouldrememberthatthe instantaneousvelocity

vectoratanytimeistangenttothepathatthe positionoftheparticleatthat time Ifyouareaskedforthedirection

calculate thecomponentsofitsvelocity for thattime

thevelocitycomponentsare given,can becalculatedusing

v = ^v ^vl + vl

Accelerationand AverageAcceleration

n Interms ofthe velocity viattimet^ andthe velocity V2at

fromti to<2isgivenby

V2-vi Av

Interms ofvelocitycomponents,thecomponentsofthe

av-erageacceleration are

a-r = , di, — ——% ana a-r = .

^ At '

^ At' ""

At

n lbusethe definition to calculate theaverageacceleration

at thebeginningand end ofthe interval This may mean

time

and

Theinstantaneousaccelerationaatanytimet isthelimit

oftheaverageaccelerationoveranintervalthatincludest,

asthe duration of the intervalbecomesvanishingly small

Interms ofthe velocity vector,it isgivenby

dv

"=d7-Interms ofthe velocitycomponents,itscomponentsare

n Tousethe definition to calculate the acceleration,you must

knowthe velocityvectorasa functionoftime Theterms

"instantaneousacceleration"and"acceleration"meanthe

n Anon-zerovelocity indicates thatthe positionvectorofthe

in-dicates that the velocityvectorofthe particle is changingwith time Rememberthatthesechangesmay bechanges

inmagnitude,in direction,orboth

4-5 ProjectileMotion

n Ifair resistance is negligible, a projectile simultaneously

horizontalmotion Theaccelerationof the projectile has

projec-tile islauncheditsaccelerationdoes notchangeuntilithits

kinematic equations can be used to predictits position at

alltimes(untilithitssomething)

D Theinitialvelocity hastwo nonzero components. Ifthe y

axisispositiveintheupwarddirectionandthexaxisisizontalintheplane ofthetrajectory,thenvq = i^ox" "•"

hor-^Oy

J-n IfVQistheinitialspeed and 6qistheanglebetweenthe tialvelocityvectorandthe horizontal,thenvq^ = t;ocos6qandVQy = vqsin0q. 6q (andVQy)are negativeifthe projec-

4-6 ProjectileMotion Analyzed

n Ifthecoordinate axes areasdescribedabove,thenthe

co-ordinatesandvelocitycomponentsattimetaregivenby

Notethatthe equations forxandVx describemotionwithconstantvelocity and that the equations for y and Vy de-scribe free-fallmotion (withconstantacceleration-g)

the highest pointofitstrajectorythe velocityofa projectile

is horizontal and Vy = 0. To find the timewhen the

y = yo- lb findthetimewhenthe projectile returns to the

launchheight, solve = {vq sin 6^)1- Igf-fort. The

mag-nitude ofthe displacement from thelaunch point to the

pointatlaunch heightiscalledthe horizontalrange ofthe

projectile

4-7 UniformCircularMotion

cir-clewith constant speed Remember thatthe velocity torisalways tangentto thepathandthereforecontinually

D Ifthe radiusofthecircleis rand thespeedis v, the erationoftheparticlehasmagnitude

and alwayspointstoward the center ofthecircle. The

di-rectionofthe acceleration continuallychangesasthe ticle moves around thecircle.

par-The term"centripetalacceleration"isappliedto this

accel-erationtoindicateitsdirection Youshould notforgetit is

D

n

n

Thepositionorvelocityofaparticleismeasuredusingtwo

coordinate systems (orreferenceframes) thataremoving

Let xpj^be thecoordinate of a particleP asmeasured in

coordi-nate ofthe originofframe Basmeasuredinframe A Then,

- X

Differentiatethecoordinateequation withrespecttotime

toobtainthe relationshipbetweenthe velocityofthe

inthe other TheresultisvpA = vpB + vba- Here vp^is

thevelocity as measuredinA, vpp isthe velocity as

mea-suredinB,and vba isthe velocity offrame Basmeasured

n T^keca rewiththe subscripts Thefi rstsymbolinasubscriptnames anobject (theparticleorthe originofframe B) and

the position orvelocity ofthe object Youshould say all

XBA you should say "the position vector ofthe origin of

getacquainted withthenotationfastandwon'tgetitmixed

uplater.

4-9 RelativeMotioninTWo Dimensions

positionofthe particlePrelativetothe originofcoordinate

fqao^the originofframe Brelativetotheoriginofframe A.

Frame A

time to obtainvpA = ypB + ^ba forthe velocity oftheparticle inframe Ainterms of theparticle velocityvpB in

acceleratingrelativetoeachother

n Airplanesflying inmovingairorshipssailing inmoving terareoftenusedas examplesofrelativemotion Theair-

wa-plane or shipis theparticle, one coordinateframe moves

with the air orwater, and the other coordinate frame isfixed to the earth The heading ofthe airplane or shipis

in the direction ofitsvelocity as measured relative to the

airorwater,notrelative totheground

4-10 RelativeMotionatHigh Speeds

n Ifanobjectis movingatnearly thespeed oflightorifwe

compareitsvelocity as measuredin tworeference frames

other,thentheresults givenabovefail and we mustuse a

inreferenceframeB IfBis movingalong thex axiswith

velocity vqj^, as measured in another frame A, then thevelocityofP,asmeasuredinA,isgivenby

vpB + VBA ypA =

1 + vpbvba/c^

Herecisthespeed oflight.

n YoushouldbeabletoshowthatifvpB = c,thenvpA = c.

then it moves with the speed oflight in all frames You

shouldalsobeabletoshowthat ifvpB and vba areboth

much less than the speed oflight, then the correct

non-relativisticresultisobtained

I

NOTES:

and

Chapter 5

FORCE AND MOTION — I

Inthis,themost fundamentalchapterinthemechanicssectionofthetext,

attentiontothe relationshipbetweenthetotalforceon anobjectanditsacceleration

Important Concepts

accel-erationofanobject,giventhe objectanditsenvironment

objectandthetotalforceonitcausesittoaccelerate Thefirstpartof theproblemisto findthetotalforceexertedon

theobject, given the relevant propertiesofthe objectand

itsenvironment The secondpartof theproblemistofind

theaccelerationoftheobject,giventhetotalforce Inthis

chapter,weconcentrateonthesecondpart

5-2 Newton'sFirstLaw

Ifthe net force actingonaparticleiszero, then the

accel-erationoftheparticle,as measured relative to an inertial

frame, is also zero Such a frame might be attached toa

acceler-ation of theparticle,asmeasuredinthatframe,iszero

n Thevelocityofanyinertialframe,measuredrelativetoanyotherinertialframe,isa constant Inertialreferenceframes

donot acceleraterelativetoeachother Theacceleration

ofaparticleon whichthetotalforceiszerois itselfzeroin

e\eryinertialframe

5-3 Force

n Aforceisapushorpullexertedbyoneobjectonanother

tothestandard(1kg)mass and measuringtheacceleration

ofthe standard mass IfSIunits are used,themagnitudes

ofthese quantities arenumericallyequal Bothare vectors

vec-toradditioncanbe checkedby simultaneously applyingtwo

forcesindifferentdirectionsandverifyingthattheresultis

thesame aswhenthe resultant ofthe forces isapplied as

a singleforce All measurements mustbe madeusing an

n TheSIunitofforceisthenewton andisabbreviated N.In

terms of theSIbaseunits, 1N = 1kg•m/s^

n The massofanobjectismeasured,in principle,by

compar-ing the accelerations ofthe object and the standard mass

whenthesameforceisappliedeachtothem Inparticular,

themassofthe objectisgivenbym = (1kg)(ao/a),where

Chapter ForceandMotion—/

qq is the magnitude ofthe acceleration ofthemass dard Theaccelerationsmust be measuredusinganinertia!

stan-frame

n Anobjectwithasmallmassacquiresagreater acceleration

than an object with a large mass when the same force is

appliedtoeachofthem Massissaidtomeasureinertiaor

resistance tochangesin motion

n Mass is a scalar and is always positive The mass of two

objectsincombinationisthesumofthe individualmasses

n This isthe central law ofclassicalmechanics It givestherelationshipbetweenthe net forceYlFactingon anobject

and the acceleration aoftheobject:

where misthemassoftheobject

isequivalenttothe threecomponentequations

n Inthese equations,^ Fisthetotal (ornet)forceactingon

theobject,the vectorsumofallthe individual forces This

meansthat inanygivensituationyou mustidentifyall theforces actingonthe objectandthensum themvectorialfy

n Note that^ F = implies a = 0. Ifthe resultant forcevanishes,thenthe objectdoes notaccelerate;itsvelocity as

observedinaninertialreference frameisconstantinboth

that act sumto zero For some situations, you may know

thatthree forces actbutaregiven onlytwoofthem Ifyou

alsoknowthattheacceleration vanishes,you cansolveFi+

F2 + F3 = forthe thirdforce

5-6 SomeParticularForces

n Theforceofgravityon anobjectiscalledtheweight oftheobjectand itsmagnitudeisgivenby V^ = m^r,where mis

themassofthe objectandgisthemagnitudeofthe ationdueto gravityatthe positionoftheobject NearthesurfaceoftheEarththedirection oftheweight istoward

acceler-the center ofthe Earth

n Besureyouunderstandthatmass andweightare quiteferentconcepts Massisapropertyofanobjectanddoes

evenintoouterspace It isascalar Weight, ontheotherhand, is aforce Itvaries as the objectmoves from place

to placeandvanisheswhen the objectisfarfromallother

objects,asinouterspace Thisisbecauseg,not themass,

variesfromplace toplace

its acceleration Weight is a force and, ifappropriate, is

includedinthesumofallforcesactingontheobject This

sum equals ma andifotherforces act, then a is different

n TheSI unitof weightisthenewton

oneach witha forceofthesamemagnitude,calledthe

transmitting a force from oneobject to the other; the

sit-uation isexactlythesameifthe objectsareincontactand

exert forces on eachother Strings pull, not push, along

their lengths, so astring serves to define the direction oftheforce

n Asurface may exert a force on an object in contact with

it Ifthe surfaceisfrictionless,that forcemust be

perpen-dicular tothe surface It iscalleda normalforce Unlessadhesivegluesthe objecttothe surface anormal forcecan

surfacetowardtheinterioroftheobject

—