Fundamentals of Physics 8E

TO THE STUDENT

You have the potential to make a difference! W;!I )'07

12-4

The ne Center of Gravity 308

9-10 9·10

Colli.ion. DimenM.J arbonatord ~I ccalMS ..... th. the fog!h.t fog th.1 IIppHn drink;' opened? drirJr. ;,~7

19-1 Ph~ia? Sa! S08 What II PhY.le.? 19-2 Avogadro's Number 508 A~.dn;l" Numblor 19-3 1de,1 19·3 Idea l (in., Gases 509 19-4 Pressure, Tempe Temperature, RM5 Speed 512 P,.nure, rature, and RMS 19-5 Translational Kinetic Energy 514 T'an,l.tIO".1Kine tie 19-6 MeanFreePa Mun F... P. th 515 19-7 TheDistributio.-.oIMolec ular5peec1. Speedl 517 Th. D,Unb\llion of Molecular 19-8 The Heatl of .n an idelll Ideal (ill Gas 520 Th. Molar Molir Specific HU ll ol 19-9 Degrees Freedom and MoI.. Molar 5pce Miche Michelf>On'slnterie lf>On's Interie rometer 978

RevieN Revie N & Summary 979 Prob lems 981 Questions 979 I Problems

~I.1 ~I.'1

Diffractio n 990

What cau UUSM .... th thot.. a"esting blu. color of the ffacial .. cial skin of a mlltldral mandral baboon? 36-1 What Is Physics? 991

36-2 3~3

36-4 36-5 3~'

Diffraction and the Wave TheoN & SUlTll'Nlry Summary 1213 auestoon. Question. 1213 I Problem. 1214

H I Quarks. lLepto ns. and the eptons. Big Bang Ba ng

12 18

How un" /y uni_.. be ~'«I c.. a pftologr."n photog,;oph 01 of dw the •• e./y t.a~ ... 11 44-1 Whit PhysICS7 1219 44· 1 Wh. t I, l. Physics? 44-2 PanIC . ., PartICle., PMlocles 1219 44·2 Pa rt icle. Particle. , PartICles 44·] No 44·3 An Int.,lud. I"teo-Iude 1223 44-4 Thel. ptonss 1226 44·4 The Lepton 44·55 T,... 44· The H.dronl Hadrons 1227 Still Anolher Co nserva tion La" La'l 1229 44·6 Another Conservation 44·7 Th. Eigh tfold WilY Way 1230 44-8 Th. Quatk Qu~rlo. Mod.I 44·8 The Model 1231 .... 9 Th. BiOIw; Fon:H M.ssenger 44·9 The B. sic Forces and MelS8f>ger Part Part"ICle. . . 1234 APIiUMfOfReoflKbon 44-10 A Pause for Reflectoon 1236 ....... Is &panding ~ndng 1237 44-11 The Uno Universe CoimIC Background Radiallon RadoooDOn 1238 44-12 The Cosmic 44· 1J 44-13 44-14 44-15

Daric M,," .. 1238 Oatk Matter 123B Bog Bang BIIng 1239 The Big ASummirogUp A Summ,ng Up 1242 ReYMl'N & & Sul'T'VNry SulTll'Nlry 1242 1242 Revie-N a..-toon. 1243 I Problem. 1243 Question.

_

Appendices Appe ndices

The In ternational System of Unit. IntlfTlauon.l Unlfl (Sf) (51) A-1 A. l Scom. Fundament.al Fund"mental Constanta eon.tantl of 01 PhYlics PhysIQ A_' A·3 Some s.:.m.Aslronotnocal Dati A-4 A-4 Some AstlOflomica i Data o Conv ConYlf'llOfl Fac;~ M A·S D .... ion Factors E Math Malhematio bolh aoo.'SSible and in'·an,bi.:l. If we dofi"" 100 lenglb 51all""',d as Iiii' l,n", ool",,"n o"e's nose and Ibc indox fingi)' on a" oUlSltelch.~ a.m. "'e "''' la;nl)' h3\" an acressiblil SIa"d of abom 23 noos I"" bour (,;dc'Slb ). The ride is an aocicnl G"'.cnn is. Brilisll voIu",,, for f,.sIlly mug/II hrrings: 1 Cfall - 170. 47~ lil.r.; (L) offish. OOoot 7SJ oorrings. Suppos.:! {hal. 10 tX' d,c " ..'lOOlS in Saudi Arab... _" shipment of 1255 m..,;l 00 ooI:.!inal , .. uiu of rolrwatioru art of teo ,ounded ' 0 m.tcb ,be Ie"" rrumbel of ,i.!."ilk.,,' figur .. in ,b, gi""n d., .. (Howeve,. ,om" ..... . . .,,'" 'iylifir"'" figure ;, h pl.) IVn.n 'be Itf'mOO1 of the digi" '0 be di"" .. ded io 5 or mor •• ,be 1:0. 10 the ~q"a1or . Laler . 101 pnctical rea",,,,, IhlS Eanb sla"dard ""as 3OOodo ....>OCunda,y slill moOro """""hI.>. ~ a","' rd~ so Ihat ulllm"lol}' e~,,)' "",-"suring d..",ioo "",ive tango olleng!b,.fro", Iha! 01 !he universe (lop I;",,) 10 !bose 01 so",e very sOIali objoc1s.

j

Some Appro.im .... l.e"9'h. Leogt. io M ..."

M" a.. ,em,n'

fi,,,

Oi".IlC' to tbe gawie. fofmod O;,!>"", to tbe Amdrornod. golny 0;',. • .,.. to tbe o ... by"", Pr,m"" C.nuuri 0;',."", to PI.to R:odi", 01 E .,.th Heigl" 01 Mt_ Ew,e" llOCkn"" 01 ,b;' pas, Leng'h 01 . t)piocol

m•

2 x II" 2 x 10"

4 x 111' 6 x 111' 6 x HI' ~ X

10'

I x 10- ' I x II}-"

Rodiw 01,

hydro!"n "''''''

5 x IO- H

R:odiw 01 , Jlf ot""

I x 10- "

PROBLEM-SOLV!NG lIlCTlCS

Tactic 2: Order of Magnjlud~ The ON" / "'~gnN"J< of , o.mo., is ,be powe' 01 "n ..·ke. ,b, ."",!>eo- is .xpres>tnd .ce"", prole .. ioo:w wiU esti_

m". tbe "",Ill, 01 • ,.kulatio. '0 ,b. ",arl or"", 01 m.y,;_ ,»de, FOf OW" " """ple, ,be ne .... ' Ofck, 01 m'gW,IKle ;. 4 fOI A ODd 5 for R SIOclr COII1DIOO wben ""tailed Of prod .. do .. '''i"irO liSI" p.1ot. Lifetime of 'he m",' 1LII",bIe panicle The 1'I>nd rim ..

2 X 10' 9 X JI)' S X IO-L

1 X 10-' 1 X 10-1<

I X 10-" I x 10-4>

tho mlic>, tim, o/tIeOl ..... propooed in I ,h. bour ,""'lioN to J'fovkle . W_bow cia)'.

,n.,

m.

The ide.didnOl catch 0 ... u.. .,"er 01 tim IO-bour "",,,b ";re]y pm _

vid. d .... .n dial 'h" l.,. 000""._ .ioo al Il_holU ,...... Do ,be , .... am'" baw 00 01 Ib.1 d.:>nsil)': plahn" m has a don,ity tha! i, about 21 !JUlilS Ib.1 of w.{('r,

'ill

A be. ,')' ob/,-'C! can 'ink into Ihe ground du,iog ao

e.. l hqu. ko if the sha king c."ses the ground to unde,go Ilq" ,'"Id rali" e for a ""'pIc of too ground:

v_

'.-

< - - - -.

(1-9)

Here, V..... is lhe !Olal Vo! "flle of the sand sr""' in lbe sampl , 00 V..... is 100 lotal volume ""IWOOO ttl< grains (in the ,ids), If ~ excililds a cmical n.l u" of 0,80.

EJep/l'.' Gn, Sped of dusliluling Ihi:! oxpression inlo Eq. 1· 10 and Ihen sub· sliluliDg for v.. _ f.om Eq. 1· 11 lead 10 P-J - ~ V..... _ ~

v.._ (l · ll)

From Eq. I-&Ihe 10lal mass m ..... 01100 ,",ad g.ains is Ih. product of Ihe dellSil}' of silicon dioxide and 11K> 10lal volume of lhe sand g.ains:

V,''" 1 + ,

1 + ,'

(1 · 13)

SUt>sh IUI,n g PliO, - 2.600 X 10' 19lm' and 100 Cfllical val"" of, - O.ilI . .." find Ihal bqU.Oipul.,«I Iil:. algebnJ< quan'iti..

Wl'~

""If'''' de .. red uni" ",maiD.

Lengt h The "",,'er ~ defuted .. ,he !omic docm

1" ~ ..- t .-

• _••• _"'_,.,..,. ....... " ' _ .. _

,.s:;: _

.,"' _

_

M1.«l....

Problem 6-

. oo#1y .... ""' ...1.... ..llh • ,adi .. d .!Wl~ ..

.

~

n

(F\&. I.,';). n.. ..... ~ n W"'" is .lOOO

,~idcn ... (If;lI

m. 110'0' "'any ""bic « n,ime'en 01 k OCODd .. _ d " b..od . . ,.w-.. ......,., are roo .. in~ ....,""" "an (hip.ly","",fICl " ... ,eat. Jw 'ify )u, cboke. .. ..

-

""'* ,-,

A"",,,, ,,"",. tBN' "

WO>,

'moo. ' '10''

'2:I'>!:rI

It""" ". "","57 '2g. in di.. pp..'~ JI the .I • .,..d ,im. '" _ 11.1 ~ u-hal ;, .b. radius rof E.rth?

..e, ' ·7 MI..

atom. :tnd a c,.,bon "''''" .,. 1.0 u, 16 u, and 12 u. respectively. (I/jnt: Cat. are """.. im .. krI""",, '0 kill a mole,)

..27 0 ••• pe.ding SP'" in Mahy .... fOIl buy on ox wi,h • weigbt 01 2S.9 p;cul. j. ,b. local URi, ol ... igh t ~ I pirul _ 100 gim. 1 gin _ 16 rahil>. I "'hil _ 10 c""'~ md 1 diet _ 10 hoon~ The weight of 1 h""" """"poods '0 a ..... 01 0.3779 g, Wh.n yoo anans. '0 5IIip ox Ilome '0 fOUl . "oni,bed family, bow mIlCh.,.., in kilogram. mo" yo. ded .. , o. ,b •• hippin~ m... ~... tl (Hmo Ser up mul.jple ,h"'n _ ~nk ronvtJOioru.)

,be

•• 28 0,,,,., 01 fi"" Califomi. be",h .and

.re

appro,i_ m.lely .pbe,.. ..-no .n ave ...ge ,adi •• of 50 "'" and are ",ade oI .iIkoo dioxide. ",lOCh h .. a d.nsity 01 2600 kslm'. Who, m,.,. 01 >and gr';". woold bave a 'OI>! ..... f""" ""' . (,h. 1OIa1 "T" 01 all .h. individual sph .. ,,) «luI '0 ,b •• urf",. ore. 01 • ""be. 1.00 m "" .. .qe? ~

run.. """j""

•• 29 Doring b• • vy 01. lll' 0.80 kIll up >lOll! tbe d 2.0 m deep .hl" in.o a vail.y i • • mud . lidoe.A,,,,me tb>l ,be mud.nds up uniformly diIT" ....",ri.g 14.0 an by I 7.0 an .. bein~ Iill.d ..-on id.ntic>! pfi.. 01 '''''< wat., fill1 (c) W'Ie' b .. . de .... y 01 lOCO kYm-'. How mucb rn . .. does , •• ...,t.. in ,b, doud ha",,1

,Ila,

,...Ie,

•• 25 IfOIl bas. den"'y 01 7.87 !i!om i. 917 X 10- " kS If tb, orom • .,e ",berical and 'ishlly pocked. (.) "hal is .h. volume of an i,OIl >10m and (b) "hat ~.ne di ,uffel, ,,'inS b" ruret. ..,d "h,y, >long 'f'ide' who ", down t.e.ide b ... .. ." The ,pOd" ,,, clown .01 bt",,,,,,, of 'h' d (e) bl"" ? 14 An ok! man"",,,po Ih". l>ndo ....,.., i. Ih . lime of King ATth., b.1d 3.(10 >CTe. of plowed 1.00 pm. 0 li,.._ "ock .... ol !.I.O perche. by HIO pelebt. Wtr ...... lhoe '0101 ..... i. (.) lhe old "nil of rood. and (b) doll b.,.... 10 iiI "';,bin • doll howe) b. , lhoe seal. of 1 :144 of. ,,>I b"" ... Suppooe. ,,01 boo", (Fig. 1_7) II"" .Iroo' len~'h ol2Om. • depoh of II m•• beigh' ol6.0 m, and . .. . . d.,d ,loped ,oof (""n ....1 '''''''gular f""" 00 , •• endo) of beighl 1.Om . J. JIY of each ,h",,1d be plUeb,e;:I and. 0"" lhe ,,,,. pt,,,1e "filled. how mucb wi"" i. lef! ""er in I",,,,, of (b) ":rnd,,d boo,,,", OlId« ) b .. ,,?

.,"ead

I "",d.,d boolle

I lIIagnWD _ 1 """dard

"""j..,

I je,oboam _ 4 ,,:rnd .. d boolles I "boboam _ 6".ndard bool'"

1 »Inr .... 7.1I

n """dard """I",

_

I balrllazar _ 16 ,randard bood .. _ I U56 L I ",buch.>dntzz.. _ 2Q .. :rnd>rd bollles

"'ga'

41 A I)'pkil cube b.. on edge jeoglb of I em. If )'011 bad a cubical box 11.." contained . mol. ol "'gar ",be~ whal .......Jd in . dge longlh be ? (One _e _ 6.02 X 10" ""il~ j 44 UoinS """ven ions :u>d do,. in ''''' chapltl. delenni.o. ,b. numb..- of b)'llr"!l.n >1011" ""Iu~td 10 oo,ain 1.0 kg of bydrog •• , A h)'lely 1.50 X 10' km. The ,petd of hSb,;, .boUl 3.0 X 10' mit.. Expoe .. l!roe 'peOO of Uglr' in ""ro.omkol ""its pe' minul.. .."

Earl"

B B B B fIG. 1

41> Wb>1 m"" of ...... , ... feU 00 ''''' b"" adeMiry of 1.0 X 10' kg''''.

Probl,m.lK

'own in i'rOOl.m 'I? W",o,

47 A pe""" II .

.>-

S5 A ... ndard in","" .. .u-.-... b .. 110!" .acb with . ri" (b.igh. ) of 19 ClI1 and • ''''' (borirootal depth) of 23 em. R... ",b "'Sg"''' tb at the .. ",. "1"

"airc...,

56 lb. con"""" Eastelo _ ' •• mammal. t)llically has • m. " of 75 ~ .midi con .. poods to about 7.5 mole. of .. oms. (A mole of " om, ~ 6.W X 11)'" .. om~) In atooDc m... \!JIits (D) . .. bat;,.1>0 ~ .. age "',." 01 tb. i. the commo. E:utem moIe1

"om,

(Jlint: S.. !beM,." ,.hle in Appendix D.)

y"" ..... orn din.", 101400 poople., . OOoo:< are oerondo io • ~,,? (b) H"",aru h.ave ".;tOO 101 about 1(1 ye",...·b...,. tho unive,,,, i,.bout 10" Y"" old. 11 the.ge of tbe .oiv,... i. d,fioKl .. I -unive, .. day:· wb .... unive .... day coru;... 01 ··l1.JIi"" .. fffOIIdo" . . . ... rn.al day 00IIKOo:>ds b.ve hum.:ur, .:w.. d? 52

A..! . ,," 11, . .. "",'1""" !be old and the IJIOdem ",d

t..t~

the I..-ge .00 tl>o "'nil. ooruid" tb. 10Uooring; I. old ,ual Eogland I hide (bet ..... " 100 ond 120 """) ,.'" tb • • re . 01 1.00 ... ded to ,wtain o.e family ..;,b • ".gle pI""Sb!Of I!_ing .IJW_ let'" a bam door. "tUClr can hardly be oW..,d.) What io tb. r>tioof21 ,..pe.tal:e< to II ba,,\l 1

.re .

53 A to:adi.ional ""~ oIleogth in J. pan ~ ,b. hn (I l .o _ 1.'17 m). Wb . . .f< tb .... ti", of (.) "I..... hn. to "I"'" ",,,,,,,.00 ( b) . U .. !be l."sth "bl. in Appe E.nb _ Su. dinanc. in (a) P" """ ""d (b) ~ght _y.ar.

P''''''''

,b.

.o.

sa

10 pe.ip 01 tbe middle fins" 01 ,I>e A..!.u"", ,h.at ,1>0 distaD'" "",god from 43 to 53 em. .Dd , uPI""" that anci,", du"':o!.' iodicate that • ,)'lindrical pillar was to h.v •• I,"!tb 0(9 cobits aDd a di,,",_ tff 01 2 ",bi.. fi>r!be "ated ''''ge.",b., ... tb. lowe< .al"" .nd tb. uppc' vol .... '''p«tively.loo (.J 'he C)'lind,,·, 1'_S,b in ........ (b) .1>0 C)'line mOOS old (1""''''' _ rir) Briti>.b me"""". and an"",! oomJI1OII (>tiU premttrir) us. m ....."" (Th ......... ""''''00 !Of metric..ion.) fi>r liquid " "'''''''- I Britislr ""'I""'" - I u.s. .. "'f'OO'L fi>r..-y me .. "".. I Brirub ""'JlOt1 _ I us.q">t1. ln us.m .......... bow mud! (.) "oci.{b) ",.Ie tops. (e) rice.aDd (d) salt of. "'l"ired in th, 'ecipel

m.",,,,,,,

h.,..

w,..

Old Briti>l! Meaowe>

...,pooa _ 2>al"I"""" de.."t"""", _ 2 ...,pooa. .. bl"poon _ 2 de"",n.pooo> ..",up _ 8 ..1>I"pooB' bo-e.kf .."up _ 2 te>cup<

t.bIe sbown. A chaoge from pos;l;on x, 1o posilio"~-, "calJed a di"Ia«'"",n, .1x. wh •• "

FIG. Z.1

.1., -

x, - x,.

(2 -J)

(Tbil'ymbol .1. Ih. G,,,,, k uppe"a", d"II'. '"pro",n" a .ha"g" in a qua "lily, aod

T""- "-orld, a nd oWr)1bing io il. m""os. Evo" ",.",ingly "a' io"ary Ihings. such "' • roa""-a)', 010'" ","ilb Eanh's ,otolio", Earlh', o,bila.OII"d 100 Suo . lh" SURS 0'bil around lhi' ""Ow. ollhe M,lty Way gala.ty. "ud lb., ga l.l Iy", migfJlion ro[atiw 10 o,her gakaxiposili'''' d .. "Clioo (IO Ih ~ righl'" Fig. 2 -1) al ...a}'s co""'s o ul posil""'. and" dis-

2-4 I "", ... _VkxhyOcales Ihal Ibe molloo is in Ihc nesalive dimelion. Thc aetual numoo. of n"'I~" cov.,cd for a \rip" irr"lev, nl ; dispLaeemcm involve. onl)' Ib, origin,1 an d final poslliOffl. RI. ~IJmp"'. if Ih' parlide mOV(lS from x - 5 m oul lox - 200 m and loon bact 1o x - 5 Il•. Ih' displa"""",,n l from ,la,llofinish isginnmg 01 you, d"," 10 }'our arrival al (h , 'talio"?

8.4 kO!

70 k mlb

- 0.12 h.

in tbe problem? Wri1f down the ¢Ven data. wi,b unit~ ""OS the ')'fllbok of the OO>eIim .. Di>Od'oroovm unit~

Tactic 3 : Is YOUI AM"' ...· Rea.onabl.., dx " - • 11m - - - -dl-. • _ . t>.1

Fisu, .. 2-1. 2-3., 2-I , . nd

2_5 .... graphs )'OIl s.boukl be . bIe '0 "ad •• siI),. lo ....... y . ph, tbe v:uioble "" ,be bOO"""al i, ,h. ,ime " ,.j,b 'be direc_ '''''' 01 in", .. inE ,ime '0 ,he !ish" 10 .a.cb, ,b ••ari.bl. "" tbe unicoi oxi. ~ pooi'ioa x of the moving p.nicle wit • re'f"'CIlO ,he origin, ..ito the p qui,e d,ffe,"", .) A , ... Iocny of + 5 fils and one of - 5 mfs oolh h3ve an """"ialed SJ"l"d 01 ~ mt.. The sp.",oo"",(e, in a car ",.asu",.. sp....m. nol wlcity (il cannot d 0): (I ) x _ Jt - 1; (2) x _ - .\I' - 1: (J) z _ :u,~ and (4) z _ - 2. (oj In which ,;,u,1ioD ~ u.. v.loc_ "r v of lb. panicleroru","'? (b) In wbicb" v in , ••• e~ . ';'ex dir«tiOll?

Sample Problem

m •

Figure 2-6.. is"n ..-(.) plol 101 an ",,""",0. cab lba, is inilially Slalionary. Ibon moves upward (,,-hien we 13 ~ . !O 00 lb. posiliVi' di'I'C,;on olx). "od Ibo" SlOps. Plot vCr).

•

2

We can find {he veloci!y a! an)' !i"", from Ike ,lope of lh. x(t) CIHW a! {k .! !une.

j ""

Calculatio ...: The slope of x('j. and so"ko !Ilok>foti.", ~

1M posilioo 01 a p.1.lie'" mo\1ng 00 IUl .< ,,-,is is giV('n b)x - 7.8 + 9.21 - 2.11'.

- 2. 1 mI,'.Takiog Ibe d",ival;v~ 01 Eq. 2-5. W~ 1... il~

'"

(2-5)

,,0

..ilh -< ill ,,,,,lets and ",",nds. Wbol is iI' ,-.Iocily al I - 3.5 s1 Is Ihe ,-.IOCIly conSlIUlI. ot ts il conlinuousl}' changing?

i Vcloc,I}' ts I.... firsl "",i' -ali,.. (,.-.Ih m'p.lCI 10 Ii,,,,,) of lhe posilion IU OClion x(,). u/culationo: Fot ,implirily. lh. unils have b,.-;eo omil IN from Eq. 2-5. bU I you can i~'1 lhem if }'OU like by cbanging Ih~ COi'foc..nl' 10 7.8 m. 9.2 mls. IUld

" - T, - d, " (7.8 + 9.2f - 2.11'). whieh tK>, Whon • is conslaol (al ei lhe, 00' 4 mI').lh~ ""'ivali", " Zi'ro and so atlo;, lho> ooc.l "'ralio" . When lhi! cab fus1 begin' 10 mo,,". Ihe ,-(I) eu II... ba:s a polili"" derivaIi.. (Ill« ,Iop into tbe page). (C.."",y U.S. Ajrf"omo)

n." ""O""oos IOU would 1",,1 .,hilo riding in Ihe cab of FIg. 2.fi .w ",dicalM b). tho slotchoo figuros at the bolton!. Wh"n lhe cab fi.st acede"tes. you fe1 as Ihougb )"ou 3te p,o",,'«,1 .. >,;"" ;. posUi'~. but tb. speed It .. d",,..,.,.d. Th. " ...,., i, tbe in .iyu: tbe di'efl graph' of }'Oll' posilion. wloci'y."M """"Ie,alion would resembl~ Ib"", in Fig. 2-K (NOlO Ihal a(1) in Fl.!;. 2-& is oon SI.nl. " 'hirn ''''Ill''''' 111.1 1'(1) ,n Fig. 2-& ha~" a conSlnn! slope.) Lal", when }'OU bla ~ ~ ,00 ~ar 10 a Slop. Ihe acc"",ralioo (01 1k'CCt."lion in oommon language) mighl .Iso be 'I'ploximal"l)' conSiant. Such cases a,e so common IIID! a 'p"cial sN of equD!ioft'l bas ""en do,ivoo 101 de'ling ",illl 100m. On" apPloarnlo Ihe do.,,-alion of Ihlds d,M, - a, ..-k"", is I"" di>finili"o 01 a. Figure 2-Sb sho .... " plot of Eq. 2-11.1ho \'(,) fUlKtion , Ih. Illocl;on is Iia;>Jf "nd thus thc plot is • Slraigbllin". III • similar maone, . .." can ,.,,·rite Eq. 2-2 ("ilh a low chang"" in oomion) .. v , .. -

x - x. , 0

and Ih"" as (2-12) in •.-bich x. is the rosi!iO" ollbe panic"'., , - 0 and ", .. is 100 avc .. g~ yelocity ""1 .... ~" I - 0 and a law. 1;""" . For Ih. Ii",,", v.loci! YIll"Clio" in Eq. 2-11.lbe a"Hag' velocity ""C, any Ii"", interval (,"y. from I - 0 10 a later l;n", ,) is thc aver.go of the velocity.l Ih. 1>ginning of 1M interva] ( - "oJ and 100 velocily al Ibe .nd 01 Ihe in!",,-al ( - ,'). For Ihe inlorval from, - O!O !b~ la!", !ime 'IOOU. Ihe aw,"g.. "lid asked !o find Iho fourlh. Equations 2-11 ""d 2-15 coc h co"lallllour of the"" qu"ntilie" bU I "01 lhil "'"'" fou,. I" Eq . 2-11. Ihe "m.-;si"g illgtedi.",., is Ih. disp]"",nwnl -' - x ~ In Eq, 2-15. il is Iho ",Iocil)' " 'Th an equalion frOlll Ibis ~SI Ii/you hove lhe ~Sl " 'ilh you). ChOOS !he oonSlan1 -acl'S!ood, but !boom !"'O main argun ... (1) n... ...oodp.'der", molion is ,Imoo alo"g a Slraigb! line.Some r.",an:b"fS bel",,,"" !ba! ooocu"'oo can occur III "uIllaos .nd anima" ..nc" the ooad is rapidly ro1a100 arou"d!he neel (o "d b,"in Slem), 00llhal il is "'"s ~k.I)' in SlIaigll! lin;> motion. (2) n... "'uodpilcl ds brain is anocbM S() "'01110 100 "'Ulllha! lhore is linl.:l re,idu,l ",,,,,,men! or osciU .. i"" 01 !he b.ai" jus•• Iu', lhe impact and nocha"", for tOO lis,u. oo"U"",ing !h . skulllUld b,ain !olcar.

",e

""s

"'1SU' its posilion

Figure 2-9 gi'·.s a parlide's ",[ocily " "' il m",'", along an x axis ~iilh ",,"slanl What " ,IS ",I0> ,/Od'Jim'" ;""grol ('" II",id,,;mlil'd horizonlal and ,-..lical mol ion, ore examiuedj (2)"J"b.> I •..,-Iall ace"Ii'''lio" i, uegah"e-Ibal is, """-""'~rd on Ih~ y axis. 10-"'ard Earlh's cenler-a"d so il has In. "alue - g ," Ihe eq"aIlOn~

'I_d,

. - The boe_falt """,ler.,ion .... Bank', .",[ace ;u _ - g _ - 9_8 ",!fore;1 is ca ug/n).!1Ie eq"aliollS of Table 2-1 apply!O il' mOliofL The a.ocolofJtion " al .....ys a - - g - - 9.8 ,n"'. lli'gal;W arid Ihu,

"-

2· 11 alld 1·16: ,juring th~ .. oce of .... ,Joe f... her 'M ,pple fall . scen!. lbe magnilude 01100 posil;", '"locily d walSl!ioll'> lrom (b), lot a - - g io Eq, 2-11 "nd then substitute. in lum , thi! valu", 1 - 1.0 .. 2.0 .. and 3.0 ~ is oXllmpl~:

(c) Wh.1 WM Munday', velodly a, ""

w. ,".I.",,?

,.""h..-..:I Ih. "'.-

w.

H.," '0

' - '. - 81 - 0 - (9.8 ml,')(1.0 s) - - 9.8 mls.

Calculation: To find 18.> velocity froRlIl\;> "rigi"al data withool u.ong Ih~ tilTh' of faU from (a), we rc"~;I' Eq . 2 -16 in J DOlalKm and Ihen ,"t>sli!Ul" ~"OW" d",":

,.' - \'1 - 2g(y - "0) - 0 - (2)(9.8mls')( - 48 mI. • - - .10.67 mrs _ - 31 nlfs - - IlO ~m lh . (Ao".-,,) We eb""" 100 negalive 1001 hefC D..'Call'" 100 ,,"iomy

so

"-as in the !Ird. Th~ ;. Hue no Ill. " ..... hoe .. ,b. body;' Ioca,ed, We ,.te d.. """,Ien';oo '0 be DeS,live ( - 9.8 mi.') in all problem. d.oIi'g ..-i1h CallinS bodia A "'g.. i"". "",ele..,;oo """0", 'bat, »';m, goo. 0., v,loci')' 01 'be body beoom",

.ui,

,bo,

,b.

eitbe' .... pos;';"" or ..... ""sa'i",,_ Th .. ;' nu. "" "",.. , wb".lI.. body "Iocated:ond "" maUo, bow /,." 0';' IOh:o,

dir""io. i, .. DlO'-in!o In S..nple Probl.m 2--8. ,be """,Ie' . ';011 of ,be baU ~ n'l" '" {dow",..,dl 'b,OII£O"'" ;" !iglr', wbtlher ,lie bill ~ ,i,inS or Caning,. Tactic 7:

Untop""c.b. g.A .. .... ..,.,.." .;bili';e~

,b.

2-10 I Graphical Integration in Motion Analysis Wh o" "" hav~ . graph 01 an obj.."C!'s accek:lralion ",''us lin",, "''' can inlogral~ on Ibe graph 10 filld Ih" Obj..'CI'S WIociI}'"1 gi'"en lim •. Soc,u", acee",,",io" a is dofi"'-'gan 1o aroelofJle?

From 100 ms 10 110 ms. ,egio" C bas Ih e sll'pilof . ,,,,,. {aaglo...'i{h a,~a a,,,a c - (0.010.)(50 mls') - 0.50 \l\Is. SubshtuhGg ll1c5o "olue, and I', - 0 iOlo Eq , 2·26 givo:; us

", - 0 - 0 + 1.5 mi. + 0.50 I', -

",Is.,

2.0 mi. - 7.1 k mlh.

(An' .... e')

Com""",tI: When 100 h~,d ;. just Slarl'l\g to mov~ (o ...·,'d, Ihe 101S0 .I,,,ad)' bas a sr-tioo of . " ..·end ooIliC'em.nt i. a """'Of quan.ity.1t iT pooit; ..,

~

tb. p.,.tiocle

b., Mv.d '" .h. positive direclioo of Ihex """ ""d •• saliv, if tbe. particle II .. lIlOV,di. tbe. ""!iui"" dir«liOIL Average Velocity \\0." a p.njde h., mv.d I",m pooi . tiOA x, '0 poo.;tioo. •• 'be ocreler"iOll;' .'" OOIK1>n'. F.... Fa ll Ae.-I...."'lon A. important exomple 0( Itroiy.t_1ine mOlioa witb occe .... tioo .. that 0( ... object ,;'log or fallin! f,,,,1y E arth·, ,wi""" The COII_ " . n' . ere" "'tioo "'Iu"i"". d.lCJibe ,~ .. DlO1iOOl. 00' we mak. ".0 chang .. in nor.,ioo: (I) we the 11101 ;. ,be , iY' of ,he pa""' .... poo.iIi",,?], ,he p .. t~ , .... velocity pooi'i ... neg. tiv •. orO>1(b)' _ 1 ,,(e) t _ 1o.and (d) , _ 3.1 (e) How m.oy ,im.. doe. tb. panic.. ~ ,be poi.' x _ O? 1 Fig"" 2_17 giv .. tho v.1oII x ..is is given by x _ .lr - 4P + ,,_u-h"ox;, in mot." >J>dl in >oooodri. Fiod the po>i'iOil 01 'he object" 'he loU"",ing vol." 01 t. (a) I .. (b) 2 ~ 3 .. ODd (d) 4 ~ (.) "''bar ;, tb. obj.tioo. .hould fiod overag' of Jit, ",d dil, (method I) Of .bould we divide d by tb. ,v""Ee 01 and "l (h) Wha, is 'be l,actio...1 diff.rt"", in 'he metbod. wb,n • wiod blow> alons th. c,,', ,oute >J>d the 01 ' h, "';ll(h~ ". !. (a) FOf lOtI» ..pauo diu:uu d btllWtll I'" IU'H Qn 'Iou , •• """"'..,,~ , ....iII ....ion .. )'! II 'hI "p""'lion io I....;". IM1 """""'~ ....... or. ,be (b) .p«d - ' Ie) lIi,ect, ... (lIpm..... "dowmu..",)of'b. ohock ....! ~

,,,,bo ....

, ..!Mr' whor:r ,he ... loaIy ......,1 If 00. "". ,be ..... " if .... 11.-' ..... (I) b ,10y .- - 1!>or' D ......... , • io _ Ho. "" io ... .!«or.,.I...,.. lhc "'iv........ ~ .......... "",1

..,ly ..

TH politi ... of • I"'''id. """;'! - , 110 . .. oxitr t>y.r _ ~,75 + 1.50" ....... '" , io ;" """""" CakW ... (0) 'boo ue"!' ~ Iolword ... ,be ,,"P'.'- _ iod..... ,10.< ..... +JJtor 'loc ..... -JJt i. W 0 - lh.. ...t..... ioat..., '" pbx """u.,.nd ,. 'm ... , for !be muoo.

.25 Suppose , rock .. oIlip in....,p'pace 1IlOVe "';,h ,. _ S.70 X 11)' mf~ lI'hal ;, it> """)tT"ion , II1>nt 7 ....

Nooo .. « . ... dng AllmS oIf iIl,o the air. Typically. the .pm...""b. . . >peed 01 1.6 mi, ill a 5.0 pm I,uncb: its .peed it ,he. re dwo.d to "'to in LOrn", 1»' ,be .... u ... s th., d.,a and "'UmiD! co",ta"' occtl .. .,iOlU. lind the """,Ieration ill, .."" of g dur_ ins (.) the 1.lUICh and (b) the >peed reducti"". ~

"ache.

.29 An . I«"k .. hid. "am fTOm ... , and """"Ie .."" ., • r.le 0I 1.0mJs' in . "raigh' Ii"" uMil i, "oclte •• >peed 01 20 mr.. The v.hide ,b ... >low. >< a oi l ,0 mi.' UD _ ,il i, stop< (a) How moch ,1m •• I. poe' f,ol[l "OI' '0 "op7 (b) H"... f .. 00.. ,b. vellkle " .",1from '0"'",,7

"",.""It .."

.tar,

• 30 A world', land 'I""'d ,,in .Iev"", XiOlwo .potecelerat" fTOm , .., .nd ,b •• back '0 ,," ., 1.22 H""" for do., cob ",ov< ...-kile ,,,,,,,lfIa'ins 'olull 'p'edf,om .. ,,? (h) How lon~ 00.. i, take '0 m:a. ,h. """,,"" 190 m ,un,,,ani.~ and .nd_ inpt , ..,7 • •

,_i.oce

mi.'. (.)

,h.

. 32 Th. bo-ake< on )'Our ,,ar can sl".. you ., • "''' 01 5.2 OIi". (.) If)'Ou . re SoinS 1J7 bnlb and "-_Iy >t • • i. the minimum time in wlliorh you caD set yottr cor undor ,he 90 balb >peed limit? (The ......." ,ev.>!, ,he fu'ility of bo-akin! '0 kNep )'O"r bigh >peed flOlIl be;nS dpeed 01 ,he ,rnn.f,om """ .... ,_up '0 tbt l1ext? (d) a ..pb x. ". ond .""".. ,,'" the iIlt"">! an_up '0 ,h. next. • • 37 Con A and B """" ill tbt ...". dirtclioo in ~, !>rI.. Th. pOiOOo x of n, ! givffi ill Fig. 2_26.trom time, _ 0 • '0 , _ 7.0. The figun'" ",nn! ocaIint .. •," by x, - .11.0 m. A' , _ 0. car B is atx _ 0. with , ve_ 0(" locity 01 12 ..~ >Ad •• 'g>'i", FIG. Z_Z6 ProblemJ7. .,ioo '" (.) \\b., mmt _. be sud! """ Ib..c"", (""""".,.,ily) "'" by side (""""".wi!)'" the ...... voIue of x) '" ,_ ,t o.? (b) JU """ val"" of _.. 00... ..,.,)' tim .. ore the can.ide by side? «) s.t,etch the pOitioo .T of cor B"' ..... time , txt R~ 1r2b. H".. """')' times will ,he " ... be side by side if th. ""!niIade 01 """,I" ",ioo _. .. (dJ ,hl1 .. d (.) "'" til .. th. (.)1 • • 38 you .... drivillg 'word a "affio .ignal ,.,1teo i, "" •• yello .... Your.peed i. ,b. Ie!>! 'p'.d limit of ''0 _ 55 kmlh, YOllr be" dodthe ~ . , x, _ Dl m. If ,be red C'>I Iw • """".. t vekxity 0110 kmlh. ,b. can p....odr ott... . , x _ 445 m.... d ~ it ... . """"an' ve· Iocity 0I-W kmlb. ,k,y P"" e.dt ott... " x _ 76.6 .... are (0) the initial ",loci!), and (b) the 1tik outside duri.! •• oinsto..,,?

I'robl. m.40 ond41.

Figure 2·18 .ko,., . rod ""' .nd . cor ,nat ,"d. ,he •• gino •• i. ,boded '0 .... ,nat a looomOl;'. bas improp1 istb . spud 01 tbe boat1 ...... .. 52 At time I _ O. • ppI. I .. dropped /r"", , bridge 011'0 • rooclw.y be.t >t h the bfidB'; """""'hot b'tT.• ppIe 2 ;, 'hrown doMt 110m th . ...... bfiSh'. Figure 2·31 gives 'be wniocai po>itio .. y 01 tb. apple< durin! tb. falling. un· ,il both apple. nave ru' tb. roadw,y. ",tb opproxirn , "l y ... nat '1"''''' ~ .pple 2 thI","" t 110 kmilL Your "gum,", e1i",r" your . ",ruioo 110m the pobc< Ie< 20, (Jong rnooglr for yoo to look .. the phme and )" U. - [ "",,'t do that!-). A! tbe beginning oI:!Iu, 2.0 ~ ,h. police oIiX" begir>; bntinB,uddonly" 5.0 mI,'.(., Wb>t ~ ,b... ~'ion bet"""'" the when )'OW ."",tion finally r tb. ",,,kal .. Ioci'y 01 the packa~. .."." 'ime. from belore i, br. rn fr .. '0 wh,. it reaclt .. 'be ground. (. ) Wh.u """,imum

o

0r-, 1C-'l,-;- T'" 10 fIG . Z·J2 Problem 53.

_ _ Ch.opt ... 2 I Motion Akmg. Strlligtlt (j,..

,bow.

bdgh' ,b. bf.ak_lroe poin' doe. i, ri .. ? (h) How rugh i. ,be bI •• k_f, .. point , ..."" the YOWld?

ve" ••

•• 54 Fi!,," 2 _33 ,h"", ,h • • peed \' height y 01 , baH ,oo .. d directly up,. .. d. oIoog • Y Hiam< lim • . (.) n.", i. 'he i"~ i .1 speed of ,toe """"'" "one ? (b) Plot oeloot)' v ...... ,ime 00 • yap!> to. . ,.,h " one. 'akinS zero time .. tho in" • ., ,b. firs' "OIIe;, released.

•• 57 To ,.., tb. q.>Ii,y of. to.o;' 00ll, fOIl drop ;, 00'0 ,be floor f,om a heiEh' of 4.00 m. It ,ebooodo to. h' igh' d 2.00 m. 1f ,b. ball ;, ill oo,,, :oct ",til ,i>;, 1I0OI to. 12 .0 ..... (.) lObo, i. ,he .... sni'ude d it< """.g. "",.1 . .. ,,,,,, dwill8 .h>1 rootact . Dd (b) io the "" .. "!!. "pOf_1 •• 58 A 'OC. ~ ,hro..... vtItic. lly upward Irom grounod 1e".,1 >! o. At' _ l.5, i, !W'" ,he ,opol > tall tow" . • Dd 1.0. I.,,, i, , ..ebe. iu maximum heigh', lIb., ;, ,he h.igh' d ,he t"....,.. 1

"""I".,,,,,,

,ime , _

""zzI.

•• 59 W" ., drip f,om ,he 01 a showe, onto tile tloor lOO cm below. The drop< 1000.t resub, ("1" 01) in'ervak d time. ,he /int drop .. rikillg tbe ftoor at the ;""ant lourth drop begi .. to loll. wtr...,he fi", drop ",ik ... the ft OOf. bow far below tb. i>OzzI. are ,b. (. ) .. ooad .. d (b) 'hird dropOCepl . bIe soIu,ion d the GU'dratic "Iu>tioo in ,til" you obtain .

.b.

,Ir.

•• 61 A dr"",y "', >pOt! • _rpot thai ,oils fir" up ond,hen doom pail ... ope-rr 'Olin.CCe1e"uion a{') 01, o 0 does i, fir .. re >eb .. ~ e of ,h. ",,,,,.1

,hi.

·.at

i."••,..

'pol"

,im.

,hi.

•

""'1

""',b.h.,

FIG. Z.JI Problem 69,

••• 70 Two pan iod .. mv, along." x >xi. Th. poo>'ion of pankle I ~ Yv'" by x _ 6.00r' + lOOt + 200 ('" mele" """ .. «>D< , ign.1 wi~ ,b • • u'ornobile ke ,b. u""kl (bJ How fa" willtb. au,omobil. be ,,,-,,,,I"'g" 'h " ::lI

000lIi ' f>«'d wh •• i, pa>oe> ,b, halfw.y poin' 00 ,h. '''ydown' (d) How lOllS b.. it """" falling wh •• i, ""... ,lie h:rlfw:oy poin'?

,.,t"

,pan. How loo.!; of,,, ,b. fir .. diamond beEiru '0 fall Mil ,h. ' ..0 di:tIllollOd. be 10 m . """ 77 If , b_ alI pilcOe' ' hrow> • f ..,ball .. a horizon'a1 ' 1"«101160 kin/h. bow IonS dou ,h. ball,ake10 "aclr born. pi ... 1M m . w:oy? 71 A proroo., ..... aloog ,h. x..;, >OCO!. (e ) lndic. " ,h ....."'. " '0 (h) o. the y . ph. (I) 1'101 Yv",... t>M indica .. 00 i, ,he .... ,.," '0 (' ). 79 A rnorOlam. velocir~ II "",ne. ,h. bon"," 4.1.0 I ai, .. i, ~ dropped. (0) How d"p" ,he Iak.? or. ,he (hI and (eJ direc'io. (up or cIoron) of Dve ..!"- v. loci'y of ,b. ball for ,h •• c,ire foll l Suppose ,"", all ,he ",ate, i. dr.in. d fr om ,h • I. ... Th. ball ~ no..- rb,o..-n from ,h. diYirlg !>o:r.rd so ,na. i, . gain reaches ,b. bonom i. 4,80. Wb" ar. the (d) m. gni_ ,ud • • nd (e) diro:r.rd dit is !be I.....", """,J"..;oo • ...t..d!of takeoff/rom, 1.80 km .. oway?

oo.""nt

U An automobil. driVel iocr."",. ,.,. speed .. . const ... , r.t, f,om 25 kmlb 10 55 kmlb in 0.50 mlo. A bkycle rider "",ods up " • 000>1..,' "',. f,om ,," 10 .lO l mlb in 05() mi • . \\nat ... , ... m>gll;tu"". of (0) ,h. drivd. """,,10..,;00 . nd (b) ,b. rid,,-, "",.lfIatioo?

ti", di"""""

85 To st"l' • Ih" dol ("" .he Iefl mFig. l-ll). !>om! eti"" ,ito., (a) H"", f.. from ,b. low" dol >llould you pIace.be 50.0 .... m"k? H""" moch hiy.er ,book! you pi""" oil. m:uks !of (b) 100, (eJ ISO. (d) 200.... d (.) 2,';(j ""? (For eumple. >lloold ,Ire (00 m. ffi>rk.r be 2 om .. . . far from tho dol '" In.. ~O m. .... l" l U '" y"" ." an"." 01 2 'ime~ 0", you fi.d >my 'un", ,be ..,we,,?)

.r",

p.'_

.,..we,

1>."""

,

•

,

- ,

II A rock ;' >1101 ""nically op·. ,.rd from ,be edg< 01 'be lop 01 0 I. Ubuilding. Th. rol ..-bo dr"". ,he 100km tbe Buffolo ."lran« ond tbe N.", York Cily .. it ., the legal speedlimil? ....

.""'go

!>O,,,,,,,,,

A " OIor'YoIKt wbo i. moving >I",,~ on x aJlis directed ton:.. >II >ch hio 'op ,,,,,,,d in • di .. """" of 12.0 rn. He .. ,b •• able '0 maio'",n 'OP 'p".dlor ,h. ,emainder 01 • 100 II'h.,;. hi> ,ime lor ,he 100 m ,acel (h) I. or< ... , " •• n • sudd,n gu" of wind ,~".., the iceboa' '0 b •••• """_ """,Ie, .. i"" 'oward ,he .... lor. p",iod 0110 ~A plo< ol x ven", ,~ 011""", in FiS- 245, wher" _ 0;' lak.n '0 be ,he in,"n' ,be wind " .. " '0 blow """'be pooi';" x >:Iio i. ,,,,,,,,rd 'be .,.". (aJ Wh., io ,he occel" .. ioo 01 ,be iceboa' d\lIinS , ... 10, in,,,,,. I? (b) lIRa, ~ ,he veloci1y 01 the iceboa' .. ,he .nd 01 !h. 3.0. (c) If ,be o=le, .. iOll Itn.';", 000"an' for IllI oddj' ;OIIoJ 3.0~ how fOT doe. tb. iceboa' Ir""el durm3'hiooeoond3,O .in"rval? ....

"aD'

i.,.,,,,,)1

•

" "

",

" "

"

, " ,

1,00l • b:m.1 olleng,b 1.20 IlL A".mi.! ining ... expelimenu l p,kag. =0 be dropped. (.J How long i ...... pIleTe i. f,ee foil? (bJ WIu, is i.. 'I"',d j"" ar i, """he.. ",,,h,,,! devie... bot'om 01 ,,,,,,,,,t? (,) When c. ugh'. ,he 'l'h"e expeli.."" '" . ve.,8" d".I" .. ioo 01 1'ig .. i.. "",od ;" reduced '0 ",ro. Thr""gh w!la1 diou"", 00...;, , ..,..,1 during ,b. doc.Ie, .. iOll?

,im.

i,"

,h.

,b.

112 A b.all ;" 'hrown .w..'n v,,,icall)' ",!h .., ini,ial ~,J ol I,lrom , heigh' 01 h. (.J WIt" ;, it! "",od ju>1 belore i, >trikes 'he g,ound? (b) How lOllS doe. ,he b,,1 ,ak. '0 Ie""h 'he YOWldl be ,he (c) pan • and por' b ~ ,he baU "'.,,' ,hrowa "P~~,J fr"'" 'be ...... heigh'."" wilh the same ini,iol "",ed? Befor. ,.,Iving any «[ ... ,ioni oJ", pl.)," face • • ball .. speed and invoiunt>rily bIi.h h. black> out the ICeD. !Of 100 ..... H.,... for ""'" ,h. ba.II "'I" during ,h. black"", ?

,bo,

Vectors

, ,

,~"'"

,

,

"

, ,, ,

,i cC ,

;/

,, ,,

,

,, , ,

" The desert an! Cataglyphis forti. lives in the plains

of th .. Sahara desert. Whe" on" of the ant. forages for food, jt travel. from it. home nest along a

haphazard search path Uk .. the one shown here. The ant may travel more than 500 m along sud! a com_ p/jcated path over flat, featureless sand that con· lain. no landmari< •. Yet, ....nen the an! dedd ... to reo tum home, it turns and the" run. directly home.

38

How does the ant know the way home with no guiding clues on the desert plain? The answer is In t hls chapt ....

}'3 I Add;"gVKlorsGoomCIt,ic:.lty ~

3-1 WHAT IS PHYSICS? Ph)',;c,; deals with. gre.1 many quantilWs Ihal h"... bolh sire and di.ection, aOO il """ds • ""..... '1 fIlJlhomalicallanguage-lhe langu'ge of veclors - Io CIion. ThUs. Ihey s""cify ioomical di'pJ""""",nl we· lOIS and '"pfOscnl Ibe 5aJl1C changt.

Eq.J.1.

i" 'ii. If w~ w,""o indiea . e 0"1}' , .... m,soitude Ollhe Voct01 (a quantity that lad. a 'igo ot dircClioo). we • hall "'" t .... italk symbol. as in a. h. and •. (YO" can "S "ith 1!Ii>

""'"'0/" 'qro""',,,, 7 - li + b.

(3-1)

a

FIG. J.5 The veClon b . ad - b h. v. ,be ...... m. gnitude ..d opposite di· rocUo...

,,-hien ,")', th't Ih. vecto, 7 i, Ih" vOOo, ,urn 01 vOOors aod b.Th~ .ymool + ," Eq. 3-1 and II> wo,d, "sum " and "add" have diffe'''"t meanings fot VOCtors than the)' do i" too usual algebra OOcall'iol Ihey involvu !Jolh magniluc:l;> and diroclion. Figu,e 3-1 suggests a prOCi'du," fot adding Iwo-di"IC",io",i wClots Ii and b goo""'1rk-~lly. (l ) 0. pap"'. s~Nctt ,-octot --.; to some COIIve"K-"' "",10 and al Ihe P'''P''' "ngkl. (2) S~,,'ch .-000' b 10 Ih ... me seal ....lIh its 1ail allh. ""ad of v.c101 --.;. again at the P'''P''' angle. (J) The wetor ,un, 7 ipla",n",m veClo," he"' . lhe ,ulos f01 addition ,nd ,ublraClioo hold 10' "OClors of all bllds ••,helhe, the~' '''P,eSnod ? _ iI + 5. Coo,iderinS""'ious Of;' . " ' _ of iI .oo S. "b.r .. fA) lhoe maJIimum """sible magoirude for? and (b) the mioilllUlD poosible m'Y'i,1Ode7

Sample Problem

.--....--,

m

10 ao ori~ nl""'ing class. ~'ou h,,'e Ib~ goal o f mo",ng a. far (Sirughl· Lne d .'l1a"",,) 1'001 baSIll P i< y eat." if yoo wwle,.o:o di.· pI.« ...." iI. b,.nd - ?, io "Y Ofd". a"ang~monl 01 ' ''''Iors"if. b. and - ... Th~y call t>c in an}' 0''''''_ becau,., 100;, vecto, sun, is Ihe sa 'ni' for an}' 0''''''. Tlhl 0''''' ' sho ..n in FIg. J-7b " 10' Ille vecto, sum

d - b + a + (- 7 ). U~ng Ih~ scal. giwn in Fig. 3·7". w~ d of Ihrs voClo, sum. find,ng d -

"'0""'" Ibe kon gl h

a m.

(An ...... ')

•

3·4 I Components of Vectors Adding ' "OClors gei, arrowheads would poinl low,"d negat;'-o x and y. Rosoh'; ng "CClor b;n Fig. ] ·9 }'idlk a pusi"ve compoaonl b, and a ""gal;ve compon~nt by. In g~noral. a voclo, h:rs Ib'N compon e nl s.. all ho ugh 10' Ih ~ case o f Fig. 3-&

a

a

a.

>lid.,

J..

FIG. (o)The"""'p"""nt5 . , of v«ta' if. (b) lb, """1""""1' >Ie undwI,ged ~ ,hoe """"" io 1oot '" ,he ....gnitude aad orien, .. ;.., >re m ... ·

_od...

tain,e:

" . - aoosB ,nd a, - a.i0 8.

(J.5)

wb.,. 8 i< l/Ie anglillhallh. ,,,,,,ot;; mak., ...ilh Ih. posil,,"" dir"",io" of lbe uis. ,nd Q i, lb. magnilude of a. Figure 3-& sbow, Ibot a "nd its.< and )' .om· pon"nlS form a nglll Iri,ngle. II also sho,"" bow .... ~~n ,coon,lIuel • '·i'Clor from ilS oomronents: w~ a"~ng. tbose compo""n" Mr.,/ 'ail. Thon " .• oomplo le a righl triangle ""h tb. \\.'""" 01 veSag ... !raveling a ,,~t 26 ~ ", ,... "'I .... 'd. 3.9 hn soulhward .• nd 25 m upwar d. Il>.bowing d• • Ddlbeteam·, o ..nIl di< . bIe '0 jIodg. ",b.,he, • cokul>lOf re.ult ~ Eve. k.oo..iog ,b . sign< 01 the IUncli"". iD ,he .",iom quadran" can I>< 01 help.

/unci"'.'

,,,,,,,,,,.bIe.

Tactic 3 :

Inverse Trig Functions

",

"'be" 'he inve'''1r~

f""CIiow sin- '. 000- ' . and , • • - , >I. . . ken"" , calrul>1Of. fOIl m ... ' ,b, ...""",bI"""" 01 ,he ""''''''' )'00 ,he .. i...... Uy . oongod and ,h.

,

.,

,,,go

i-

,

II =

-

I

r

,.,

FIG . 1-1l ThI .. =Cut curv."o rememl>nd kde_ line ,he direc'ion< of . IiYI1_b...ood """,din". '),",01.

A uni, "ecr,,, is " '"OC101 that bas a magn,tude of ~xaclly 1 "od po,nt' in a particula, di''''''ioo. It I,ds bot b ~im.n"on "nd uru1. 11, .ok> P"'p1 '" W';I~ Ih. dispLac.:lmenl J of Ih e sl"lun king leam of Samplo Ptobl.:>m}'3 ill I"mrs of "nil wclors. Firsl. "'P",impos nel displOC>lm'"1 11_. we 10 .u mln.. fiv. ,"d,,"du.1 di-'

,., •

• '". ''''

' '--' - --"T'

'" (o)A

,.,

FIG. ,." ~h p.. b of!ive run> (b) Thu and Y compon,"" of".... (e) Vector " __ poiD" ,t.. -..y '0 tho hom......

- 11,

+", +11, + 11, + J,.

(2) We e,'"I""e In;'; sum lor the x compo"","I' "Ion". Ii .... - db + d" + d" + w, us d-", - + 3.8 an. V.ctor d _ "nd ilsx "nd )' componenl' are ,how" ," Fig. :J.-17b. To find Ihe magnilud. and '"glo of J ... from ;1, oompo""nl', w~ \ISO Eq. 3-fi. Th . magoilude ;,

d_ - ~d;" .• + d!...,

Ihe dirl'Clion of J_ is io Ihe fou rlh quad""1 of ou, xy COOf.r~u ... w. do nOl l """, Ihe valu~ fo, Ihe angle

BI

en""", Ibal a_. is

HoI'''''''', we call exp ' e..s Eq, 3-16 in I"",, of compofo, ei lh ~, llie -" alis 0' 100 Y alis.

Calclht#ono: Since (: is d lf('CI"d along Ihe ..- ax's. 1L"

(A "",,,,)

."0 ",iIO C. - A, + B ..

We ""xl "Ipross ead! x oomiX'"e"1 in 1M form of 1M

-' pari of Eq. J.5 and ",b;l;IUIC l "",,-" data. Wo In.,, M"" C cos 0" - n.OCC,

,

..-

C

. W~ noxt mowd t>ac~ 10 Ih. A' uis. 10 ~valual~ C

3-7 I Vectors and the laws of Physics So for, i" eV"f)' figure Ihal welu"", " coordioate 5j'S1"m. Ih. x ood)' aXilS ar~ par· aUd !O tile edgos of Ih. boo~ pag.!. ThUs. ".-h." • ,-octo. Q " ,ndu""d . lts compo",,"IS ", .nd a, ar. al"" p" •• liollo lb. odg"" (as in FIg. 3-19.» . The only reason f"r Ih.1 "",,",",ion of Ih~ u.s IS Iha, il looks "Propil' "; 'b.:>re is no d""p'" r•• son . Wecould.lIlS1e.d. to!>,. Ihe axe. (bul nOlloo vOClor II) Ihrougb ,n 'nglo Ibas in Rg. J.19b. ", "'h ir h case tbe compo"o"t. would ba ve new ' -alues. call1''em and a;" Si""e them arc an illfinit~ "umber of Cboir5 of . Ih...'" are an infinit. number of djff.. ~"t pairs of compo"""'S fo, II. Whirh toon is Ibe ··rig"'·· pair of compon~nts"l The an'WIlr is Ihal tooy are aU "'IuaUy v,l,d becallSi' each p,i. (~jlh it' u.s) just g>'"eS u. a diff..ent way 01 ~ ­ scribing too same ve and J6O' - lb. Eil"'" can be "",d in Eq. 3-20. I'oecaIlSo' lbcir a,~ the sam •. N o{~ lhal {he," a,~ only "",lars 00 Ihe rigbt ~ j"" of Eq. 3-20 (ineiuding Ibe valu" of cos 011). Thus a· b on Ibe l"f1 ,ide a ,cala, quantity. B.."'CJu"" of the oOlalion, II· b is .110 ~nOl'.. n as Ih~ dot prvda,'1 "nd" .poh" as ". dOl b: ' A dol p.od"e! can N "'g.1.d.."' bo;>lW""";'~-:;';,ili; ' ~-:-;',;";l:.:";'~';-:-:-------:":_::';";'0;,:.;,:-;';,0;);':_::',:00:'-,--:,~,-,;,;) - 2.oi + 3.ot? Althougll many of Ihe foUow(C",,,,o~:

iJlg slep' can 00 bypassed ..itb a wClof- and sin(:l6O' - o j differ in algcbrak sign .) &""' .... oltbe o01>1ion. Ii " b;,; abo ~ n~'n as th. rf" " prod" .... and in 'f"l"Cb it ,,"a rf"" b:

'0'

. - If if ond"liOl. p.",IIe! Of ""'; por.tllel if ,,"Ii _ o. The mogni'udoe of if x b. ,.Ilkb can be .... in .... ji1 " lil ;, """;"' um ",n.n if . Ad li are l""l"'ndicul .. ,o.arn "'her.

a

-- -

The d'r"",ion 017' i,; Jlr>d eo. VEflor produ"~ (0) s,."P """or . Th. thunm "nds up in Ih. oppo>ite directioo from p.evlOusl}'. and so it must "" th"?' - - .... Ihat i~

F"a - - (Ii >d n ccof< j as being in 100 posih.e d"., - b,u, ); + (a» , - b.a,)k .

(3-.lO)

A dote.minanl (APP"odi x E) or a vector-capable calculator cau a!:so he used

Todlecl: "'tlNh'" ao)' xJ: rootdiIL:lIe ')'>tem is. right -h.:uJded wonlin,t •.• ):sIem.

usc too.tight-band .ok< for the "'oss (>!oooo ; " j - b';lh trut s)'>1Om. If your fingers "'''''p i (piriS your bmd. (4) Roih".,o wOIk wi,h , ri!,h'-1wlded OOOIdio"e .y.. t.m , .. un. lObe. you illlge' how '0 d",,. •...,b a 'l"""" . Se. Fig, 3_14 iIII ooe l""l""'ive. Pr:rC1ko dn"';"g ",he, p..-spe1 1"" ~ • is ",!"ive, To divi.d. "by •. mul';ply V by II~ The Scalar Product The . ....... (or do.>t) product of ,..., V«tOf' it and 5 is ""it... it· . .. d ~ tbe u al., qU,,"'ity gi"".

"

(J-20)

ill which oj.;' ,hoe ""90- bot,.....,. the dire1he ""aller 01 U..... gl •• bet .." . ,b. diren' 01 it md S. Th. diren 0I ? ~ perpeDdicul:u to thoe pi""" defi_1w it and b and is given by • 'ioht_hand mk all ,00,.." ill Fig. 3:2 1. Nor. that it ~ b _ - (Ii ~ Jt). I. urut_veII aI_ ~gatOf plt. Figur. J-2J .... OM ... over_ bead view 01 011. p"ttmg ,b.n.nge 01 the team; ODX)'1 are ,be ,;gr.. 01 tbe x and y compoaentue.JII .00 de"""in. (.) ,be m'f!'litude ond (b) ,b. an_ gie 01 ,be car', from i" " ",ti.! point. He.

lOt" dispI""''''''"'

~ Chap_ l I Vectou .9 (a) In ".i,_vector ."' '' '''''_ "bat il tb .... m it + ri if ii _ (4.0 111)1 + 13.0 111)] and -,; _ ( - lJ.OlDji + (1.0 m}j ? W,.." ne ,., ( b! m. yllrud. and (e) directioo 0171 + bl ....

1'"""'"

.10 A 1"''''''' ",ill.. i. the following J.l km .orth. tbe. VI Om "'"". and 6n>lIy 51 'm ""'tb. (.) Sketch tb. vec_ tor di . g.nrn rtpruenl.! ,hi< mOl;OIL ( b) How for and (e) i. ..-bat dir«1ioa would a bird fty i•• "'aigb, line from the .... . " ",ting point 1 'be .. m. final point?

,b.,

t." ;,

.11

A person 'rom htl 1"'''''''' ""'atioa and in. dir.Clioo that ;' 35.(]' nonb 01

>1",... """irI..

e. " , Howev". sb. m.... ,,,,,,I oI 011~ to .. Ol' OIi. nted ,itb" nonb _",uth or .... _ "'.... Wha, ;,. ,he minimum cIi ...""" ,b. could Iravello ",.,h her io.1

.12 IU ,b. vWon it _ (l.0 m)1 + (4.0 m~ .. d b _ (5.0 m}i + (- l .Om )l. ~ive it + ri in (.J unit·vonor .", .. ioo .

(0) . the. (b, 1UId - 70). th •• (- 20 ""d e, ). tbe. (_60 >nod - :ro). \I'h" are (. ) OOnlpo!>nod (b) component e,? Wbat are (e) th. rn~n"oo . , nd (d) ,b. >fly.. l rel"i"" to tbe posit;"" dir«,ion of ,be x .. ill) of ,be overoll dkplocem. OI?

Th, .. V",'Of. if.•. >nd i" .:ocb have. magni'ude 01 50 IJI . .d tie in ... .T)' pl ..... Th.~ diroctioru ",I., ... '0 ,be posi'i", dir...,iOll ot ,be .T >xi. are 30'. 195' .... d ll~ · . 'ively. Wb .. or, la) maE.i,od. and (b ) ,be , ogle 01 ''''' v«''''" if + b + 7 ... d ,h. m.gni'ude and (d) ,be msJe 01 iI - • + n \>''h .. at. m:ogoi'ode >nod (I) any. ot • I"""b """or d.uclJ ,na, (il + b) - (7 + d) _ fJI .... .19

«)

,h. (.)

.20 (a) Wh" is 'he , um 01 ,be foliowinE lou, """or. i. uni,_ vee'"," nou,i",, 1 For tb. , '1UIl. wha' at. (b) ,be ...yti,.de. (e) ,be >llgIe in d.y.. ~ ""d (d) 'be angle in radi.",?

""S";''''''' ..

. 1Id "" {bJ • d (e) an OIly.. (re1>ti"" to I). Now give b - ;J i. (d) ".it _veclor ""''';011. and .. Ie) • mag_ .irude and (f) an :m~e.

•n

Twovtc!or .. regiv •• by it _ (4.0 .,)1 - (1.0 .. >1 + (1.0m)t.

.Dd

b _ (- 1.0m)]

In uni'_v.nor nOl,,"'" find (.J if + b. (b) if ,hird vector i" ouch if - • + i" _ O.

,h.a,

b. >nod (e)

•

Find the (.) x. (b) y • ... d (e) , """'poIIe.'" 01 'he >lim 7 01 displ:oc ..... n'" i" . nd d whOSf compoll.nts in met." . Iong ,be ,hr...." or. " _ 1.4" , _ - 1& " _ - 6.1; d, _ 4.4. d, _ - 2.0. J, _ 13.

.14

,b,

.15 An "'. crazed O}' ''''' Sun "" a 00' T• ..., of"... ooa. d . n. OVtr >J1 xy pi"". II-0). (b,. - 70.0). ( - 20.0. c,J. ( - &0. - 70.0). Th. over>ll di>pf>et ar. (0) b, >nod (b) e,? Wh" Ole ,h. (e) m' g""oo, and (d) .. gIe ("la'ive '0 ,he popf>ee ... n, 1

=

.16 I. ,be.urn A + 'A _ ~. v.n,.. A h .. , magni'ude of 11.0 ., and i> ",,0.1«1 -lO,O" """.tl ""gIll'udes of 10.0 m and ,be mgl .. • re ~ _ lO" .nd t>. _ 105'. Find the (0) x and (bl y """'POn •• " ot ,b. ir vector ... '" 7. (e) ,be m. gni'od. 0I7. • 11d (dl ,be

, ogl. 7 ",ok.. wi,b ,toe polly tr,veled 7.8 km ., ~ . north of doe ..n. (0) How lar aod (b) in ",nat dire,,1on mu" h. now ,,,,vel ,ore:ocbb,..""",p? ~

150 mok ... run ~.'h, is U; m oJ 01 due \>''h" mu!! be (. ) ,be maY'i'ude . nd (bl ,be dil' ~ i1 is '0 end up " the ...... 10 are ,h . following. r..,pectivel~ in met ... , (2tl . od

•• 27 T)'pical backyard """ oflen cr •• " , ""work ot chemical C"," guidonce. Ext,ndin~ from ,be n.". > ,nil braD ".U. i, II tell tbe ....y '0 ,be """ at . . y braocb poirn: If i, i> moving ,,,,ay frOID ,be n." . i, h" '\00 choice. 01 patb r"lw,ing' om.all turn in ito ",,,,I dir'lisle ~ _ 56.!)" oouot.....cloct_ wi! 'heo j, 11 in uru1-Yenor o"'>!iooifB, - B,1 C ••• 3 The ,br ... vector, in Fog. 3_l'i have JIla!IIirud" • _ 3.00 Ill. b _ -I.m m. and , _ lO.O m ",d >ogl' ~ _ JO.O'. lib>! ... ( ' j ,he x rom"""e", and (b) 'be y """pone.' of if: (e) ,boo x com""""" and (d) the Y com_

1" _ -I.~ - lOj +

,

121.

,

-~- . fiG . J-35 Problem 4J.

~ Chap_l I Vectou pont"' of O, ... d (.) ,be Xoompoo,n' >lid (f) ,b. Y componen, 01 n If i" _ pil + qt. ,.h, m the values of (g) P .Dd (h) q1

.....

•• 44 In. """,tio ~ 01 ~. mim.' I goes 'hroug/l a di._ plaoem,n' ilL_ (4.0m)i + (5.0 1D)j .Dd mime 2 soe. through. di,pI:=menl "il, _ ( - 3.0 m); + (4.0 .. ,). Wh., .re (0) ;I, ~;. (b) at' J,. (e) (aL + !I,), II,. ,lid (d) ,b. oompone"' of J, .Ioog ,h. dir«.;oo 01 J,7 (H."" fur (d), >t. Eq. 3-20 .Dd Fig .!-20.) "dd~ional

Probl.m. Rock faullS or. "'pow.. al oo~ '"toeh """",i'. foee.. 01 rook ••v, slid pa« .orn ",h... I. FIg. 3--"6, poi... A and B ooiDcided ""for. tbe Joct in--ilie for.you"" ,lid down to tb. lith,.The "" di5f>lacem.", AS ~ aloo~ ,be pi.... 01 ,he f.1lIt The oo..izoolal-S"'P""""' 01 Ajj i> ,b. strib..up AC The """'po.,,", 01 AB 10>1 i. dir,('!fd dowo tbe plaD.e of the f, uh .. the Jjp->IiJi.flD. (. j YOb., .. the "",£ni,.hp i. 17.0 m l (b) If the plaDe of bull i. ilOdimd "' angl. i.52.11' '0 d•• hocizOll'al. wb .. to vertical com""" ... ' of AB 1 45

pi"",,,,,,",

01 the p. nide lor 'be rh,ee """",i"" w.pI:ocem'"t~ Wb, . re (l) tbe x""",,,,,,,e.,. (h) tbe y""",,,,,,,e". (i) tb. magni_ tude ... d (j) ,.e diregni'udoe ond (j) dir«1iono/J, ~ (d,l4)1

«)

,l .r< (a) th. m".!."i,ude .nd (0) ,h. diroctioa of ~.011 "'b>l are (eJ tire Mog.o"ude .. d (d) tlr. dir.Iem. Ako. !! _ (11.0 + (8,0) OIl tb" • ..... ~"e 'Y"om. We .""""'a" tire 'l"'om oountercloct· ....... boo. the orig.. by 2OJ)",0Iorm ""x'y' ,y5t ..... O.thi ..... . )'Stem. wb .. ore(. ) A .oo(b) 11.blioo?

mp

'")1

woJ'.

1M> A '"""""" 25() "' '" ,Ire diroctio. 30" ... t 01 .onh. 'be. 17~ m lid Fig. 3-10: fof (d) .... Eq, 3.27.) " lfit - ]j _ 1? it + t _ 4? and? _ .Ii + .\j.,h •• ,.hal or. (. ) iT .. d (b) b1

Jr.,,,

,bo,

.b.

&9 A 1""' .... , "arri.. his 'igo 01 1"0"". " " ting from the oriE'" of aD xY' )" "m. wi,h 'he xy pl,ne noriz",,· 'aJ. H. move-s 40 m in tire .'g"i", diTau,,",;. a pkok ... a. EO. n","e .. ;1S posihon veCio. ch.nge, in ,uch ..... )' Ih.! 1110 '-ocl", poi'" (the origin). lillie pos~ always eXlond, 10 100 pa.!iclo 110m lhe lion v"""o, changos-sa)'. 1.001 7 , 10 7, dlUing " ""flain Ii",,, i"I""'"I- tbon Ibe par uc],,', d;; p ~......,m.. ' fJ.7 during Ihat lime inw,,·.1 is

.eI",""""

fJ.--; - 7, - --;,.

,

(~-2)

Using Itie UID1 -verl'" nolaloon 01 Eq, 4- 1. ".~ can " ..-.ilO this displace"'""l as fJ.--; -

(x,; +

y,j +

z,k) - (x,; + y,j + z,k) (4-3)

"'00" OOO1 silo"",, Ihe pkJls 101 five "alu", 01 ,and lhe pain coooochng Ihem . w~ c,n ,Iso plol Eqs. ~ - 5 aod ~ - 6 00 ,calcul"or.

4-3 1Average Velocity and Instantaneous Velocity If. p'rlide ",oves hom 0 00 poinllo a"Olnor .... " mig/ll n."d 10 know bO~' I,SI;1 JuSi "' in o.apl'" 2, we can Ikfioe 1"0'0 quanlilies Ih al dal .-jlh "ho... r",I" , OCeIe,.,,,,. oompoo'o"

liOll • 010

par'ide ",d !he ""aI.. OOOIpooe." of • .

~ Chop.... I Mo'''''' in Tw_ Th , ... Oim'il tb. lime '-anable, does nO! ap1"'"' tU tbe expwssion for e;llIe. ooc~"',"lio" rompo""", . Equation 4-17 .oon yields {j - ( - 0.61

tn!s';

-t

(OA~ III

!")i. (ARS.....)

" 'bidl " ,u"".imp0Si'd on Ihe rabbil". palh in Fig. ~-K To gel tbc magnimoo and angle 01 a.ei1be, .." "'" a wClor-capablo calculalO1 or we follow Eq . J-fi. F01 tbe ""agoit,,"" we

h,,-.

a - ~,,; + "; - ~( 0.62 mls')' + (O.4-lm li)' - 0.76 ml". (A"'.....,,)

Sample Problem

Beeall5C

Ib~ .=I~.alion

is con't.nl, Eq. 2· but ~.• musl u.. it Scllic motion of a ball . • nd nloc h 0110.1 i. Sl"'"! In Irying 10 ooouolibol motion fo. '" alorl~ w;,h tbeir rornpooenlS.N01e 'h>1 11", boo-izoll1aJ veloriryoompo... Ill " .. aio, """',.. , bu,

.!!}, 10.1>0 light. "",ical mOl;"". are identic>!.

The~

s"",,,. Rd u,,d M.p»! n-w",.". PII"'Ofl"I~M

Figure ~ - 11 shows. ""monmo"o" Iha, has .nLV{'noo m,")' a ph)"iCll lecture. It in'-ol,,,,, a blowgun G. using a rull os a p1'O)ooil"- The ,a'gel is a can su'pended from a n"'gR"1 M , and Ih" lUbe of Ih" "Iowgun;,; allned di'''''11y allh" can , Th.:l experim"ol is a"ang"d so Ihal Ibe magllel relxi,,, up"'>rd, ODd" is i. mOl", pe>'

>c"COmll Ihe equalion of IIIolic,

flG . .a.U Tbe .. """"",,,",pon launcb b~>8hl . NOIe Ihl 11. ,n Eq 4-26 " ... its mu,mum VIIlu~ ...lI~n .on 16" - I. ~blrll rofr~,;ponds 10 260 _ 90' or ,. _ J 5' .

flG . .a.U (I)Thof'"'bohlyball cU:ui>IN by lKloS"" ruutaK< ;''''"",,"Ol { IIJ pm. II>< ball '"'""" 101.,.. .................... i>IN by I..... tl>odo 01' dlioobopon-. Sec TobI... , for "",wI"""'">! JIimlUll fOJ • "'~lICb "'ste 01' 45' . Ilowe'·cr...·hen tbe la"ncb and land log beiShlS diIIC'. ". '" . hol put, h.m""". throw.ud b.l,kclbaU,a laun"" nngle ol 45' d;,1....,"0 our calnrla· llOllS looille actual mo"o. of I"" fIfOIoK'Id;> can 00 Ia.go Ilea .... "'" arr ,~s:i§.t< 1S normal 0000"""'. of pI.y, til alf Path" (the physocs r rofessor. fty roD) '" Ihe palh Ihe ball " -ould lollow io a ""mum.

It'II''+

.. ..... .,. .,. Two> Fly B.U..

""

(Air)

Palh II

(V..,..,,,,)

m•

MuiJDo.,

"-

o I'fbpl

~'"

",

.Am 7.9.

"St. r.......Th. _ _ ok "6O'.d th. _ _ 'P'«I" "'.1 -..

~ Chop.... I Mo'''''' in Tw_ Th, ... Oim ..........

X - .... - (I'.=6,).

,-

(Answo,)

(b) A, 1M C3"",h:! reacM' Ille waw, .•,h.1 is iT'; ",,'ocily V

in uoil -'"ilC1o, "OIalioo ,nd io m,gOlluoo-angk nOla!ion?

(4-28)

Here ,..~ ~now !hal x. - 0 t>.>c.ause the o,igin" pl'Ci!d al Ih~ pOlO! of '"'~'s.> . B.causc the capsule IS ",1.G..d and nO! snO! f,om Ih. plane. its ,",!i.1 "eIOCl!)' 1'. IS equal 10 Ih" plane', wlocny. ThUs. .... ~ "0 ..· .1so !ba! the inilial ,..,looly has mag""uM I '. - 55.0 01/' "nd .ogl~ 80 - (I" ("",asur.'d ,el'h"~ 10 tho po ,..hen i! hi" lh" ....He') and I, - SOC! m. W" ,hould 00 abk> 1o find x ..i! n Eq. 4-21 :

,

" 1', CaU"OD. Iocaled al Si" le-vel,fir'" baH... 1inilial s!",oo I'. - 82 ml~

(a) AI "hal a"gle a, f",m lile firM lohil Ih. ship?

~OrilOl\lal

must a baU be

(I) A fired «Inno"ball is a projoc1U,,- W~ '.-aul 0" equalio" Ihal r"L:H~s Ib~ lau"m anglo a, '" lile ball's horiT,OIliol displ""",,","1 os il m"".s 1'001 "",,"on 10 shIP. (2) B.«Ius.> Ihe ",,"non and 100 ship a,e al 100 same heigh!.lh. horizoolal di'l'I"""",,,nl is 100 range. Cakcations: We can ",Iale III vcrs\/'; I,n", ,.assuming Ihal 100 oUlf",ld", is alre"dy posilio"M '" calch Ihe ball. is 6.0 01 100 cloSi' 10 Ibe bailor. and is 6.0 01 100 far aw.y. .""..,

(Answer)

As Ibo pira!~ ship sailyco,d Ihal distaIloCe Ill of Ihe ball in Ih. oUlfiolde,', ,·i.w is give" b)' Ian d> - y/(R - x ). r", the he.ghl .", "-e us.> Eq . 4-12 (y - )'. - (I'. sin 00)< - ~gr'), seni"g y. - 0. For 1M

R -

..1. , S ""·60 -

..,,,w

(4Oo1(s)' . ,_, ,"., 91l mIS' SIn (u - "" .~_ m. ( ~ - 32)

~

Chop .... I Mo'''''' in Tw_ Th , ... OimsI ,IU!O ""n Eq. 4-21 (x - -'0 - (I'G cos 80)/). Sl'nin8 ... - O. Thus, uSlIIg I', - -10 and /10 - 35 ' . w~ ""'. , (-10 sin 3Yj, - 4.9,' (4-33) - 1"0- 153.42 (4000.35°,, "

"li,

Graphing Ih" function V"tsU' I gin', !be middl" plol in Fig. 4-17b. W~ """ Ibal 100 ball'. 'ngl~ in III~ oU lfield .. ·, ,·io ... iOC1c","" a! .n ,101051 steady rale !itroughoulloo ftiglll. ]{ 100 oUlfidrlor i, 6.001 100 doso. 10 Iho ball .. , we I~place lhi>. di'l.n"" of 15:JA1 m in Eq. 4-33 "';111

153.42 m - 6.0", - lHA2 m. Regr.ph,ng Ih . [UllCtion

gi"" th. "Too dOSi''' plot in Fig. 4-17/>. No ... lb •• "" -alio" ""S'" of the ball "pidly iocr~aS Ih~ di'{JUCIl of 153.42 m in Eq. 4-33 ,,-jIb 159.42 m. Thi! ""ulling plot i, laN-l.d "Too fa,- in th o fig"r~: Tho angkl fim anJ Ihoo rap>r to ,lOy p"LfIln toward ,II bane •. or bad: 'W")' from In" b.II",.

ilK""•..",

AI IUnc I - O. a golf boU is ;/JOI from ground levd jnw lhi>. air. as indicaled in Fig. 4- 1&,. Th •• nglo 6 Nlw""n lhi>. boll's dir""1Ioo of lra",,1 and Ih. posi!i'" dir.Clion ollh • .< is given in Hg. ~- I8b as a IUnClio" 01 lim" I. Tho balllanJs al I - 6.00 s Wh,l is Ihe magnilUrlo "0 01 lhi>. b.Ws laullch "'loci I}". al .. h.1 h.,ghl (y - )'.) aoo", 1hi>.launch lowl doos Ih. ltaU land. and "'hal is 100 b,U', dileelion of Ifavd jusl as ill,ods?

.xi,

,;- •",-, 01--", - \:- ,

".

d

, ('1 (' )

101

'"1'

(I) Th" b,ll is • [l1oj - 8 ... h",h means !hal --0 is di"_ ,,,(. ]lO'ilKm of I"" ca, al • give" u"""","t From Fig. ~ -21 .." .... Iha ! (4-.j()

,

f ...m ••t

f t"'. MOf""'inT_Di""""ion' _ _

He,. "~conside, only f.. mes Ibal mO\-~ aI con,laol ",Ioclly ,.lall'-~ 10 eocb ol tler. In 0"' "lample. IhlS "",.n, Ibal B.mara (frame B) d,'w" an--ars aI cou.lOal wlocily ,-&< ,ebb'" 10 AKl-x (f.,me A). eM P (Ib~ mO\'iag par1icl~). ho ..·evo,.cau ehange 5p ,al~). To relale"" 3CCi'[;"allon o f P '" "",",urod by B"ba.,a and by AIi>l . w ~ I" ~ ~ Ih~ lime do,'valive of Eq . ~- ~ I:

B inl!lal WOO!)" of P ,d.:Hi,,, 10 Alox i, " .... - - 78 " mlh a"d the final velocity is O. ThUs.

(a) If Alox nJ.""""" a rod>!an! "", - - 78 l mlh f01 ca. p, ..-h.! ",loCHy "n w,U Barna" r""",ll1el

y - " -~ -

We

, - 2.2 m/';.

aBaeh a f,ame of ,olo,eno> A 10 AIoJ and a I"me 01 ,deron~ B 10 B.,ba,a . Boca"", tile fran",s """'" a! rons!anl '"locil)' ,.lal;'-~ !o~ocb mile, along 0"" axi~ "" can u,," Eq. 4-~ I ("", - "", + yaJ 10 ,.Ial~ "n 10 "",- and "& Willal WOO!)' of P ,dalive to Barbara I.om parI (a) (,'''"' - - 130 bnlh ~ The final ""loci!y 0( P ",Iall", 10 Barlxtra is - 52 l mllt (Ihis is !he ",locily 0(100 Slopp..-.J ca, ,d:H;,,, 10 tho moving Ba,bara). Th""

Comment: II car P "W~ con"'-'C!c Iranles A Dod B, whikl B """,es a! a oon~arH ",locil}' ,,,lalive 10 A. (Jb.:l cor,esponding axes of !h..,., lwo f,ames romain pataUd.) Rgum ~ - 22 ,ho~,. 0>,!aln iIr>lanl dUflllg !be nlOlio" .AI !hal in orig>o of A AIso, Ine post!!OU voctors 01 ",,'he" Pale -; p~ «>!3,i,,, 10 lhi> origin 01 A and,.."" .. lal;'" 10 100 o"~n of B. From Ih •• ft.ngm"'"t 01 bc,ds 000 ,ail:! of too... Ih'''' posilioo veClor" "'" can ,d.:u~ tOO veClors with

,

,

' . ~.

~

(4-4])

By ta ~ing th. Ii"", d"';V"li~~ of this equalioo. "-,, can ,elate and \'n! 01 ra'hel. p ,dal"''' 10 our observe",:

lh~

",k"",Ife"oo I hal mow al ooos,""1 volocily ,,,Iali,,, 10 eocb """"""0 100 "'m~ »CCo:lJ.:>1alio" 10' a moving pa'lido.

I" Fig. ""2 3. a plane n'OW5 due casl whilo Ih. pUOI POUIiS Iho pIa .... """"",,hi ""ulb 01 eII inilW .. lociIy ,,~ Du'ing i" fligh" It.. panicle. horizontal """"I,.. ,ioo i. zt ~ the free _bU acede"";011 - E. (Up">rd i. ,aken I" be • poo~I"" dirertiOll.) If,,;' exp, .... d '"' , m.gni,ude (,t..'f"""d y,) and aD aog.le", (....... "d fTOm the bOfizootol ). 'he par'icle', "Iu>1l"", "fIJlOl;oo oI011S bori_ . 011'" X aID .Dd .",ic.7 i. ,ime in"rv.1 1>.,. i1S "'~r"gt 1',lolOlLS ' cir_ cle or circular ore oI...mu, r .. ro'''''''''f''''''d 1'.1. ;, ,aid be i. u"I/Im dmJ., mol",11 and b"" >II """"1,, ..100 of ""'_ .. ,., .,.gnitucle

'0

a

' - -.', .

(4_1l)

i."..

"cp~'= 2(...,,= ~"

dil I,om ,b. I, unch point t" ,h. poin' " .mieh tbe panicle Ttt""" 10 ,b. launcb !>eigh'. is

(4-11)

A, :., in Eq.4_8 is d .. u.l 100,1'_. ,.",,1Ies . Iim.. c.Ued ,i,b., ,b. 1~Io01,,100 .

if - o) + oJ + a,k. "ber ••, _ JI',ld" 0, _ Jy,ItA and a, _ ,w,ldi.

(4_17)

'"1''''' ,,, A .

-..'her. 1'... i. ,b. . .Ioci'y 01 B ,..j,b oo.erv", ",e.. ute ,bo urn. >«"""""' .:. n

«)

,hr..,

o 0-

....,r' , ",l • 'm

FIG. 4-Z.

,he cr.am '''''!,Tm. , .... 10 p"" 'hem :u>d (d) i velocity YKlon: (I) 1', _ 201 t 7(ij,j'2) 1', + ~. (3) 1', - 201 - ~. (4) Jlgle~ Th. projin. h"""", ... R ..... ,be OOCOIding '0 ,b. fin al speods of,b. proj«1~ ... ju" before they 1>Dd £l'"'''' fir ...

poi."

,od". -m

gr'"''''

'h,

"'na'ion<

,..

ve:r'ica.ll)'. Rank .hooe 'hr", window, according '0 (a) ,b. ,ime ,he cr.am ,,,,,!,,m. , .... 10 p"" 'hem :u>d (b) tho . v,,,age speed of ,h. cr.am 'anS"i.oe dwing ,be pa>!ag' windows 4.5 , , Dd 6. ,.Ilie •• re id'"''''01 in ,ize and inegularly 'l'aced bOli_ zo"'all~ Rank . b~ windo ... ocro,ding '0 (0) ,b. ,ime

..

FIG. 4-U Oo."io. 4.

,wioc.

J. fig, . _17, . cre ... ''''SOlin ... ,hr ...... tIp pa>' ,..-in,do,., I. 2. and 3, ,..-hidt are iden,ical in .in :md TeguJarly ,paced

6

• '\

• flG. 4-27 Ooe""",6.

7 Figw • • _1.8

>I>oIn

'1

'''eo

pa,h.la- a football tickod I,om yomid level, I~ 'he .1_ 10(," of air. nnt 'be p"h> """",ding '0 (a) rime of Aigh •• (b) ini,ial ve:rtical v,loci')' d i.. """," ,a'io. """or during !liSI... (0) \>''bicb of ,Ite Ie" .. ed poi"" 011 . hal co",. rone_ spaOSlo "I,,;,.., '0 'he """itiv. direc'ioo 01 ,.e x:OOo. (d) Sbtcb ,..., """or 00 • rigb,_b.ndtd coordinat. ')'''TTII_ If ,he ...,d ;, mv«I '0 ,I\< xY' COOIdin"es p.OO m. 0 m. 0 mi. wb" ;, i" eIi.pl"""."., (e) in ".i'_""" 01 n",,,ion.ad as (I) • rn.!";_ t.d. aDd (V an ang" .. I.,iv. 'h, """;,;",, .xi • • which i, ,ola'et. The m'gni'ude and "!Ie of ,..., displ....,m.n' YKlor 01 'ip or. be det,,_ mined for ,hr •• ,im. in"",,,,,. Wha' .re !be (I) m"!'l;'ucle .. d (b) .. gle from • q".ner afte, 'be boo, '0 b.1f (e) mogni'oo.e and (d) angle lor ,be "u' hall boo, .• ad ,b. (.) m' gni'ude ODd (I) .. gIe lor ,I>< bow ..,., ,bat7

boo,

,h.

'0 paoI_'""

Nlid .. (e) • m' gni_ ,ud. and (d) .. >ogle ",I"iv. '0 ,b. """i'iv. directioo 01 'bex

6.01

(.)

~,

.7 A traiD ot • COlI".'" 60.0 tOllh move> .a>1 fOI -10.0 min.. ,b •• in, diTplac,,,,,,,, .:.7._ 2.!Ii;10; + HIk. e"din! ..-ito !be posi'ioo YOC'OI -; _ J.Oj - 4.ot.

(b) ... gIe 01;" ave..ge velocity during ,hi< trip?

,b:::;;1-1~'1..

Jj:l"-sr-I::i:

.boo,

1

'be p.,h of • oquiI ..1 mVing 011 lev,1 - " 'Voond. from poin' A (ot ,ime , - 0). ,,, ---W poi.", B (., • _ 5.00 min). C (>t • _ lO.O FIG. Problem 9. min). and fi •• Uy D rot' _ 150min). Coosider!be average ",loci,;... 01 !be "IDirrtl ITO.. poin' A '0 .",b of ,he O1her ,hr •• poin' ~ 01 ,ben •• wh.a.' . re ,b. (. ) "'''VI;'Dde and (h) . ogle of, •• one "';'h 'he I... , m. lI,nitoo.e and ,Ile (e) m>g.;'ucle , ad (d) angle of !be ""e with ,he Dlagoitude1 ••• 10 111< posit"'" _'or ., _ 5.(0 '; + (t< + .Ii')] loca'es ' I""icl. ... 11IDUII._

14C. . . 4

...,,,,.I.r.tion

A ponidc .. ~'" , ..... ;0, pool!"'" ti!' ....1• .,' .. Ii"", (;" - . ) iI -; _ i .. ~j .. It . IVri .. nprnoioM lot (0' il> ........ , _ (b) ill """lor",.,., .. 10.,. ·1 '

• fImd (d ) 01" ........wlI. (II ...."11 .. 'h. _ ....... ,"" .. booo I _ .l.O o1(b) ........... (ll ..... 1'ul>< ro? (.) .,I ib ... (iI' 'YO'I 10."" YOI~.oo.1

".yU,.i""

,Ioo,

,1>. H".. "'..,.. .... _ !\lor•• ', tlot ...1Ii..... ~bIe ~8" fOf I par'>l _ ... , ,100 .... -amo< b.:o!l ,Ie., it 0IId (b) w.... io .... diot .... bt, ...... tboo . .... , of ,lot b.:o!l....a 1M lOp of ,b •• ,,7 5. _ , ......... nld II>< Ib", ,,,... eI ifllot IauJ>dt anpe "" i, (.) 45.00' ond (b) U OO"? Tho .......... mdicate 'hOI ,t..

,04" •

..s .. oi 45" ..."hkb maOot.iz...... r""SC!IIi PR" ~j .. IM!""Iad? " ...

"""'u>

__It

Yooalhrooo.boIl_

• _ >l1J>ft"d 15..0 ..... ODd ... ..~" _ .tIlO" ........ ...... ZOIIIoI (1'1& 4.38). The wall .. diot ..... of _ 220 Of froo:> .... . - polo' of • boL (a) H_ far .bo", .... ....... AG._ ~ .. 38. poiDI doeo the boll kit the .....1 Wh.. are tile (h) boritiontoIODd (e) v.uiolle .... -1611 .... io to be . ""ed .. • 'US" 45.7 m a.... y. H , ... omt.. nf 1M W ! lf maJIimuM l>eigh')? -:::.:

,b.

••• 49 A ,k.illed , ki" bI ....... '0 jump up,.....-d befol. Ie,.,h· ing a doIo-n....-.rd..tope. Con,ide, . jlUllp in wbich 'he buncb . peed i. ' ., _ 10 ml~ ,n. I, """h an!ie i.~, 9.!J'. ,he ini,i . 1 cow .. ;, :oppro>:im,tely fta' . and !be track b:.. a .Iope of 11.3'. Figu .. 4--150 a p"f"mp 'bat oll ..... ,h •• kin '0 1.00 0fI the ' '''' poniOrl of ,b. ""'per u ,.,k. Figwt 4·45b a jump " tdfle 01 the u ",k. In Fig. 4·450. ,110 ,b., lands" _o,;.,,, . ly the lauocb I. v.l. (. j I" !be I:UldioS. u1\a, is , 110 ... y.." h1 befol. I. oding ~ ~ 60.0" ,b. root: (.) Roo 'he hOfiroru.J dis· tance d i, tr"",1I. (So. ,he hin, '0 Problom 41.) W.O! lIle 'he ( b) m' gni'ude and « ) .. y. (rel.,i ... '0 ,he borizo.'aI) 01 ,b. baWl initi>l ""loci! y?

1

I

,.Ie... ..,gIo-d " _ ..-i,"

•• 47 A b. u., hi" • pitched ball ... b,. 'be ""nte, 01 ,be boll ~ 122 m abov. 'be !found. Th, ball I•• """ !be b., a' an ""!ie of 45 0 wi'b ,h. ground. With 'hat l.1uncb. 'be b>ll ,book! have • bori"""al .... se (rt""", 'be •• 48

(0'

fIG. 4-4S Probkm-t'l.

, /"" ..;g

"I""" ....",

""til

,.,

-.

ProbI , m-l4,

g"'"

fiG .....

1'robI< .. 50.

,,. ",

••• 51 A lootball kick" can glv. 'be boll ... ioi'i .J .f>I bar is H·I til above ,be ,",ouncl? • •• 52 A boll i, '0 be . bot from level growd eill! poci!O'II.II, ond SJ .... noolO' lUi, ..... ~ Wb'" a.. (.) ,b. po,iod oIlb. molion. ''''' i1>J "''P'i'u'''' ond i1 ;" 55 km _ DOI,h 01 ,b. " .. ,inS poin" Wh>l i. ,b. >peed 01 ,be oirpl .... ,.la,i"" 'o,be air? •• 76

.. n

A , ... in !Javel> _ JOO ,h ", J(I", /s Irt]>'iv, '0 ,b. ground) i. a r';n ,Ila, ~ blown ,,,,,,, .. d the ..,."b by ,be wind. Th. pa,h of " "b ... iodrop m>k .. or> ongle 01 70' wilb ,be .. r_ tical ......"""ed by ... observes "ationaryoa younod.A. oo..rver 00 ,b. mio. bowev ....."''' the drops fall .,.,-f«ily venie in'mect1oo ODd movi.g " 'pe.d" .. _ 60 kmlb. (0) In 1lIIi'_v«tOl DOtalioo. " b., ~ v.loci,y oI 'be mOlOl;,t ,.;,n r .. p" " ,he pobe< cor' (b) For ,be inst .., 011""", in Fig,. 4-19 .... bat i. ,b. angle the velocity founod in (.) and ,b. 'iDO of 'igh' be,....,. ,b. '''''''

,b,

'0

,b.

1>rd >Ad OIl lw:om The loIlo win~ !a ble giv.. five p>in of 1.1UIC1r . peed! ond angle> (/,om ,he lrorizOll'oI) fill m"b ,od~ bao ond :mgle~ ... oIro lau,dled.) Suppose that )'ou ... at x _ 20km ",he. lhe "rike. lb. you.d .. ,ime , _ O."d pooil'OII x _ 0 (Fig. 4_51 ~ (. ) At , _ 20 ~ ",lr.at .... lIex .00 y roordinat .. of ,b. ,oct.. b •• der '0 illd.d. roch willi i.. launcb ~ ond angle. The ewve ,b",,1d iodie . ...,b" fOIl ..oold ..., '" you look up i.,o ,b. 'pp,oachi'B Iocb •• d wh .. din"",... mu" Ii"" '0 (oJ the [O!"ound ond ( b) lb. c. b Ilooled move> in 'be neg. ';" x directiOll at 00.""" speed I', .. hile • boll 01 ice ~ .bot I,om Ibe .ted u-i'b • ""Icily Vo _ ,.,I + I~ "I"ive '0 the.ted. V"b"n '"" ball lond, iI. blllizorolol diopl:ocem.n, dio'>nl iDS" 01. ","",' i. il> velocity i, omi,_vocto, .",,,i001 87 I. F'E- 4_53.• "U i. 010", directly upward ITom lhe yound "i1h .. i.iI,.1 'pe,d of '" _ 7.00 mi. Simul .. oe"",'~ • oorutru! Frt""h t."",... ,be TOY (T, "" • G, .. d. \"t,. ...) bas • ",h. dul«l . ve'''S' .p«louO1 ,hat .J>f'I'd IlIId ,b. masnitud. 01 ,be >t .pt«! row' 'be go ",ound a CIIrve "';,b • 1.00 km r..aim '0 be '" 'be """,leratioo 1imi1?

">r'"

,h.

,mi.

lila!!",''''

93 A field can ro.oe • ohar!"" particle to """'. in • circular path. SUP!""" ,h., aD .1 ..... ,011 movi.og i. • cilcle ",p,,,ie,,,,,,, • rodi.al """"ler"iOll of m. srutod. 1.0 X LO" in a p..,icular m"V'.'ic field. (" Wh ., is opeed olll", ,1«trOll if ,b. ,adi., 01 il> cirrular "",. is IS nd ,be" I,,,, is'" _ - 2.01 + ~.O; + 2.ok , aU in """" .. ,(.) Whal ;, ,h. prOlOO', di,plooement vOClor . ... d (b) '0 ,.hot pi .... " ,h.:i, """or palaU.!1 95 A panicle P , ... vek wi,h CODS'..,' ,p'"b '0 O. lind , .. panicle', pooi'ion ""'. flG. 4-54o Problem 95. ' 01 >< , ... ,iI.... , of (. ) 5.00 ,. (b) 7.';O . . . d «) 10.0 ~

,b.

mi.'

6.01

,

,i",. , _

~

"spec'

~

)

- --"'ol--- -.

(dJ For ,b. 5.00 . iolervai f,om ,h• • nd 01 'he tif,' .. rod

'0 ,•• end of ,h.

" .. b semnd. lind ,be panicle , d;,pl"". _ "' ...,. fix !h>1 in' ... oI. tiad (.) iu , v"'ge veloci'y :md it, v. _ loci'y >! ,b. (f) be~nin! .ad W . 0CCe1 . ... ,ion., lb. ini,i ol "",«I. Wh" is , •• ini,iol , peed"!

,b. boo,.,

,hiI,. ,••

.,.."g.

99 A proj1:" 1re in 71h Mre board> ,be oideI.! velo< ""' . If ,be in"ial v.locity 01 ,be ball is borirorr_ '111.(. ) u-hal minirrlUln m.agoi'ude m .. ' i, h.:ive if,bo ban i, '0 de .. the "," and (b) ..-h., maximum mayrnud. """;, h.:i .. if , ... baU is 'o"rike ,be fIoOI irrd i, level. (.) WIr., rnaJlimum heigh' above yound level i, ,,,.d compe,ed in".ad in 19'i6 at M.lbown. (..-b.re g _ 9.7'I9911l1,' )7 .:s:;:

,.oont

109 Dwing volc .. ic .ruptions. lid rock can be bI .. ,ed "'" of ,b. volc. nO". ,b..., projwilH are called ,'ok""", bomb. Hsur. 4-58 a "'00' Jfi'1ion of M,. Fuji, in Japan. tal A, ,.,1", initi. 1.f"""d would . bomb b.", '0 be .joct.d. at >ogle 80 _ 35" ,o'too borizOllt>Urorn ,too v.n' . , A i. orderro Iall at ,too loot of 'b.volcano at B. at venkal h _ 3.30 om .. d bor'lOrlta! d"u nceJ _ 9,wtlll? [yror •. ["" ,too moment. ,be .11....... of >iI "" ,be bomb', tr.vel. (b) Wha, woWd be ,he ,im< 01 !liS.'? (cj Would ,be .lIed 01 ,h. air incr . . .. Of n of !be je' "r""" .00 Ie", wh.n ,he High' .. oppo>ire je' "ream. Suppose • row>d_'rip Kigh' ~ ",_cl.d be'we. 11.. A • • Iev.,or wi,b"", • ceilin~ i, ascendin~ wi,b • COII_ of IOm l~ A boy 011 ,be .I.va,or >hoot. , ball dirKrl y uP""">'d. f,om a beWU 01 2.0 m . bove ,hoe . levat"" floor. iuI,

11&

A,

'O/b"

initial

.pet', yOll ""'" ,0t1>OW 'be width 01 ,ho bo._ car ?)

11. YOII are,o,ruow . ball "';,h. 'f"'ed of 12 .0ml•• , • 'arSe! , b" ;, heigh' h _ 5.00 m . bove 'he 1,,..,1 at...rucb yoo " Ie ... ball (Hg. ~ .(;O) . You ,.."n' ,he baU ', velocity '0 be b""izOD,aI " ,be in""", ;,

,h.

" ach ..

'h, , arg.L (. ) A, 'O/b "

'I!u~_"\

-. A-"

•••....

1

,I

-------------

FIG. 4-60 Problem llS.

angle ' . bove ,be bori"""al mu .. you ,hrow!be baD? (b J \\'b., is ,be horiZOfl,al di ..... '" from !be " Ie ... poin' '0 'ar!" ? (C) Wha, ;, ,hoe , peed of ,h. baU jw' ., i, r.:r.ch .. ,b. ' arS,,1 119 Hgme 4_61 .... .,.... !be path ,ahn by a drWL ~ .kunt ove, level youod. f,om initi. 1 tool poin' J The :inV'" are" - 30.0'. '" _ 50.0'. 0J>d ,, _ SUO'. ",d ,be ,,,,,.d. with .. initial 5f>t .. ,be >poe'5 An equal-arm balance. When the device is in balance. the gravitational force F;L on the body being w.. ighed (on the left pan) and the total gravitational force F;R on the reference bodi", (on the right pan) are equaJ.Thus, the mass me of the body !x-ing weighed i, equal to the total mass "'R of the rderenc~ bodies.

To weigh a body means to meaSUre its weight. One way to do Ihis is 10 place the body on one of the p..lns of an equal-amI balance (Fig. 5-5) and then place reference bodies (whose masses are kn own) on Ihe other pan nntil We slrike a halance (so thatlhe gravitalional forces on lite Iwo sides malch). The llIasses on the pans Ihen malch. an d we kn ow Ihe mass of Ihe body. Ifwe kn ow lite value of g for the local ion of lite balance. we can also find the weighl of Ihe body "1th Eq.5-12. We can also weigh a body with a spring scale ( Ftg. 5-6). The body slrelches a spring. moving a pointer along a scale Ihat ha s been calibrated and marked in eillter mass or weight units, (Most b..1lhroom scales in Ihe United Stales wor k Ihis way and are marked in the force unit ponnds.) If the scale is marked in mass unils. il is accurate only where the value of g is Ihe same as where the scale wascalibrmed. TIte weight of a body musl be measnred when the body is nol acceleraling verlically relative 10 the growtd. For ex..lmple. you can mea511Ie your weighl on a scale in your hillJtI'OOlil or on a filSl train. However. if you repeal the measurement with Ihe scale in an accelerating elevator. Ihe reading differs from your weighl because of Ihe acceleration. Such a measuremenl is called an apparent weight. CaU/ioll: A body's weight is nol ils nl1lSs, Weight is the nwgnilude of a force and is related 10 ma,s by Eq. 5-12. If you move a hod)' to a poinl where th e va lue of g is different . the body's mass (an inlrinsic property) is nOI different but the weighl is, For example.lhe weighl of a bowling ]).11 1hilving a mru;s of 7.2 kg is 71 N on Earth but only 12 N on Ihe Moon. The mass is the same on Earth and Moon. but the free-fall accclentiion on Ihe Moon is only 1.6 mis'.

The Normal Force If you stand on a tnal1ress. Earlh pulls you downward. bUI you remain slalionary. Th e reason is thai Ihe mallress, because il deforms downward due 10 you. pushes up on )'ou. Similarl y. if )'ou sland on a floor. it defo rms (it is compressed. benl. or buckled ever so slightl y) and pushes up on you. Even a seemingl)' rigid concrete floor does Ihis (if it is nol sitling directly on Ihe ground . enough people on the floor could brell k it). The push o n )'ou from the mlll1ress or floor is a lIo rn ml f" r~~ F;,.. The name comes from Ihe mathemalicallerm normal. meaning perpendicu lar: llle force on you from. say. the fl oor is perpendicular 10 Ihe floor.

Scat,· m,.,-Led in dth« ""i~ht or

""". ""i1..

, . . When a body presses against 3 surface. the surface (e,·e'!.a seemingly rigid one) deform, and pushes on the body with a nonn31 force F.v that is perpendicular to th~ surface.

r;,..

~ - ff'K FIG. ~6 A spring ,cale.The reading is proportional to the ,w;gh/of the object on the pan.and the scale gi~", that weight if marked in weight units, If. instead.it is marked in mass units, the reading is the object', weight only if the value of g at the location where the scal~ is being used is the same a, the nlue of g at the location whNe the scale was calibrated.

r.

Fi gure 5- 7a shows an example. A block of maSS til presses down on a lable. on Ihe block. Th e deforming il somewhal because of Ihe gravilational force lable pushes up on Ihe block wilh t~rrnal ~rce l1le free-body diagram for Ihe bloc k is given in Fi g.5-7b. Forces F8 and F,•. are the onl)' Iwo forces on Ihe block and they are bolh verlical. Thus. for lite block we clln write Newton's second lilw for a posilive-upward y axis (F",ing thrown forward by a catapult mechanislll installed ill the carrier deck. 111e resulting high acceleration allows the plane to reach takeoff speed in a short distance on the deck. However. that high acceleration also compels the pilot to angle the plane sharply nosedown as it leaves the deck. Pilots are trained to ignore this compulsion. but occasionally a plane is Hown straight into the ocean. Let's explore the physics (X>hind the compu [sion. Your sense of vertical depends on visual elues and on the vestibular system loc..1ted in your inner ear. That system contains tiny hair cells in a Huid. When you hold your head upright. the hairs are vet1ically ill line witb the gravitational force on you and the system signals your brain that )'our head is upright. When you tilt YOUT head b.xkward by so me angle . the hairs are bent and the system signals your brain abont the tilt.111e hairs are also bent when you are accelerated forward by an applied horizontal force F'lock S

, • l'

•

/Cpr

fbng in ~

b loclll

,'>

\ ')

FIG. 5·, 5 (a) A free.body diagram forbtock S of Fig. 5· t3. (b) A free-body diagram for bloc k H of Fi g. j · [3.

Sample Problem

(5-20)

Putting in the numbers gives. for these IWO qnantilies.

(5-17)

f agree. HoI\' do I apply p." = ma to Ihe hanging block? We appty it jnst as we did for block S: Draw a freebody diagram for block H. as in Fig. 5-15b. Then apply F"", = rna in componenl form. TIlis time. because the acceleration is atong th e y axis., we use th e y part of Eq. 5-1 6 (F"".~ = may) to write

Mill

M + m g·

T

T _

=

2. 1 kg ') 3.3 kg + 2. 1 kg (9.8 ml,.;·

3.8 mls" Mill

- M +m =

Q

m M +m g

( Answe r) g _ (3.3 kg)(2.1 kg) (9.8 m/sl)

- 33 kg + 2. lk g

i3 N.

(Answer)

The problem is /101\' so/red. right?

That's a fair qnestion. bnt the proble m is not rea Uy finished until we have examin ed the result s to see wheth er they make se nse. (If you mnde these Cd the Icn.ion In 1"'0 of the: cords are "'I = 12 kg. m) = IS kg. "" = 20 kg. T, = III N. and T, = 222 N. Find Ihe pc-ngu,n m"",m, th at is not ",·en.

AG . §..44 Prablen, 45.

--46 In Fig.5-45.e1evator rnbs A and IJ are conocctcd by a .hort ca ble alld call be I)u ll cd upward or lo ....ered by the cable .oo,·e Cllb A . Cab A has lIIaSI 1700 kg: cab B has ",aM 1300 kg. A 12.0 kg OOX of calnIp IJC5 on Ihc Roor of cab A. Th e len.ion in Ihe cable conn«ling the cabs IS L91 X 10' N. What" the maptl1tK1e of lhe normal force on the box from lhe ROOf?

()

FIG.

I~oblc",

Proble m SO.

"51 In Fi g. 5-49. Ihree connected blocks arc pulled 10 the righl on a hon zonlal fricl;onle.. !abl~ by a force of n1agn;lude T, = 65.0 No If "'I = 110 kg. m, = 24.0 kg. and "'1 = 31.0 kg. calculale (a) the ma .... ilnd~ of Ihe ,)".tem'5 "cccl~ration.(b) the !ension T"and (c) Ib~ ten,ion h

, AG. §..4S

§..

46,

"47 In Fig. 5-46, a blOC"k of ",ass '" = 5.00 kg II pulled along a horizon lal fnct ,onless floor by a cord that eJeTt. a force of mallnilllde F = 12.0N at an anye 8=-25D" . (a) Whm is the mognllude ofthe block', accclcrnllon? (b) Th e force magnitude F i•• Iowly in_ ere.sed, What i. its value just I} F before the blork IS hfted (cont• ple lely) off the Roor? (c) What i, the magnItude of the block', AG. 5-46 l~oblcms 47 acrcleralion JUSI before it " hfted (completely) off the and 62. floor? --48 In earltcrda)'" hOl'§CS pulled barges down canals In the nlllnner sho"'11 in Fig. 5-17, SUPIK"'C the: hOrle pulls on

G

T,

.

G

r,

FIG. ~9

Problem 51.

T,

• F.

• -52 I n Fill. 5·50.,. a coostUllt horizontal forCllhe tension.n Ihe connecl· ing cord? (b) \\'hal II Ihe large., value Ihe RlagnLlIKk of may ha,.., wnhoullbe cord b«oRlmS slack?

r

"'67

FlI!ure 5·59 pves,. as a funwon of 11....., f. lhe force componCUleraled by only r, = (3.0 N)I + (4.0N») and = ( - 2.0N); + (-M N)j. Ihe .. whal is (3) in un,l·vector notation and as (b) ~ magnllude and (c) nn angle reial,ve to th .. posi llve.T dlrection7 Wh~1 are th e (d) on~goilude and (eJ angle of a?

92

r..,

r,

93 A nucleus IhM captures a stray n.. U1ron must bring the ""u· tron 10 a $lOll ",;th... lhe diameter of the nudeus by means of the Mrottg for«. '!bat forne. "hich ··glues" tI.., nudeus tog;!ther. IS ar>P""'Iimntcly urOOUts"" the nucleus. SUI'~ thai a may neutrOll "'l1h an trllhal speed of 1.4 X lOr rn/s ;. jllSt barely """lUred by a nudeU'S ""th wameter d = In X 10- 1< nL A$WlnU!& tlte Strong force' on Ihe ,..,Ulroo IS IXInstant. find t .... magnllude ofllial force. The neutron·snwu.I.67 X 10-" kg.

94

A 15000kg heltrnpter lifts a 4500 kg Iruck w,lh an upward i>tCelerallO .. of 1.4 mIs' . Calculate (a ) Ihe nel u",,·ard forfe on the helicopter blades from the air and (b) the tension "I Ihe C1lble b-el ..·ceo helicopter and truck. 95 A motorcycle and 60.0 kg rider acce lerate at 3.0 m/s 1 up a rDmp inclined 10' Dbo~e Ihe horizontal. What are the IImll"i· lude, of (a) 1he nCI force on 1he rider and (b) Ihe force On 1he rider from the mo torcycle ? SSM 96 An trltefStellar .hlp has a mas. of 1.20 X 10" kg aod IS 1m· nally 81 ft'S1 47 ProbLem 9&. Figure 5.68 slIo," a box of dirty n,oney (mass "" = 3.0 k~ ) on a fnctionle .. plane indIlled lit ~ ngle 8:t = JOO. 'll'e bo~ is connected ,",a a cord of n egltgible ma.. 10 8 box of lllundered mo .... y (mass FIG. 5·68 Problem 99. m, = 2.0 kg) o n a friclion· ~ plane i"dmtd at angle tI, = 60". The pulley is fnctlOllle,os and has negligible m:us. What IS Ihe le nsion Illlhe cord'! ss.

99

•

100 Suppose Ihe I kg standard body """"leralC'S 81 4.00 mI,: a t 160" from the poIlII\"C d,rectlon of an x axis due 10 1""0 force.; one ,s ~ = (2.50 N); + {HiO N)l. Whal IS Ihe olhu force (a) in 1I11l1·.·«tor nOlahOO and "" (b) a magllilllde and (l:)anangle7 101 [ n Fi~. 5-69. a 1111 of an· tioxidants ( "'1 = 1.0 kg) 00 ~ friclionless indined surfl'cc is connected to n 1111 of corned bed (m, = 2.0 kg). lb~ pulley 15 rna ..les. and fnclionless. An u p""ard force o f mJ",Ill.lde ,.. = 6.0 N aci. on the corned 1I«f till. which has a do ...·n ...lIrcl K«Lera· lion of 5.5 mlr. Whal are (a) tbe lension ,n Ihe "",neellng cord and(b)angle/p. ss • 102 A rocket and its payload h ave a tOlal rna.. of 5.0 X FIG . 5-69 Problem 101. 10' kg. How Inrge is the force produced by the e "gin e (Ihe Ihrust) "·hen the rocket IS (a) ··ho.·cring" over th .. launchpad just after ignilion nnd (b) aocel .. raling upward PI 20 ntls l 7

91 For sporl. D 12 kg armadillo runo 01110 n large pond of levdo f nctionless Ice. Th e armadillo', Initi.1 '·doc,ty is 5.0 10/. along Ihe positive direction of an x axis. Tak e ;1. initial posi· lion on the Ice as being Ihe origin. It .hJ1'6 0\.... Ihe Ice while being pushed 11)' I wmd with a force ofl7 N in Ihe poSlll'·e direc_ 11011 of the y axi .. ln uml·,"Cctor notallOll ....·hal arc lhe alllm31". (a) wloctty and (b) flOilUon '"ector "hen 11 haul'" for 3.0 11

103 A m Olo rcycie o f ""e'ghl 2.0 kN accelerale. from 0 to 88.5 kmfh ,n 6.0 .. WhJ t arc the n.aglllludC'S of (a) the constant a «ekrallon and (b) Ihe nel force cau""ng Ihe a«eluanon?

98 A SO kg passenger rides in an ele""tor ClIb that Stan. from r~1 on Ihe ground Hoor of a build,ng all = 0 and niICS 10 Ihe lop Aoor dunng D 10. inlenal. Th e cab·~ aecekrallon as D funeloon o f Ihe hme IS ohown in Fi&- 5.67. where pasol"'" ~al·

104 An IIIl1tally statlOllary electron (rna... = 9.11 X Io-ll kg) u ndergoes 8 ,0nSlanl acceleration through 1.5 mi. re1>Ch'''1! 6 .0 x 10° m /s. Whlll are (a) Ihe magnilude of lhe force aa;'t'l. erating Ihe electron and (b) tbe electron·, weight?

Force and Motion-II The Great Pyramid, built about 4500 years ago, consists of about 2 300 000 stone blocks, most with a mass of 2000 to 3000 kg. How did the

engineers and workers manage to lift the stones into place to construct this pyramid, which is over 140 m high? Some researchers argue that during the construction a large team of men would pull a block up a giant earthen ramp that ran at a modest angle up one side o f the pyramid. However, no evidence (such as rubble or painted pictures) exists to support this theory. Other researchers argue that a spi-

ral ramp ran around the pyramid. However, such a ramp would have been highly unstable and, besides, maneuvering a 2000 kg stone around the 90 corners along the ramp would have been daunting, if not impossible. G

How did the ancient people move the blocks up and into position? The aflSNer is in this chapter.

116

6 -2 I Friction

6-'

WHAT IS PHYSICS?

I n this chapter we focus on the physics of three common types of force: frictional force, drag force . and centripetal force. An e ngineer preparing a car for the I ndianapolis 500 mUS1 consider all three types. Frictional forces acting Oil the tires are crucial 10 the car's acceleration Olll of the pit and out of 11 curve (if the car hits an oil slick. the friction is lOS1 and so is lhe car). Drag forces acting on the car from lhe passing air must be minimized or else the car will consume too much. fuel and have to pit too early (even one 14 s pit SlOp ca n COSI 11 driver the f 0 as x --> ~ and thal .T is po,itive. (b) How much work is required to increase Ihe separalion of the particles from x = .~, lo x = x, + In 115 A ppro., imately 5.5 x 10' kg of waler fall. 50 mover Niagara Falls each second. ra) What is the decrease in the grayitational polential energ)' of the water- E arth s)'stem each second? (b) If all this energy could be con"erred to olec_ trical energy (it cannot be). at ""hal rate would electrical energy be supplied? (The mass of I m' of wa ter i. 1000 kg.) (c) If Ihe electrical energy were sold at I cenukW· h. what would be the yearly income? 116 A 1500 kg car starts from rest on a horizontal rood and ga in. a ,peed of 72 kmfh in 30 .. (aJ What is its kinetic energy al the end of the 30.1 (b) What is the ",'erage power required of Ihe car during Ihe 30s interval? (c) What is the instanta neou, power at the end of Ihe 30 s interval. assuming that Ihe acceleration is consmnt? 117 A particle can move along only an x axis. where conser_ vati \'C forces act on it (Fi g. 8-69 and the follo ...·ing table). The particle is released at x = 5.00 m with a kinetic energy of K = 14.0 J and a potential energy of V = 0. 1f its motion i. in the negative direction of Ihe x axi.. whal are its (a) K and (b) Vat x = 2.00 m and its (c) K and (d) Vat x = O? If its motion is in Ihe positi '-e direction of the x axis. ""hal are its (e) K and (f) Vat x = Il.Om . its (g) K and (h) Vat x = 12.0 m. and its (i) K and (j) Vat x = lJ.O m? (k) P iol V(x) "ersusx for the range x = 0 tox = lJ.O m.

Pro ble ms

Ii

Ii.

~'i

r.

_L::i=L}_L~_L~_L:;:=L:;::J~l_ x (m) o

2

.j

FIG . 8-69

6

LO

~

12

Problemsl17andllg.

Next. Ihe panicle is released from resl al x = O. Whal are (I) ils kin~lic en~rgy al x = 5.0m a nd (m) Ihe maximum posili"e posilion x ..., il reaches? (n) Whal does Ihe panicle do afl~r il reach"" x ... ,?

------Range 0102.00m 2.00 m 10 3.00 m 3.00 m 10 8.00 m 8.00mlo I I.Om 11.0 m 10 12.0 m 12.0mloI5.0m

(b) direction of F(x) at this position? Between what positions onlh .. (cl left and (d) righl does Ihe panicle move? (el What i, the particle's speed at x = 7.0 m1 SS/ll x (m)

" r:::"~~----';'-------""C'----~;;'-----~

"

IlliCEH

::; _10

Force

F, = F, =

+ (3.00 N)l + (5.00 N)l

_w

F=O

F, = F, =

- (4.00 N)l - ( 1.00 N)l

F=O

118 For Ihe arrangemenl of forces in Problem 117. a 2.00 kg panicle i, released al x = 5.00 m wilh an initial velocily of 3.45 ml, in Ihe negali"e direclion of Ihe x axis. (a) If Ihe pani de can reach x = 0 m. wh at is ils speed Ihere. and if il ""nnOI. whal i, its lurning point? Suppose. inslead. Ihe partid~ is headed in Ihe positive x direction when il is released al x = 5.00 m at speed 3,45 m/s. (b) If Ihe panicle can reach x = ]3.0 m, wh at is ils speed Ihere,and if il cannOl, whal is its turn ingpoim? 119 A 0..12 kg shufAeboard disk is initially at resl when a player use, a cue to increase ii, speed 10 4.2 mi. at constam acceleralion. The acceleration la kes place o,'er a 2.0 m di,· la nce. al Ihe end of which Ihe ene lose. coniaci ",ilh Ihe dis k. Then the dis k .lides an addilional 12 m before 'lopping. A"Ulne Ihat Ihe shufAeboard coun is le,'el and Ihal Ihe force of friction on the disk is conSlanl. Whal is the increase In Ihe Ihermal en .. rg)' of Ih~ disk - courl 'y,tem (a) for Ihal addi tional 12 m and (b) for the emire 1 ~ m distance? (c) How much work is done on Ihe disk b)' Iheene? SS/ll

120 We nlO"e a partide along an x axis. first out",ard from x = 1.0 m to x = 4.0m and Ihen back lo x = 1.0 m. while an external force acls on il. Th al force is directed along Ihe.T axi" and ilsx component can have different values for Ihe oUlward Irip and for Ihe return Hip. Here are Ihe ,.,.lues (in newlon,) for four situations. where x is in meters: OUlward

Inward

(a l + 3.0 (b ) + 5.0 (c) + 2.0x (d) + 3.0T'

- 3.0

FIG. 8·70

Problem 121.

122 To ma ke a pendulum , a 300 g ball is anached to one end of 3 string Ihat has a length of 1.4 m and negligible mass. (The other end of the 'tring is hed.) The ball is pulled to one side untillh Siring ma ke, an angle of 30.0° with Ihe vntical: then (with the string laut) the baH is released from fi nd (aj the s~ed of th. ball when thc string makes an angl~ of20.0" with the vert ical and (b) the maximum speed of the ball. (c) What is the angle bet",een the string and the vertical when the speffi of the ball is one·third its maximum value?

,,,,I.

123 A 1500 kgcar bdocity of 16 mls at some un known angle aoo..'e the horizontal. A short time later th down lands in a nel that is 3.9 m ,'enicaJly abo,'e the do"",'s initial position. Disregard air drag. What is Ihe kinetic energy of the down as he lands in the net? 125 The maximum force ) 'OU can exert on an objeci with one of ),our back teelh is a bout 750 N. Suppose that as you gradually bite on a dump of licorice. the licorice resists compre"ion by one o f your teeth by acting li ke a spring for which k = 2.5 X 10' Nim. Find (a) Ihe distance the licorice i. com-

L ,

+ 5.0 - 2.0r + 3.0r'

L

Find the nel work done on Ihe partide by the eXlernal force fo r (he rOllnd (rip for e ach of the four situalions. (e) Forwhich,

if any, is the external force conser""ti,'e?

121 A conservati"e force F(x) acts on a 2.0 kg particle Ihat moves along an .r axis. The potential energy U(x) associated with F(>:) i, graphed in Fig. 8-70. When the partide is al x = 2.0 m, its nlocity is - 1.5 mls. Wha t are the (a) magnitude and

the the en(b)

""

~

3

.j

56789JO

P~II~" . t;o"

FIG . 8·71

dep,h (JUm\

Problem 125.

IJ

12

Clwopt. r 8 1 Pot ent i..1Energy and Con ...... at ion of Energy preswd by your tooth and (b) the work the tooth don on the licorite dUring the oornpr~on . (e) I'lot the ma",itudt of your force "e"u' the rornpreso'i. of a parhcu la. T,kmllops din~u' woo' found to ha'-e deep bite ma, ks. The shape of th e nt~ rk! suggestcd that th ey were made by a T)-"""'Q$(lllflLS r~x dtOo~uur. To test the idea . research ers nl.1de a repilca of II 1: rl'.f too th from bronze and "Iuminum lind th en used II hydraulic pre.. to , raduallydrive the replica intooow bone to the dept h 5«n ,n the T';UrrllQPS bone. A gr"ph of the force required \'Cuu. depth of pen etrat ion ,s g,ven ,n Fig. 8-71 for o ne tnal: the reqUIred force mcreased ,,·,th depth bec"ause. as the nearly conICal tooth penetrated the bonc. mo'" of the tooth can'e In oontact with the bone. (c) How much ""Qrk was done by the h)·draulic pre..-and thus presunlably by the T rt... -in such a penetration? (f) Is there a potenllal enerD asSOCtated ""h th .. penetration? (fhc lar,e btllng fo r"" and enefg)' upendnure attributed to the T. rrx by th" ",search suggesl that the animal "-as a predator and not a Kavengcr.) 126 A 10 kg firefighter .Iides. from re.I.43 m down a verti CAl pole. (a) If the 6re6ghter hold5 OntO the pole lightly. so that the frictional force of the pole on her is ncg llgible. wh.t is her speed just before reaching rhe ground 1I00r? (b) If the fire_ fighter grnsps Ihe pole more firmly as she sltdes. 50 thai the average friction al force of the pole on her il SOO N upward. whM is her spoeed JUst befo re reachm, lhe ground lloor?

121 A 15 k, block is accelerated 1120 nih! along a horIZon_ tal fritilonkn surfa«. with the speed ,"crcasmg from 10 ml5 to 30 mIs. What arc (al the chan,e to the bl