Ruina/Pratap, problems only - Andy Ruina

118 Pages·2002·6.48 MB·English

by  Andy Ruina

Introduction to
S
TATICS
and
D
YNAMICS
(cid:2)
Ms
(cid:2)
F
s
ˆ
k
ˆ
ıˆ
F1 F2
N N
1 2
Andy Ruina and Rudra Pratap
Pre-printforOxfordUniversityPress,January2002
(cid:10)c Rudra Pratap and Andy Ruina, 1994-2002. All rights reserved. No part of this
book may be reproduced, stored in a retrieval system, or transmitted, in any form
orbyanymeans,electronic,mechanical,photocopying,orotherwise,withoutprior
writtenpermissionoftheauthors.
Thisbookisapre-releaseversionofabookinprogressforOxfordUniversityPress.
Acknowledgements. The following are amongst those who have helped with this
book as editors, artists, tex programmers, advisors, critics or suggestors and cre-
atorsofcontent: AlexaBarnes,JosephBurns,JasonCortell,IvanDobrianov,Gabor
Domokos,MaxDonelan,ThuDong,GailFish,MikeFox,JohnGibson,RobertGhrist,
Saptarsi Haldar, Dave Heimstra, Theresa Howley, Herbert Hui, Michael Marder,
Elaina McCartney, Horst Nowacki, Arthur Ogawa, Kalpana Pratap, Richard Rand,
Dane Quinn, Phoebus Rosakis, Les Schaeffer, Ishan Sharma, David Shipman, Jill
Startzell,SaskyavanNouhuys,BillZobrist. MikeColemanworkedextensivelyon
thetext,wrotemanyoftheexamplesandhomeworkproblemsandcreatedmanyof
thefigures. DavidHohasdrawnorimprovedmostofthecomputerartwork. Some
ofthehomeworkproblemsaremodificationsfromtheCornell’sTheoreticalandAp-
pliedMechanicsarchivesandthusareduetoT&AMfacultyortheirlibrariesinways
that we do not know how to give proper attribution. Our editor Peter Gordon has
beenpatientandsupportivefortoomanyyears. Manyunlistedfriends,colleagues,
relatives,students,andanonymousreviewershavealsomadehelpfulsuggestions.
SoftwareusedtopreparethisbookincludesTeXtures,BLUESKY’simplementation
ofLaTeX,AdobeIllustrator,AdobeStreamline,andMATLAB.
MostrecenttextmodificationsonJanuary29,2002.
722 APPENDIX1. Contact: frictionandcollisions
Problems for Chapter 1
Introduction to mechanics Because no
mathematicalskillshavebeentaughtsofar,the
questionsbelowjustdemonstratetheideasand
vocabulary you should have gained from the
reading.
1.1Whatismechanics?
1.2Brieflydefineeachofthewordsbelow(us-
ing rough English, not precise mathematical
language):
a) Statics,
b) Dynamics,
c) Kinematics,
d) Strengthofmaterials,
e) Force,
f) Motion,
g) Linearmomentum,
h) Angularmomentum,
i) Arigidbody.
1.3Thischaptersaystherearethree“pillars”
ofmechanicsofwhichthethirdis‘Newton’s’
laws,whataretheothertwo?
1.4Thisbookorganizesthelawsofmechanics
into4basiclawsnumbered0-III,notthestan-
dard ‘Newton’s three laws’. What are these
fourlaws(inEnglish,noequationsneeded)?
1.5Describe,aspreciselyaspossible,aprob-
lemthatisnotmentionedinthebookbutwhich
isamechanicsproblem.Statewhichquantities
aregivenandwhatistobedeterminedbythe
mechanicssolution.
1.6Describeanengineeringproblemwhichis
notamechanicsproblem.
1.7AbouthowoldareNewton’slaws?
1.8 Relativity and quantum mechanics have
overthrownNewton’slaws.Whyareengineers
stillusingthem?
1.9Computationispartofmodernengineering.
a) What are the three primary computer
skillsyouwillneedfordoingproblems
inthisbook?
b) Giveexamplesofeach(differentthan
theexamplesgiven).
c) (optional)Doanexampleofeachona
computer.
ProblemsforChapter2 723
Problems for Chapter 2
(cid:2)
2.8 Find the sum of forces F = 20Nıˆ − (c) HowfarisAfromB?
Vectorskillsformechanics (cid:2) 1 (cid:2)
2Nˆ,F = 30N(√1 ıˆ+ √1 ˆ), and F =
2 √ 2 2 3
−20N(−ıˆ+ 3ˆ). 2.17 Threepositionvectorsareshowninthe
2.1Vectornotationandvec- 2.9 The forces acting on a block of mass fi√gurebelow. Giventhat r(cid:2)B/A = 3m(12ıˆ+
tor addition m = 5kg are shown in the figure, where 23ˆ)andr(cid:2)C/B=1mıˆ−2mˆ,findr(cid:2)A/C.
F1 =20(cid:2)N, F2(cid:2)=50(cid:2)N,and(cid:2)W =mg. Find B
thesumF (=F1+F2+W)?
2.1Drawthevectorr(cid:2)=(5m)ıˆ+(5m)ˆ. F(cid:2)1 F(cid:2)2 ˆ
2w.e2stAatvaenctaonrga(cid:2)lei6s02omfrolmontgheanndorptho.inDtsranwortthhe- 4 3 4 3 ıˆ
vector.
2.3ThepositionvectorofapointBmeasures C
3 mandisdirectedat40o fromthenegative (cid:2)
W
x-axistowardsthenegative y-axis. Showthe
problem2.9: A
positionvectorvector.
(Filename:pfigure2.vec1.6) problem2.17:
(cid:2) (cid:2) (Filename:pfigure2.vec1.7)
22.N4ıˆD+raw2Naˆfo+rc1eNveuckt.orthatisgivenasF = 12.1to0G4,iviesn(cid:2)zthearot,thwehsuemreoFf(cid:2)f1ou=rve2c0toNr(cid:2)sıˆ,FF(cid:2)i,2i == 2.18 Inthefigureshownbelow,theposition
2.5Representthevectorr(cid:2)=5mıˆ−2mˆin −50Nˆ,F3=10N(−ıˆ+ˆ),findF4. vectors are r(cid:2)AB = 3ftkˆ,r(cid:2)BC = 2ftˆ, and
threedifferentways. 21.01N1(Tcohsreθeıˆ+fosricnesθFˆ(cid:2))a=nd2W(cid:2)Nıˆ=−−520NNˆ,ˆR,(cid:2)su=m rr(cid:2)(cid:2)CADD.= 2ft(ˆ+kˆ). Findthepositionvector
2.6Whichoneofthe(cid:2)followingrepresentations uptozero.De(cid:2)terminetheangleθanddrawthe B r(cid:2)BC C
ofthesamevectorF iswrongandwhy? forcevectorRclearlyshowingitsdirection.
a) b) (cid:2) (cid:2)
2N 2.12GiventhatR1=1N(cid:2)ıˆ+1.5(cid:2)NˆandR2= r(cid:2)CD
ˆ 3N -3Nıˆ+2Nˆ 3.2Nıˆ−0.4Nˆ,find2R1+5R2. (cid:2) r(cid:2)AB
2.13FindthemagnitudesoftheforcesF =
ıˆ 30Nıˆ −40Nˆ and F(cid:2) = 30Nıˆ +40N1ˆ. D
c) √ d) √ Drawthetwoforces, re2presentingthemwith kˆ
13N 13N theirmagnitudes.
2 (cid:2) A ˆ
2 3 2.14 Two forces R = 2N(0.16ıˆ +
3 0.80ˆ)andW(cid:2) = −36Nˆ act on a particle. problem2.18:
problem2.6: (Filename:pfigure2.vec1.5)
Findthemagnitudeofthenetforce. Whatis
(Filename:pfigure2.vec1.2)
thedirectionofthisforce?
2.19 Inthefigureshown,aballissuspended
2.7 Thereareexactlytworepresentationsthat 2.15 In the figure shown, F1 = 100N and witha0.8mlongcordfroma2mlonghoistOA.
dtuersecsr.ibMeatthcehstahmeecovrercetcotrpiincttuhreesfoinlltoowpianigrsp.ic- oFf2F=(cid:2)3−00F(cid:2)N..Findthemagnitudeanddirection (a) Findthepositionvectorr(cid:2)Boftheball.
a) b) 2 1 y (b) Findthedistanceoftheballfromthe
4N 4N origin.
ˆ (cid:2) (cid:2)
F -F y A
30o 30o 2 1
ıˆ F
c)2N d) √ F145o 30o 2 2 m 0.8m
2√3N 2N(-ıˆ+ 3ˆ) problem2.15: x B
(Filename:pfigure2.vec1.17) 45o
e) f) 2.16TwopointsAandBarelocatedinthexy
O x
plane. ThecoordinatesofAandBare(4mm,
problem2.19:
3Nıˆ+1Nˆ 3N(13ıˆ+ˆ) 8m(ma))aDnrdaw(90pomsmiti,o6nmvemc)to,rrsesr(cid:2)pecatinvdelr(cid:2)y. . (Filename:pfigure2.vec1.20)
problem2.7: A B
(Filename:pfigure2.vec1.3) (b) Findthemagnitudeofr(cid:2)Aandr(cid:2)B. 2.20 Acubeofside6inisshowninthefigure.
724 APPENDIX2. Contact: frictionandcollisions
(a) FindthepositionvectorofpointF,r(cid:2), y y
fr(cid:2)rFo/mc.thevectorsumr(cid:2)F= r(cid:2)D+r(cid:2)C/D+F C T2 T1
(b) Calculate|r(cid:2)F|. B 60o 45o
(c) Findr(cid:2) usingr(cid:2). x
G F 30o
z y
(cid:15)
E F h
O x W
D C problem2.23: problem2.29:
H
G (Filename:pfigure2.vec1.27) (Filename:pfigure2.vec1.12)
A B x
problem2.20: 2.24ˆExpressthevectorr(cid:2)A=2mıˆ−3mˆ+ 2ea.3c0hI4ncthmefilognugreasnhdowlien,arloodnsgAyBaannddxBCaxaerse,
(Filename:pfigure2.vec1.23) 5mkintermsofitsmagnitudeandaunitvector respectively. RodCDisinthe xz planeand
indicatingitsdirection. makesanangleθ =30owiththex-axis.
(a) Findr(cid:2) intermsofthevariablelength
AD
(cid:2) (cid:2) (cid:15).
ˆ 2.25 Let F = 10lbfıˆ+30lbfˆ and W =
2.21 FindtheunitvectorλAB,directedfrom −20lbfˆ.Findaunitvectorinthedirectionof (b) Find(cid:15)andαsuchthat
poyintAtopointBshowninthefigure. thenetforceF(cid:2)+W(cid:2),andexpressthethenet r(cid:2) = r(cid:2) −r(cid:2) +αkˆ.
forceintermsoftheunitvector. AD AB BC
3m
B
z
2.26Letλˆ =0.80ıˆ+0.60ˆandλˆ =0.5ıˆ+ D
1m A 2m 0.866ˆ. 1 2 (cid:15)
(a) Showthatλˆ andλˆ areunitvectors. A 4cm B
1m 1 2
y
(b) Isthesumofthesetwounitvectorsalso 30o
4cm
x aunitvector? Ifnot,findaunitvector
ˆ ˆ
problem2.21: alongthesumofλ andλ.
1 2
x C
(Filename:pfigure2.vec1.25)
problem2.30:
(Filename:pfigure2.vec1.13)
ˆ ˆ
2.27 For the unit vectors λ and λ shown
1 2
below,findthescalarsαandβsuchthatαλˆ −
2eBx.,2p2r(cid:2)reAs/FsBitn,hdienpatoeusrnmitiitsovnoefvctethocetrouarnlooitfnvAgecwsttroitirhn.greBspAecatntdo 3λˆ2=βˆ.y 2 2o.f3t1heInpPorsoitbiolenmv2e.c3to0r,fir(cid:2)nAdD(cid:15)issu6chcmth.atthelength
y
λˆ (cid:2) (cid:2)
2 2.32 LettwoforcesP andQactinthedirec-
1.5m 1 tionshowninthefigure. Youareallowedto
A 60o cthheanagnegltehseαdiraencdtioθnwohfitlheekfeoerpciensgbythcehmanagginnig-
1 λˆ1 x taunddeθsififxtehde.mWaghnaittusdheooufldP(cid:2)b+etQh(cid:2)ehvaaslutoesbeofthαe
2.5m 1m x problem2.27: maximum? y
(Filename:pfigure2.vec1.11)
3m
B
z 2.28IfamassslidesfrompointAtowardspoint Q
problem2.22: Balongastraightpathandthecoordinatesof
(Filename:pfigure2.vec1.26) pointsAandBare(0 in,5in,0in)and(10in, P α θ
0in, 10in), respectively, find the unit vector
ˆ x
λABdirectedfromAtoBalongthepath. problem2.32:
(Filename:pfigure2.vec1.18)
2.23 In the structure shown in the figure, √
(cid:15) = 2ft,h = 1.5ft. Theforceinthespring 2.29 Inthefigureshown,T1=20 2N,T2=
aisspurFin(cid:2)nigt=vfoecrkctor(cid:2)erAF(cid:2)λBˆA,=BwhFaleλoˆrnegk.A=Ba1n0d0lcbafl/cuftl.ateFitnhde z4fβoe0rsrcNoue.c,shaentqhduaatWlαsTzi(cid:2)se1r,souβ.cTIh(cid:2)f2t,Whaantidsthd2eoW(cid:2)usbulmsetdilo,lffisutnhmdeαuthparnetdoe 2bsvte.yr3ci3nttwogroA.BsF1triminsg×2s0A1NEm.asnEqdxupaBrrFees.bsTothaheridsteitnsesnsisouinpopnionarttsehdea
AB
ProblemsforChapter2 725
y y
kˆ Q P
2.5m ˆ C
F ıˆ AQ45o C 6" D λˆ F=100N
E 6" 30o 10o
1 2 2m 12" D x
m
1 O problem2.45:
1m x x (Filename:pfigure2.vec1.33)
A B problem2.35:
plate 1m (Filename:pfigure2.vec1.24) 2.46 Find the angle between F(cid:2) = 2Nıˆ +
(cid:2) (cid:2) 1
2.36WritethevectorsF =30Nıˆ+40Nˆ− 5NˆandF =−2Nıˆ+6Nˆ.
problem2D.33: C x 10Nkˆ,F(cid:2)2 = −20Nˆ1+ 2Nkˆ, andF(cid:2)3 = 2
(Filename:pfigure2.vec1.21) −or1c0oNluıˆm−ns1).00FiNndkˆtahseasulmistooffthneumfobrecress(ursoiwngs 2.47Givenω(cid:2)=2rad/sıˆ+(cid:2)3rad/sˆ, H(cid:2)1 =
acomputer. (20ıˆ+30ˆ)kgm2/sandH2=(10ıˆ+15ˆ+
6kˆ)kgm2/s,find(a)theanglebetweenω(cid:2)and
2(cid:2).37(cid:2)Let αF(cid:2)1(cid:2)+βF(cid:2)2 +γF(cid:2)3 = (cid:2)0, where H(cid:2)1and(b)theanglebetweenω(cid:2)andH(cid:2)2.
F ,F , andF areasgiveninProblem2.36.
1 2 3
2.34 ThetopofanL-shapedbar,showninthe Solveforα,β, andγ usingacomputer.
2.48Theunitnormaltoasurfaceisgivenas
fitlhegneugprteho,ionisftstthoAebsaetnritdniegBdsibAnyDtshtaernindygzBspDAlaDunseia.nngFdvinBedcDttohtroes 2θn.38=Lθe0t−r(cid:2)nn=(cid:25)θ1.mU(csoinsgθnaıˆc+omsipnuθtnerˆg),ewnehreartee noˆn=thi0s.7s4uırˆf+ace0.a6c7tsˆ.inIfththee−wˆeidgihrtecotfioanb,lfioncdk
r(cid:2) andr(cid:2) . therequiredvectorsandfindthesum theanglethata1000 Nnormalforcemakes
AD BD withthedirectionofweightoftheblock.
B (cid:20)44
r(cid:2)i, with(cid:25)θ =1oandθ0=45o.
y n=0 2.49Vectoralgebra.Foreachequationbelow
statewhether:
2m
2.2 The dot product of two (a) Theequationisnonsense. Ifso,why?
D (b) Isalwaystrue.Why?Giveanexample.
vectors
(c) Isnevertrue. Why? Giveanexample.
1m 2an.3d9(cid:2)bFi=nd2tıhˆe+doˆt+pro2dkˆu.ctofa(cid:2)=2ıˆ+3ˆ−kˆ (d) wIsasyosm.etimestrue.Giveexamplesboth
Youmayusetrivialexamples.
A 2.40 Find the dot product ofF(cid:2) = 0.5Nıˆ+ a) A(cid:2)+B(cid:2)=B(cid:2)+A(cid:2)
1.2Nˆ+1.5Nkˆ andλˆ =−0.8ıˆ+0.6ˆ. b) A(cid:2)+b=b+A(cid:2)
(cid:2) (cid:2) (cid:2) (cid:2)
30o x 2.41FindthedotproductF(cid:2)·r(cid:2)whereF(cid:2)= c) A·B =B·A
(5ıˆ+4ˆ)Nandr(cid:2)=(−0.8ıˆ+ˆ)m. Draw d) B(cid:2)/C(cid:2)=B/C
z thetwovectorsandjustifyyouranswerforthe (cid:2)
problem2.34: dotproduct. e) b/A=b/A
(Filename:pfigure2.vec1.22) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)
2(cid:2).42 Two√vectors, a(cid:2) = −4√3ıˆ +12ˆ and f) A=(A·B)B+(A·C)C+(A·D)D
b =ıˆ− 3ˆaregiven. Findthedotproduct
o|a(cid:2)f|t|h(cid:2)be|tiwnothvisecctaosres?. How is a(cid:2)· (cid:2)b related to c2o.5s0ineUs’s;ei.thee.,dotproducttoshow‘thelawof
2.35 Acirculardiskofradius6inismounted 2.43FindthedotproductoftwovectorsF(cid:2)= c2=a2+b2+2abcosθ.
oinntahxelefixg-uxrea.tthTeheenddisaknLis-stihpappeedd4b5aroawsisthhotwhne 1(cid:2)0lbfıˆ−ˆ20lbfˆandλˆ =0.8ıˆ+0.6ˆ.Sketch (Hint: (cid:2)c =a(cid:2)+(cid:2)b;also,(cid:2)c·(cid:2)c =(cid:2)c·(cid:2)c)
horizontal bar AC. Two points, P and Q, are F andλandshowwhattheirdotproductrep-
markedontherimoftheplate;Pdirectlypar- resents.
b
ahlilgehletsotthpeoicnetnatebroCveinthtoetcheenpteargeC,.anTdakQinagttthhee 2.44ThepositionvectorofapointAisr(cid:2)A = c θ
basevectorsıˆ,ˆ,andkˆasshowninthefigure, 3√0cmıˆ. Findthedotproductofr(cid:2)Awithλˆ =
find 3ıˆ+ 1ˆ.
2 2 a
(a) therelativepositionvectorr(cid:2)Q/P, 2.45Fromthefigurebelow,findthecomponent problem2.50:
(b) themagnitude|r(cid:2)Q/P|. offorceF(cid:2)inthedirectionofλˆ. (Filename:pfigure.blue.2.1)
726 APPENDIX2. Contact: frictionandcollisions
(cid:2) (cid:2)
2.51FindthedirectioncosinesofF =3Nıˆ− 2.57ThenetforceactingonaparticleisF = 2.63 Write a computer program (or use a
4Nˆ+5Nkˆ. 2Nıˆ+10Nˆ. Find the components of this canned program) to find the dot product of
force in another coordinate system with ba- two 3-D vectors. Test the program by com-
sisvectorsıˆ(cid:8) = −cosθıˆ+sinθˆandˆ(cid:8) = putingthedotproductsıˆ·ıˆ,ıˆ·ˆ, andˆ·kˆ.
2.52Aforceactingonabeadofmassmisgiven −sinθıˆ−cosθˆ. Forθ = 30o, sketchthe Nowusetheprogramtofindthecomponents
asF(cid:2)=−20lbfıˆ+22lbfˆ+12lbfkˆ.Whatis vectorF(cid:2)andshowitscomponentsinthetwo of F(cid:2) = (2ıˆ +2ˆ −3kˆ)N along the line
theanglebetweentheforceandthez-axis? coordinatesystems. r(cid:2) =(0.5ıˆ−0.2ˆ+0.1kˆ)m.
AB
2.58FindtheunitvectorseˆRandeˆθintermsof
2.53 (a) Draw the vector r(cid:2) = 3.5inıˆ + ıˆandˆwiththegeometryshowninthefigure. 2.64 What is the shortest distance between
3to.5rminaˆk−es4w.9i5thintkhˆe.(zb-a)xFiisn.d(tch)eFainngdleththeisanvgecle- eWˆθh?atarethecomponentsofW(cid:2) alongeˆR and tahlleelpeopiinptedAshaonwdnt?he(UdisaegvoencatlorBmCeothfotdhse.)par-
thisvectormakeswiththex-yplane. B 1
A
C
2.54 Inthefigureshown,λˆ andnˆareunitvec- 3
tors parallel and perpendicular to the surface θ
(cid:2) (cid:15) 4
AB,respectively. AforceW =−50(cid:2)Nˆacts problem2.64:
ontheblock.FindthecomponentsofWalong ˆ (Filename:pfigure.blue.2.3)
λˆ andnˆ.
A
nˆ ıˆ W 2.3 Cross product, moment,
ˆ ˆ eˆθ and moment about an axis
λ
eˆ
ıˆ W R
30o problem2.58: 2.65 Findthecrossproductofthetwovectors
O B (Filename:pfigure2.vec1.44) showninthefiguresbelowfromtheinforma-
problem2.54: 2.59 Write the position vector of point P in tiongiveninthefigures.
(Filename:pfigure2.vec1.41) termsofλˆ andλˆ and (a) y (b) y
2o.f5ıˆ5aEndxpˆresshsotwhenuinnitthveefictgourrsen.ˆWanhdaλtˆairnettehremxs ((ba)) fifinndd1tthheecyo-2cmopmopnoennetnotfor(cid:2)fPr(cid:2)aPl,ongλˆ1. a(cid:2)2 60o4 (cid:2)bx a(cid:2) 3 105o 4 (cid:2)bx
andycomponentsofr(cid:2)=3.0ftnˆ−1.5ftλˆ?∗ y P
(cid:15)
y 2 (c) y (d) y
ˆ
λˆ nˆ θ2 λ2 430o a(cid:2) 45o 4 a(cid:2)
ˆ θ (cid:15) ˆ 30o x 5 45o x
1 λ
1 4 (cid:2) (cid:2)
b b
ıˆ x
θ
problem2.55: 1
(e) y (f) y
(Filename:efig1.2.27) x
problem2.59: (cid:2) (cid:2)
b b
(Filename:pfigure2.vec1.45) 3 3
2.56 Fromthefigureshown,findthecompo-
npeonsittsioonfvveeccttoorr)r(cid:2)aAloBn(gyouhavetofirstfindthis 2−.6200NLeˆtF+(cid:2)12=Nk3ˆ0,Nanıˆd+F(cid:2)430N=ˆF−3x1ıˆ0+NkˆF,3Fy(cid:2)2ˆ=− 22 3a(cid:2) x 22 3a(cid:2) x
(a) they-axis,and ˆ
(b) alongzλˆ. zfFoe3rrzotkh,.fienIcfdotmhthepesorunemeqnuotisrfeoadlflsFtc(cid:2)ha3el.aserefoqrucaetsiomnsutsotesqoulvael (g) y (cid:2)b=4ˆ (h) y
(cid:2)
(cid:2) (-1,2) b (2,2)
2.61AforceF isdirectedfrompointA(3,2,0)
2m topointB(0,2,4). Ifthe x-componentofthe a(cid:2)
A B foofrFc(cid:2)e.is120 N,findthe y-andz-components a(cid:2)=3ıˆ+xˆ (-1,-1) (2,-1) x
2m
3m 1m 2.62Avectorequationforthesumofforces problem2.65:
30o y resultsintothefollowingequation: (Filename:pfigure2.vec2.1)
√
λˆ F(ıˆ− 3ˆ)+ R(3ıˆ+6ˆ)=25Nλˆ 2.66Vectoralgebra.Foreachequationbelow
2 5 statewhether:
prxoblem2.56: whereλˆ = 0.30ıˆ−0.954ˆ. Findthescalar (a) Theequationisnonsense. Ifso,why?
ˆ
(Filename:pfigure2.vec1.42) equationsparallelandperpendiculartoλ. (b) Isalwaystrue.Why?Giveanexample.
ProblemsforChapter2 727
(c) Isnevertrue. Why? Giveanexample. 2.71 Inthefigureshown,OA=AB=2m.The
forceF =40Nactsperpendiculartothearm
(d) Issometimestrue.Giveexamplesboth (cid:2)
AB.FindthemomentofF aboutO,giventhat
ways. (cid:2)
θ =45o. IfF alwaysactsnormaltothearm
Youmayusetrivialexamples. AB,wouldincreasingθincreasethemagnitude
a) B(cid:2)×C(cid:2)=C(cid:2)×B(cid:2) ofthemoment? Inparticular,whatvalueofθ
(cid:2) (cid:2) (cid:2) (cid:2) willgivethelargestmoment?
b) B×C=C·B y
F
(cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) problem2.74:
c) C·(A×B)=B·(C×A)
(Filename:pfigure2.vec2.8)
d) A(cid:2)×(B(cid:2)×C(cid:2))=(A(cid:2)·C(cid:2))B(cid:2)−(A(cid:2)·B(cid:2))C(cid:2) θ B
A (cid:15)
2.75 Find the sum of moments due to the
(cid:2) twoweightsoftheteeter-totterwhentheteeter-
2.67WhatisthemomentMproducedbya20 totteristippedatanangleθ fromitsvertical
(cid:15)
NforceFactinginthexdirectionwithalever position.Giveyouranswerintermsofthevari-
armofr(cid:2)=(16mm)ˆ? ablesshowninthefigure.
A
O
x
2.68 Findthemomentoftheforceshownon
problem2.71: α
therodaboutpointO. (cid:15) α
y F=20N (Filename:pfigure2.vec2.5) θ
h
B
2.72 Calculatethemomentofthe2kNpayload O (cid:15)
ontherobotarmabout(i)jointA,and(ii)joint
B,if(cid:15)1=0.8m,(cid:15)2=0.4m, and(cid:15)3=0.1m.
45o y W
2m (cid:15)
A 3 C
30o
pOroblem2.68: x (cid:15)1 (cid:15)2 C OAAB==AhC=(cid:15) W
B
(Filename:pfigure2.vec2.2)
45o 2kN problem2.75:
O x (Filename:pfigure2.vec2.9)
2(cid:2).69 Fi(cid:2)nd the sum of moments of forces problem2.72:
W and T about the origin, given that W =
100N,T =120N,(cid:15)=4m, andθ =30o. (Filename:pfigure2.vec2.6) 2.76 Findthep(cid:2)ercentageerrorincomputing
y themomentofW aboutthepivotpointOas
2.73 Duringaslam-dunk,abasketballplayer afunctionofθ,iftheweightisassumedtoact
T pullsonthehoopwitha250lbfatpointCofthe normaltothearmOA(agoodapproximation
(cid:15) /2 ringasshowninthefigure. Findthemoment whenθisverysmall).
oftheforceabout (cid:15) A
(cid:15) /2
W a) thepointoftheringattachmenttothe O θ
board(pointB),and
θ W
O x b) therootofthepole,pointO. problem2.76:
problem2.69: board
(Filename:pfigure2.vec2.10)
(Filename:pfigure2.vec2.3)
6"
A 2.77Whatdoyougetwhenyoucrossavector
2.70 Findthemomentoftheforce 3' 1.5' basketball hoop andascalar? ∗
a) aboutpointA B
b) aboutpointO. 10' 15o 2.78Whydidthechickencrosstheroad? ∗
F=50N
250lbf
α=30o 2.79Carryoutthefollowingcrossproductsin
different ways and determine which method
O
takestheleastamountoftimeforyou.
problem2.73: a) r(cid:2)=2.0ftıˆ+3.0ftˆ−1.5ftkˆ; F(cid:2)=
1.5m (Filename:pfigure2.vec2.7) −0.3lbfıˆ−1.0lbfkˆ; r(cid:2)×F(cid:2)=?
2.74 Duringweighttraining,anathletepulls b) r(cid:2) = (−ıˆ+2.0ˆ +0.4kˆ)m; L(cid:2) =
O A aweightof500Nwithhisarmspullingona (3.5ˆ−2.0kˆ)kgm/s; r(cid:2)×L(cid:2)=?
problem22m.70: haacnadblleeb.aFricnodntnheecmteodmtoenatuonfitvheersfoalrcmeaacbhoiuntetbhye c) ω(cid:2) = (ıˆ−1.5ˆ)rad/s; r(cid:2)= (10ıˆ−
(Filename:pfigure2.vec2.4) shoulderjointOintheconfigurationshown. 2ˆ+3kˆ)in; ω(cid:2)×r(cid:2)=?
728 APPENDIX2. Contact: frictionandcollisions
2.80AforceF(cid:2)=20Nˆ−5Nkˆactsthrough 2.86 Whatisthedistancedbetweentheorigin a) FindaunitvectorinthedirectionOB.
∗
apointAwithcoordinates(200 mm,300 mm, andtheline ABshown? (Youmaywriteyour
-100mm).WhatisthemomentM(cid:2)(= r(cid:2)×F(cid:2)) solutionintermsofA(cid:2)andB(cid:2)beforedoingany b) FindaunitvectorinthedirectionOA.
oftheforceabouttheorigin? arithmetic). ∗ ∗
z
(cid:2) (cid:2)
2.81CrossProductprogramWriteaprogram c) WritebothF1andF2astheproduct
1 A
thatwillcalculatecrossproducts.Theinputto oftheirmagnitudesandunitvectorsin
thefunctionshouldbethecomponentsofthe kˆ theirdirections. ∗
twovectorsandtheoutputshouldbethecom- (cid:2) ∗
A d) WhatistheangleAOB?
ponentsofthecrossproduct.Asamodel,here ˆ (cid:2)
isafunctionfilethatcalculatesdotproductsin ıˆ e) What is the c∗omponent of F1 in the
pseudocode. d x-direction?
%program definition O 1 y f) Whatisr(cid:2)DO×F(cid:2)1? (r(cid:2)DO ≡ r(cid:2)∗O/D
z(1)=a(1)*b(1); isthepositionofOrelativetoD.)
z(2)=a(2)*b(2); (cid:2) (cid:2)
z(3)=a(3)*b(3); B g) WhatisthemomentofF2 aboutthe
1 axisDC?(Themomentofaforceabout
w=z(1)+z(2)+z(3); B ˆ
anaxisparalleltotheunitvectorλis
2.82 Findaunitvectornormaltothesurface pxroblem2.86: definedasMλ=λˆ·(r(cid:2)×F(cid:2))wherer(cid:2)is
ABCDshowninthefigure. (Filename:pfigure.blue.1.3) thepositionofthepointofapplication
z oftheforcerelativetosomepointon
2.87 What is the perpendicular distance be- the axis. The result does not depend
A tween the point A and the line BC shown? onwhichpointontheaxisisusedo(cid:2)r
D (Thereareatleast3waystodothisusingvar- whichpointonthelineofactionofF
iousvectorproducts,howmanywayscanyou isused.). ∗
4" B y find?) y h) Repeatthelastproblemusingeithera
5" differentreferencepointontheaxisDC
x 5" C orthelineofactionOB.Doesthesolu-
problem2.82: B A tionagree? [Hint: itshould.] ∗
3
(Filename:efig1.2.11) z
(cid:2) ˆ
2.83 If the magnitude of a force N normal 5m A
C
tworittheeN(cid:2)surafsacaevAecBtoCr.D∗inthefigureis1000 N, ıˆ 0 2 3 x B
z
problem2.87: F2
B (Filename:pfigure.blue.2.2)
A 2.88Givenaforce,F(cid:2)1=(−3ıˆ+2ˆ+5kˆ)N O F1 3m
actingatapointPwhosepositionisgivenby 4m y
1m D C 1m y r(cid:2)P/O =(4ıˆ−2ˆ+7kˆ)m,whatisthemo- D C
x 1m mentaboutanaxisthroughtheoriginO with x
1m 1m directionλˆ = √2 ˆ+ √1 ˆ? problem2.90:
problem2.83: 5 5 (Filename:p1sp92)
(Filename:efig1.2.12)
2.89 Drawing vectors and computing with 2.91 A, B, and C are located by position
22su.x8rf4−acTeyh..eeFqiunadtiaonunoiftavseucrtfoarcenˆisnogrivmeanlatsozth=e vcxoeycoztrodcroinso.artdTeishne(a4tpe,os5in(,0t1,O25),mis1.t2h)emo.riPgioni.ntPBoinhtaAsxhyazs vr(cid:2)eCcto=rs(7r(cid:2),A8,=9).(1,2,3), r(cid:2)B = (4,5,6), and
a) Usethevectordotproducttofindthe
2.85 Inthefigure,atriangularplateACB,at- a) MakeaneatsketchofthevectorsOA, angle BAC (Aisatthevertexofthis
tachedtorodAB,rotatesaboutthez-axis. At OB,andAB. angle).
theinstantshown,theplatemakesanangleof b) Find a unit vector in the direction of b) Usethevectorcrossproducttofindthe
60o withthe x-axis. Findanddrawavector OA,callitλˆOA. angle BCA(C isatthevertexofthis
normaltothesurfaceACB. (cid:2) angle).
z c) FindtheforceF whichis5Ninsize
andisinthedirectionofOA. c) Findaunitvectorperpendiculartothe
B
planeABC.
d) WhatistheanglebetweenOAandOB?
45o C e) Whatisr(cid:2)BO×F(cid:2)? d) HAoBwfrfoamritshethoerigininfi?ni(tTehlainteis,dheofiwnecdlobsye
(cid:2)
45o f) WhatisthemomentofF aboutaline is the closest point on this line to the
paralleltothezaxisthatgoesthrough origin?)
1m
A thepointB? e) Istheoriginco-planarwiththepoints
y
A,B,andC?
x 60o 2.90 Vector Calculations and Geometry.
(cid:2)
problem2.85: The 5N fo(cid:2)rce F1 is along the line OA. The 2.92 PointsA,B,andCinthefiguredefinea
(Filename:efig1.2.14) 7NforceF2isalongthelineOB. plane.
ProblemsforChapter2 729
a) Findaunitnormalvectortotheplane. b) A(cid:2)=2ıˆ,C(cid:2)=15ˆ+360kˆ y
∗ B(cid:2)=3ıˆ+4ˆ,D(cid:2)=2ıˆ+17ˆ+37kˆ B
b) pFlianndettohethedipstoainncteD.fr∗om this infinite c) A(cid:2)=kˆ,C(cid:2)=ıˆ+ˆ+kˆ
c) What are the coordinates of the point B(cid:2)=ˆ+kˆ,D(cid:2)=ˆ
ontheplaneclosesttopointD?∗ d) A(cid:2)=ıˆ+ˆ+kˆ,C(cid:2)=3ıˆ+4ˆ+5kˆ 60o 45o x
z B(cid:2)=2ıˆ+3ˆ+4kˆ,D(cid:2)=4ıˆ+5ˆ+7kˆ A C
45 e) A(cid:2)=√1ıˆ+√2ˆ+√3kˆ, problem2.99:
(3, 2, 5) B CB(cid:2)(cid:2)==√√74ııˆˆ++√√85ˆˆ++√√96kkˆˆ,, (cid:2) (cid:2)(Filename:(cid:2)pfigure2.4.2Dp2)
1 (0, 7, 4)D D(cid:2)=−ıˆ+πˆ+ekˆ 2.100 Threevectors,A,(cid:2)B,a(cid:2)ndC(cid:2),(sh(cid:2)ownin
thefigure)aresuchthatA+B+C= 0.You
2 (cid:2) (cid:2)
3 4 aregiventhatA=|A|=8andC =|C|=5.
Findangleθ.
5 A C 7 y
x (5, 2, 1) (3, 4, 1) y 2o.n97aIcnotmhepuptreorbtloemfisnbdealo,wbuasnedmcaturnixiqaulgeleybriaf
possible. Ifnotpossibleexplainwhynot. You
problem2.92: B
aregiventhat
(Filename:pfigure.s95q2)
(cid:2) (cid:2) (cid:2) (cid:2)
2.93Whatpointonthelinethatgoesthrough aA+bB+cC=D 30o θ x
thepoints(1,2,3)and(7,12,15)isclosesttothe A C
origin? andthat
(cid:2) (cid:2)
2.94 What is the distance between opposite a) A=ıˆ,B =ˆ,
edges of a regular tetrahedron with sides of C(cid:2)=kˆ,D(cid:2)=−2ıˆ+5ˆ+10kˆ problem2.100:
(Filename:pfigure2.4.2Dp3)
length 1 (there are many ways to solve this (cid:2) (cid:2)
problem). b) A=ıˆ+ˆ,B =−ıˆ+ˆ,
C(cid:2)=kˆ,D(cid:2)=2ıˆ 2.101LetF(cid:2) +F(cid:2) +F(cid:2) = (cid:2)0,whereF(cid:2) =
1 (cid:2)2 (cid:2)3 3
2.4Solvingvectorequations c) A(cid:2)=ıˆ+ˆ+kˆ,B(cid:2)=2ıˆ+ˆ+kˆ, 10N((cid:2)ıˆ−(cid:2)ˆ),andF1/|F1|=0.250ıˆ+0.96(cid:2)8ˆ
C(cid:2)=ıˆ+2ˆ+kˆ,D(cid:2)=2ıˆ+ˆ+kˆ andF2/|F2|=−0.425ıˆ−0.905ˆ.Find|F1|
fromasinglescalarequation.
2.95Considerthevectorequation d) A(cid:2)=1ıˆ+2ˆ+3kˆ,B(cid:2)=4ıˆ+5ˆ+6kˆ,
aA(cid:2)+bB(cid:2)=C(cid:2) C(cid:2)=7ıˆ+8ˆ+9kˆ,D(cid:2)=10kˆ 2.102Toevaluatetheequation(cid:13)(cid:13)F(cid:2)=ma(cid:2)for
withA(cid:2),B(cid:2),andC(cid:2)given. Forthecasesbelow e) A(cid:2)=√1ıˆ+√2ˆ+√3kˆ,B(cid:2)=√4ıˆ+ someproblem,astudentwrites F(cid:2)=Fxıˆ−
fileuxntpdiolaanisnangwdivhbeyi.aftploesassitb2le..IfIftthheerreeaarreenmouslotilpulteiosnos- C√(cid:2)5=ˆ+√7√ıˆ6+kˆ√, 8ˆ+√9kˆ,D(cid:2)=√10kˆ (syFsyte−m,30buNt)a(cid:2)ˆ+=520.5Nmkˆ/isn2ıtˆh(cid:8)e+x1y.z8cmo/osr2diˆn(cid:8)a−te
a) A(cid:2)=ıˆ, B(cid:2)=ˆ, azkˆ(cid:8)inarotatedx(cid:8)y(cid:8)z(cid:8)coordinatesystem. If
b) CAC(cid:2)(cid:2)(cid:2)===33ıˆ,ııˆˆ+B(cid:2)4=ˆ2ıˆ, 2eq.9u8iliTbhrieutmhr,ei.eef.o,rTc(cid:2)e1s+shT(cid:2)ow2+niFn(cid:2)th=e(cid:2)fi0g.uIrfe|Fa(cid:2)re|=in ıcfˆoo(cid:8)rs=t6h0ceooxsˆ(cid:8)6,a0nyodı(cid:8)ˆ,kˆ+a(cid:8)n=sdinzk(cid:8)ˆ6,d0fiiorneˆdc,ttihoˆen(cid:8)ss=.ca−larsienq6u0atoiıoˆn+s
c) A(cid:2)=ˆ, B(cid:2)=2ˆ, 1(cid:2)0N,fin(cid:2)dtensionsT1 andT2 (magnitudesof
C(cid:2)=3ıˆ T1andT2. 2.103 A car travels straight north-east for a
d) A(cid:2)=ıˆ+ˆ, B(cid:2)=−ıˆ+ˆ, y whihgihlewaoyn.aTdhiertcraoratdratvhealtsloenadtshetohingohrwtha-ysoduuthe
(cid:2)
C=2ˆ T northforawhile. Whenthedriverstops,the
(cid:2) (cid:2) 1 GPSsysteminstalledinthecarindicatesthat
e) A(cid:2)=ıˆ+2ˆ, B =2ıˆ+3ˆ, 30o T2 car60milesnorthand30mileseastfromthe
C=3ıˆ+4ˆ startingpoint. Findthedistancetravelledon
f) A(cid:2)=πıˆ+eˆ, B(cid:2)=√2ıˆ+√3ˆ, x thedirtroad.
(cid:2) F
C=ıˆ
2.104 A particle is held at point P with the
2.96Considerthevectorequation problem2.98: helpofthreestringsPA,PB,andPC.Letthe
aA(cid:2)+bB(cid:2)+cC(cid:2)=D(cid:2) (Filename:pfigure2.4.2Dp1) t(cid:2)ensionsinthethreestrings(cid:2)beT(cid:2)A,T(cid:2)B,and
(cid:2) (cid:2) (cid:2) (cid:2) TC,respectively(sothatTA actsalongline
with A, B, C and D given. For the cases PAandsoon). Theequilibriumoftheparticle
below find a if possible, there is no need to 2.99 Points A, B, and C are located in the requiresthatT(cid:2)A+T(cid:2)B+T(cid:2)C+W(cid:2) =(cid:2)0where
findab)aABn(cid:2)(cid:2)d==c.ıˆˆ,,C(cid:2)D(cid:2)==kˆ3ıˆ+4ˆ+19kˆ |vxr(cid:2)eyBctp|olarasnn,dew|aer(cid:2)sCcsa/hBno|wwifnrirt(cid:2)ienC,tr(cid:2)=hBe1+fi0gmr(cid:2)uCreıˆ/..BF=orr(cid:2)pCo.sFitiinodn WFs(cid:2)tirnind=gtsh.−e1m0aNgnkiˆtuidsetsheofwteeingshitonosftihnethpeartthicrleee.