Introduction to Statics and Dynamics. Problem Book

117 Pages·2001·4.869 MB·English

by  Pratap R., Ruina A.

Introduction to
S
TATICS
and
D
YNAMICS
Problem Book
Rudra Pratap and Andy Ruina
Spring 2001
(cid:176)c Rudra Pratap and Andy Ruina, 1994-2001. All rights reserved. No part of
this book may be reproduced, stored in a retrieval system, or transmitted, in
any form or by any means, electronic, mechanical, photocopying, or otherwise,
without prior written permission of the authors.
This book is a pre-release version of a book in progress for Oxford University
Press.
The following are amongst those who have helped with this book as editors,
artists, advisors, or critics: Alexa Barnes, Joseph Burns, Jason Cortell, Ivan
Dobrianov, GaborDomokos, ThuDong, GailFish, JohnGibson, SaptarsiHal-
dar,DaveHeimstra,TheresaHowley,HerbertHui,MichaelMarder,ElainaMc-
Cartney,ArthurOgawa,KalpanaPratap,RichardRand,DaneQuinn,Phoebus
Rosakis,LesSchaefier,DavidShipman,JillStartzell,SaskyavanNouhuys,Bill
Zobrist. Mike Coleman worked extensively on the text, wrote many of the ex-
amples and homework problems and created many of the flgures. David Ho
has brought almost all of the artwork to its present state. Some of the home-
work problems are modiflcations from the Cornell’s Theoretical and Applied
MechanicsarchivesandthusareduetoT&AMfacultyortheirlibrariesinways
that we do not know how to give proper attribution. Many unlisted friends,
colleagues,relatives,students,andanonymousreviewershavealsomadehelpful
suggestions.
SoftwareusedtopreparethisbookincludesTeXtures,BLUESKY’simplemen-
tation of LaTeX, Adobe Illustrator and MATLAB.
Most recent text modiflcations on January 21, 2001.
Contents
ProblemsforChapter1 : : : : : : : : : : : : : : : : : : : : : : : : : : 0
ProblemsforChapter2 : : : : : : : : : : : : : : : : : : : : : : : : : : 2
ProblemsforChapter3 : : : : : : : : : : : : : : : : : : : : : : : : : : 10
ProblemsforChapter4 : : : : : : : : : : : : : : : : : : : : : : : : : : 15
ProblemsforChapter5 : : : : : : : : : : : : : : : : : : : : : : : : : : 18
ProblemsforChapter6 : : : : : : : : : : : : : : : : : : : : : : : : : : 31
ProblemsforChapter7 : : : : : : : : : : : : : : : : : : : : : : : : : : 41
ProblemsforChapter8 : : : : : : : : : : : : : : : : : : : : : : : : : : 60
ProblemsforChapter9 : : : : : : : : : : : : : : : : : : : : : : : : : : 74
ProblemsforChapter10 : : : : : : : : : : : : : : : : : : : : : : : : : : 83
ProblemsforChapter11 : : : : : : : : : : : : : : : : : : : : : : : : : : 88
ProblemsforChapter12 : : : : : : : : : : : : : : : : : : : : : : : : : : 100
Answersto*’dquestions
ProblemsforChapter1 1
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?
1.4Thisbookorgainzesthelawsofmechanics
into4basiclawsnumberred0-III,notthestan-
dard ‘Newton’s three laws’. What are these
fourlaws(inEnglish,noequationsneeded)?
1.5Describe,aspreciselyaspossible,aprob-
lem that is not mentionned in the book but
which is a mechanics problem. State which
quantities are given and what is to be deter-
minedbythemechanicssolution.
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(differentthatn
theexamplesgiven).
c) (optional)Doanexampleofeachona
computer.
2 CONTENTS
Problems for Chapter 2
Vectorskillsformechanics B r*BC C b2e.1lo1w,Fofirndthetheunsitcavleacrstor(cid:11)s (cid:21)aO1ndan(cid:12)d (cid:21)sO2ucshhotwhant
(cid:11)(cid:21)O −3(cid:21)O D(cid:12)|O.
r* 2 2 y
CD
2.1Vectornotationandvec-
tor addition r*AB (cid:21)O2
D
1
O
k 60o
t2h.1reReedpifrfeesreenntttwheayvse.ctorr*D5m{O−2m|Oin A |O 1 (cid:21)O1 x
problem2.5: problem2.11:
(Filename:pfigure2.vec1.5) (Filename:pfigure2.vec1.11)
2.2Whichoneofthefollowingrepresentations p
ofthesamevectorF*iswrongandwhy? 2m.6DTh5ekgforacreessahcotwinngionnthaebfilogcukreo,fwmhearses 2.12 Inthefigureshown,T1D20 2N;T2D
a) b) 40N,andW issuchthatthesumofthethree
F1 D20*N; F2*D50*N,and*W Dmg. Find forcesequalszero.IfWisdoubled,find(cid:11)and
|O 2N thesumF .DF1CF2CW*/? (cid:12)suchthat(cid:11)T*1;(cid:12)T*2,and2W* stillsumupto
3N -3N{OC2N|O F*1 F2 zero. y
{O
4 3
c) d) p T
p 13N 3 4 2 T1
13N
2 2 3 60o 45o
3
x
problem2.2: *
W
(Filename:pfigure2.vec1.2) problem2.6:
(Filename:pfigure2.vec1.6)
2.3 Thereareexactlytworepresentationsthat W
2.7 Three position vectors are shown in the
dtuersecsr.ibMeatthcehstahmeecovrercetcotrpiincttuhreesfoinlltoowpianigrsp.ic- fipgure below. Given that r*B=A D 3m.12{OC problem2.12: (Filename:pfigure2.vec1.12)
a) 4N b) 4N 23|O/andr*C=BD1m{O−2m|O,findr*A=C.
|O B 2.13Inthefigureshown,rodsABandBCare
each 4 cm long and lie along y and x axes,
30o 30o respectively. RodCDisinthe xz planeand
{O |O makesanangle(cid:18) D30owiththex-axis.
c) d) (a) Findr* intermsofthevariablelength
AD
2N p {O ‘.
p 2N(-{OC 3|O) (b) Find‘and(cid:11)suchthat
2 3N
r* D r* −r* C(cid:11)kO:
C AD AB BC
e) f)
z
A
3N{OC1N|O 3N(13{OC|O) problem2.7: D
problem2.3: (Filename:pfigure2.vec1.7) ‘
(Filename:pfigure2.vec1.3) 2.8GiventhatthesumoffourvectorsF*;i D A 4cm B
* i* y
22.N4|OF;inF*d tDhe3su0mN.opf1f{OorCcesp1F*|1O/;Dan2d0FN*{O −D −15to0N4|,O;isF*3zeDro,10wNh.e−re{OFC1|OD/,fi2n0dNF*{O4;.F2 D 30o 4cm
−20N.−2{OCp3|O/. 2 2 3 2.9 Three forces F* D 2*N{O −5N|O;R* D prxoblem2.13:C
10N.cos(cid:18){OCsin(cid:18)|O/andW D−20N|O,sum
(Filename:pfigure2.vec1.13)
uptozero.Determinetheangle(cid:18)anddrawthe
2vrr**e.C5cDto.IDrns 2tahrfeet.fi|r*OAgCuBrekOD/s.h3oFwfitnnkOd;btr*ehBleoCwpo,Dstiht2ieofnpt|oOvs,eictaitnoodnr f32o:.12rc0NeG{Ovie−vcetn0o:rt4hR*Nat|cRO*l,1efiaDnrldy12sNRh*o{OwCCi1n5:g5R*iNts.|OdiarnecdtiRo*2n.D tm32h0.1eaNg4tn{wOFi−toiun4ddf0eostNrh.ce|eOsma,nardgepFn*ri2teusDdeen3sti0nogNft{OthhCee4mf0orNwce|iOst.hFD*t1hraeDwir
AD 1 2
ProblemsforChapter2 3
*
2.15 Two forces R D 2N.0:16{O C 2.21 A1m(cid:2)1msquareboardissupported 2.24 Acirculardiskofradius6inismounted
*
0:80|O/andW D −36N|O act on a particle. bytwostringsAEandBF.Thetensioninthe onaxlex-xattheendanL-shapedbarasshown
Findthemagnitudeofthenetforce. Whatis string BF is 20N. Express this tension as a inthefigure. Thediskistipped45owiththe
thedirectionofthisforce? vector. horizontal bar AC. Two points, P and Q, are
y
markedontherimoftheplate;Pdirectlypar-
2.16InProblem2.13,find‘suchthatthelength 2.5m alleltothecenterCintothepage,andQatthe
ofthepositionvectorr* is6cm. highest point above the center C. Taking the
AD F basevectors{O;|O,andkOasshowninthefigure,
2oF.f21F7D*23−In00Ft*hN1e..fiFginyudrethsehmowagnn,itFu1deDand10d0irNectainodn E 1 12m 2m find((ba)) tthheemrelaagtniviteupdoesjir*tiQo=nPvj.ectorr*Q=P,
1m
F*2-F*1 A B kO |O Q C P
F F2 plate 1m {O Q 6"
1 45o D
45o 30o D C x A C
x problem2.21: 6"
problem2.17: (Filename:pfigure2.vec1.21) 12" D
(Filename:pfigure2.vec1.17) O
2.22 ThetopofanL-shapedbar,showninthe
* * figure,istobetiedbystringsADandBDto x x
2.18 LettwoforcesP andQactinthedirec- thepointsAandBinthe yz plane. Findthe problem2.24:
tionshowninthefigure. Youareallowedto lengthofthestringsADandBDusingvectors (Filename:pfigure2.vec1.24)
changethedirectionoftheforcesbychanging r* andr* .
theangles(cid:11) and(cid:18) whilekeepingthemagni- AD BD
itudesfixed. Whatshouldbethevaluesof(cid:11) B 2.25 Findtheunitvector(cid:21)O ,directedfrom
* * AB
and(cid:18)ifthemagnitudeofPCQhastobethe pointAtopointBshowninthefigure.
maximum? y y
y 2m 3m
B
Q
1m A
2m
P
(cid:11) (cid:18)
1m
x 1m
problem2.18:
x
(Filename:pfigure2.vec1.18) A problem2.25:
(Filename:pfigure2.vec1.25)
2.19TwopointsAandBarelocatedinthexy
plane. ThecoordinatesofAandBare(4mm,
8mm)and(90mm,6mm),respectively. 30o x 2.26 FindaunitvectoralongstringBAand
(a) Drawpositionvectorsr* andr*. expressthepositionvectorofAwithrespectto
(b) Findthemagnitudeofr*A andr*B. z B,r*A=B,intermsoftheunitvector.
A B problem2.22: y
(c) HowfarisAfromB? (Filename:pfigure2.vec1.22)
1.5m
2.23 Acubeofside6inisshowninthefigure.
2.20 Inthefigureshown,aballissuspended (a) FindthepositionvectorofpointF,r*, A
wit(haa)0F.8inmdlothnegpcoosridtifornomveac2tomrlr*oBngofhtohiestbOaAll.. fr*rFo=mc:thevectorsumr*FD r*DCr*C=DCF 2.5m
(b) Findthedistanceoftheballfromthe (b) Calculatejr*j. 1m x
origin. F 3m
y A (c) Findr*Gusingr*F. B
z y z
problem2.26:
0.8m E F
m (Filename:pfigure2.vec1.26)
2
D C
B H G 2.27 Inthestructureshowninthefigure,‘D
45o 2ft;hD1:5ft.TheforceinthespringisF*D
kr* ,wherekD100lbf=ft.Findaunitvector
O x A B x O AB
problem2.20: (Filename:pfigure2.vec1.20) problem2.23: (Filename:pfigure2.vec1.23) (cid:21)F*ABDaFlo(cid:21)nOAgBA.B and calculate the spring force
4 CONTENTS
y *
2.34FindthedotproductoftwovectorsF D 2.43 Usethedotproducttoshow‘thelawof
C
10lbf{O−20lbf|Oand(cid:21)O D0:8{OC0:6|O.Sketch cosines’;i. e.,
* O
F and(cid:21)andshowwhattheirdotproductrep- c2Da2Cb2C2abcos(cid:18):
B resents. (Hint: *c =a*+*b;also,*c(cid:1)*c D*c(cid:1)*c)
2.35ThepositionvectorofapointAis r* D
A
30o 30cm{O. Findthedotproductof r* with(cid:21)O D b
p A
‘ c
h 3{OC 1|O.
2 2 (cid:18)
O x 2.36Fromthefigurebelow,findthecomponent
problem2.27: offorceF*inthedirectionof(cid:21)O. a
y
(Filename:pfigure2.vec1.27)
problem2.43:
2.28Expressthevectorr* D2m{O−3m|OC (Filename:pfigure.blue.2.1)
5inmdikcOaitnintegrimtssdoifreitcstmioang.niAtudeandaunitvector (cid:21)O F=100N 32:.454in|(Oa−) 4D:r9a5winktOh.e(bv)eFctionrdtrh*eaDngle3:t5hiisnv{OeCc-
30o 10o tormakeswiththez-axis. (c)Findtheangle
2.29 Let F* D 10lbf{OC30lbf|O and W* D x thisvectormakeswiththex-yplane.
problem2.36:
−20lbf|O.Findaunitvectorinthedirectionof 2.45 Inthefigureshown,(cid:21)O andnOareunitvec-
* * (Filename:pfigure2.vec1.33)
thenetforceF CW,andexpressthethenet tors parallel and perpendicular to the surface
forceintermsoftheunitvector. 2.37 Find the angle between F* D 2N{O C AB,respectively. AforceW* D−50N|Oacts
* 1 *
5N|OandF D−2N{OC6N|O. ontheblock.FindthecomponentsofWalong
2.30Let(cid:21)O1D0:80{OC0:60|Oand(cid:21)O2D0:5{OC 2 (cid:21)O andnO.
0:866|O. 2.38AforceF*isdirectedfrompointA(3,2,0) A
(a) Showthat(cid:21)O and(cid:21)O areunitvectors. topointB(0,2,4). Ifthe x-componentofthe nO
1 2 forceis120 N,findthe y-andz-components |O
(b) Isthesumofthesetwounitvectorsalso ofF*. (cid:21)O
aunitvector? Ifnot, thenfindaunit
vectoralongthesumof(cid:21)O1and(cid:21)O2. a2s.3F9*ADfo−rc2e0alcbtfi{nOgCo2n2albbfe|aOdCof1m2labsfskOm.Wisghiavteins {O O W 30o B
2.31IfamassslidesfrompointAtowardspoint theanglebetweentheforceandthez-axis? problem2.45:
BpoainlotsngAaanstdraBigahrtep(a0thina,n5dinth,e0cino)oradnidna(t1e0sionf, 2.40Given!*D2rad=s{OC*3rad=s|O; H*1 D 2.46 Fromthefigureshow(Finle,nafimned:pfithguerec2.ovemc1p.4o1-)
0(cid:21)OAinB, d1i0reinc)te,drefsropmectAivetolyB, fianlodngthteheunpiatthv.ector .62kO0/{OkCg3m02|O=/sk,gfimnd2(=as)athnedaHng2leDbe.t1w0e{OeCn1!*5|aOnCd pnoensittsioonfvveeccttoorr)r*aAloBn(gyouhavetofirstfindthis
* H*1and(b)theanglebetween!*andH*2. (a) they-axis,and
120.3N2kWO;rFi*t2ethDeve−c2to0rNsF|O1CD23N0NkO{;OCan4d0F*N3|O−D 2nO.4D1T0:h7e4{uOnCit0n:o6r7m|Oa.lItfothaesuwrefaigchetiosfgaivbelnocaks (b) alongz(cid:21)O.
−10N{O−100NkO asalistofnumbers(rows onthissurfaceactsinthe−|O direction, find
orcolumns). Findthesumoftheforcesusing theanglethata1000 Nnormalforcemakes 2m
acomputer. withthedirectionofweightoftheblock.
A
2.42Vectoralgebra.Foreachequationbelow B
2.2 The dot product of two statewhether: 2m 1m
3m
vectors (a) Theequationisnonsense. Ifso,why? 30o y
(b) Isalwaystrue.Why?Giveanexample. O
(cid:21)
(c) Isnevertrue. Why? Giveanexample.
2.33 ExpresstheunitvectorsnOand(cid:21)O interms x
of{Oand|Oshowninthefigure.Whatarethex (d) Issometimestrue.Giveexamplesboth problem2.46:
ways.
andycomponentsofr*D3:0ftnO−1:5ft(cid:21)O?(cid:3) (Filename:pfigure2.vec1.42)
Youmayusetrivialexamples. *
* * * * 2.47ThenetforceactingonaparticleisF D
y a) ACB DBCA 2N{OC10N|O. Find the components of this
(cid:21)O nO b) A**C*bDb*CA** fsoisrcveecintorasn{oO0thDer−cocoorsdi(cid:18)n{OatCe ssyinst(cid:18)e|mOawndith|O0bDa-
|O (cid:18) c) A(cid:1)B DB(cid:1)A −sin(cid:18){O−cos(cid:18)|O. For(cid:18) D 30o, sketchthe
*
d) B*=C*DB=C vectorF andshowitscomponentsinthetwo
coordinatesystems.
{O x e) b=A*Db=A
problem2.33: * * * * * * * * * * 2.48 FindtheunitvectorseORandeO(cid:18) interms
(Filename:efig1.2.27) f) AD.A(cid:1)B/BC.A(cid:1)C/CC.A(cid:1)D/D of{Oand|Owiththegeometryshowninfigure.
ProblemsforChapter2 5
* * * * *
WhatarethecomponetsofW alongeO and 2.54 Write a computer program (or use a a) B(cid:2)CDC(cid:2)B
R
eO(cid:18)? canned program) to find the dot product of b) B*(cid:2)C*DC*(cid:1)B*
two 3-D vectors. Test the program by com-
putingthedotproducts{O(cid:1){O;{O(cid:1)|O; and|O(cid:1)kO. c) C*(cid:1).A*(cid:2)B*/DB*(cid:1).C*(cid:2)A*/
Nowusetheprogramtofindthecomponents * * * * * * * * *
of F* D .2{O C2|O −3kO/N along the line d) A(cid:2).B(cid:2)C/D.A(cid:1)C/B−.A(cid:1)B/C
(cid:18) ‘ r* D.0:5{O−0:2|OC0:1kO/m. *
AB 2.58WhatisthemomentMproducedbya20
|O (cid:18)2n.55DL(cid:18)e0t−r*nnD1(cid:18)1.mU.csoinsg(cid:18)na{OcComsipnu(cid:18)tne|rOg/,ewnehreartee Narmforocfer*FDac.ti1n6gminmt/h|eO?xdirectionwithalever
therequiredvectorsandfindthesum
{O W 2.59 Findthemomentoftheforceshownon
X44 therodaboutpointO.
eO(cid:18) r*i; with1(cid:18) D1oand(cid:18)0D45o: y F=20N
eO nD0
R
problem2.48:
2.3 Cross product, moment,
(Filename:pfigure2.vec1.44)
2.49 Write the position vector of point P in and moment about an axis 45o
O O 2m
termsof(cid:21) and(cid:21) and
1 2
(a) findthey-componentofr*P, 2.56 Findthecrossproductofthetwovectors O x
(b) findthecomponentofr* alon(cid:21)O . showninthefiguresbelowfromtheinforma-
P 1 tiongiveninthefigures. problem2.59:
y P (a) y (b) y (Filename:pfigure2.vec2.2)
(cid:18)‘2 (cid:21)O a* 60o *b a* 105o *b 2*.60 Fi*nd the sum of moments of forces
2 2 2 4 3 4 W and T about the origin, given that W D
x x 100N;T D120N;‘D4m; and(cid:18) D30o.
y
‘
O
1 (cid:21)1 (c) y (d) y
T
(cid:18)1 x 430o a* 455o 4 a* ‘ /2 ‘ /2
problem2.49: 30o x 45o x W
4 * * (cid:18)
(Filename:pfigure2.vec1.45) b b
O x
2.50 Whatisthedistancebetweenthepoint
problem2.60:
A and the diagonal BC of the parallelepiped (e) y (f) y
shown? (Usevectormethods.) (Filename:pfigure2.vec2.3)
* *
b b
B 1 3 3 2.61 Findthemomentoftheforce
3 3
A a) aboutpointA
C 2 x 2 x
3 2 a* 2 a* b) aboutpointO.
4 F=50N
problem2.50:
(Filename:pfigure.blue.2.3) (g) y *bD4|O (h) y (cid:11)=30o
2.51LetF* D30N{OC40N|O−10NkO;F* D (-1,2) *b (2,2)
1 2
−20N|OC2NkO; andF*3 D F3x{OCF3y|O− a*
F3zkO.Ifthesumofalltheseforcesmustequal a*D x x 1.5m
zero,findtherequiredsc*alarequationstosolve 3{OC|O (-1,-1) (2,-1)
forthecomponentsofF .
3 problem2.56: O
A
2.52Avectorequationforthesumofforces (Filename:pfigure2.vec2.1) 2m
resultsintothefollowingequation: problem2.61:
2.57Vectoralgebra.Foreachequationbelow
F.{O−p3|O/C R.3{OC6|O/D25N(cid:21)O statewhether: (Filename:pfigure2.vec2.4)
2 5 (a) Theequationisnonsense. Ifso,why? 2.62 Inthefigureshown,OA=AB=2m.The
where(cid:21)O D 0:30{O−0:954|O. Findthescalar (b) Isalwaystrue.Why?Giveanexample. forceF D40Nactsperpendiculartothearm
*
equationsparallelandperpendicularto(cid:21)O. (c) Isnevertrue. Why? Giveanexample. AB.Findthem*omentofF aboutO,giventhat
* * * * (cid:18) D45o. IfF alwaysactsnormaltothearm
2*.53*Let (cid:11)F1*C (cid:12)F2 C γF3 D 0, where (d) Issometimestrue.Giveexamplesboth AB,wouldincreasing(cid:18)increasethemagnitude
F ;F ; andF areasgiveninProblem2.32. ways. ofthemoment? Inparticular,whatvalueof(cid:18)
1 2 3
Solvefor(cid:11);(cid:12); andγ usingacomputer. Youmayusetrivialexamples. willgivethelargestmoment?
6 CONTENTS
y
F 2.66 Find the sum of moments due to the %program definition
twoweightsoftheteeter-totterwhentheteeter- z(1)=a(1)*b(1);
totteristippedatanangle(cid:18) fromitsvertical z(2)=a(2)*b(2);
(cid:18)
B position.Giveyouranswerintermsofthevari- z(3)=a(3)*b(3);
A ‘ ablesshowninthefigure. w=z(1)+z(2)+z(3);
A
‘ (cid:11)
‘ (cid:11)
2.73 Findaunitvectornormaltothesurface
(cid:18)
h ABCDshowninthefigure.
O B z
x
problem2.62: O ‘
A
(Filename:pfigure2.vec2.5) D
W
2.63 Calculatethemomentofthe2kNpayload
B
ontherobotarmabout(i)jointA,and(ii)joint C 4" y
B,if‘1D0:8m;‘2D0:4m; and‘3D0:1m. 5"
y OA=h x 5" C
A ‘3 AB=AC=‘ W problem2.73:
(Filename:efig1.2.11)
30o
‘ problem2.66:
1 ‘
2 B C (Filename:pfigure2.vec2.9) 2.74 If the magnitude of a force N* normal
45o 2kN 2th.6e7moFminedntthoefpW*ercaebnotaugtetheerrpoirviontcpoominptuOtinags tworittheeN*surafsacaevAecBtoCr.D(cid:3)inthefigureis1000 N,
O x afunctionof(cid:18),iftheweightisassumedtoact z
problem2.63:
normaltothearmOA(agoodapproximation
(Filename:pfigure2.vec2.6) when(cid:18)isverysmall). B
A
2.64 Duringaslam-dunk,abasketballplayer ‘ A
pullsonthehoopwitha250lbfatpointCofthe
O (cid:18)
ringasshowninthefigure. Findthemoment 1m C 1m y
oftheforceabout W D
1m
x
a) thepointoftheringattachmenttothe problem2.67: 1m 1m
board(pointB),and problem2.74:
(Filename:pfigure2.vec2.10)
b) therootofthepole,pointO. (Filename:efig1.2.12)
2.68Whatdoyougetwhenyoucrossavector
board (cid:3)
andascalar?
2.75TheequationofasurfaceisgivenaszD
6"
(cid:3) 2x − y. Find a unit vector nO normal to the
A 2.69Whydidthechickencrosstheroad?
1.5' surface.
3' basketball hoop
B 2.70Carryoutthefollowingcrossproductsin
different ways and determine which method
2.76 Inthefigure,atriangularplateACB,at-
10' 15o takestheleastamountoftimeforyou.
tachedtorodAB,rotatesaboutthez-axis. At
250lbf a) r*D2:0ft{OC3:0ft|O−1:5ftkOI F*D theinstantshown,theplatemakesanangleof
−0:3lbf{O−1:0lbfkOI r*(cid:2)F*D? 60o withthe x-axis. Findanddrawavector
normaltothesurfaceACB.
O b) r* D .−{OC2:0|O C0:4kO/mI L* D z
problem2.64: .3:5|O−2:0kO/kgm=sI r*(cid:2)L*D? B
(Filename:pfigure2.vec2.7) c) !* D .{O−1:5|O/rad=sI r*D .10{O−
2.65 Duringweighttraining,anatheletepulls 2|OC3kO/inI !*(cid:2)r*D? 45o C
a weight of 500Nwith his arms pulling on a
hadlebarconnectedtoauniversalmachineby 45o
acable.Findthemomentoftheforceaboutthe 2.71AforceF*D20N|O−5NkOactsthrough A 1m
shoulderjointOintheconfigurationshown. apointAwithcoordinates(200 mm,300 mm, y
-100mm).WhatisthemomentM*.D r*(cid:2)F*/ x 60o
oftheforceabouttheorigin?
problem2.76:
2.72CrossProductprogramWriteaprogram (Filename:efig1.2.14)
thatwillcalculatecrossproducts.Theinputto
thefunctionshouldbethecomponentsofthe
twovectorsandtheoutputshouldbethecom- 2.77 Whatisthedistancedbetweentheorigin
ponentsofthecrossproduct.Asamodel,here andtheline ABshown? (Youmaywriteyour
* *
problem2.65: isafunctionfilethatcalculatesdotproductsin solutionintermsofAandBbeforedoingany
(cid:3)
(Filename:pfigure2.vec2.8) pseudocode. arithmetic).
ProblemsforChapter2 7
z * *
c) WritebothF1andF2astheproduct c) What are the coordinates of the point
(cid:3)
1 A oftheirmagnitudesandunitvectorsin ontheplaneclosesttopointD?
(cid:3)
theirdirections.
O z
k (cid:3)
* d) WhatistheangleAOB?
A 5
|O e) What is the component of F*1 in the 4
{O x-direction? (cid:3)
O d 1 f) Whatisr*DO(cid:2)F*1? (r*DO (cid:17) r*O=D (3, 2, 5) B
y isthepositionofOrelativetoD.)(cid:3) 1 (0, 7, 4)D
B* g) WhatisthemomentofF*2 aboutthe 2
1 axisDC?(Themomentofaforceabout 3 4
B anaxisparalleltotheunitvector(cid:21)O is 5 A C 7
x definedasM(cid:21)D(cid:21)O(cid:1).r*(cid:2)F*/wherer*is x (5, 2, 1) (3, 4, 1) y
problem2.77:
thepositionofthepointofapplication
(Filename:pfigure.blue.1.3) oftheforcerelativetosomepointon problem2.83:
the axis. The result does not depend (Filename:pfigure.s95q2)
2.78 What is the perpendicular distance be-
onwhichpointontheaxisisusedor
tween the point A and the line BC shown? *
(Thereareatleast3waystodothisusingvar- whichpointonthelineofactionofF
(cid:3)
iousvectorproducts,howmanywayscanyou isused.). 2.4 Equivalant force sys-
find?) h) Repeatthelastproblemusingeithera
y differentreferencepointontheaxisDC tems and couples
orthelineofactionOB.Doesthesolu-
(cid:3)
B A tionagree? [Hint: itshould.]
3
z
|O 5m A
C B 2.84 Findthenetforceontheparticleshown
{O 0 2 3 x inthefigure.
F 6N
problem2.78: 2
(Filename:pfigure.blue.2.2)
4
2.79Givenaforce,F*1D.−3{OC2|OC5kO/N O F1 3m y |O 3 P 10N
actingatapointPwhosepositionisgivenby 4m
mr*ePn=tOabDou.t4a{On−ax2i|sOtChro7ukOg/hmt,hewohraitgiisntOhewmioth- x D C {O
problem2.81:
direction(cid:21)O D p2 |OC p1 |O?
5 5 (Filename:p1sp92) 8N
2.80 Drawing vectors and computing with 2.82 A, B, and C are located by position problem2.84:
vxeycztocroso.rdTihneatpeosin.0t;O5;is1t2h/emo.riPgioni.ntPBoinhtaAsxhyazs vr*eCctoDrs.7r*;A8;D9/:.1;2;3/, r*B D .4;5;6/, and (Filename:pfigure2.3.rp1)
coordinates.4;5;12/m.
a) Usethevectordotproducttofindthe
a) MakeaneatsketchofthevectorsOA, angle BAC (Aisatthevertexofthis
OB,andAB. angle).
b) Find a unit vector in the direction of b) Usethevectorcrossproducttofindthe 2.85 Replace the forces acting on the parti-
O cleofmassmshowninthefigurebyasingle
OA,callit(cid:21)OA. angle BCA(C isatthevertexofthis equivalentforce.
* angle).
c) FindtheforceF whichis5Ninsize 2T
andisinthedirectionofOA. c) Findaunitvectorperpendiculartothe
planeABC. T
d) WhatistheanglebetweenOAandOB?
e) Whatisr*BO(cid:2)F*? d) HAoBwfrfoamritshethoerigininfi?ni(tTehlainteis,dheofiwnecdlobsye T 45o m 30o
f) WhatisthemomentofF*aboutaline is the closest point on this line to the
origin?) |O
paralleltothezaxisthatgoesthrough
thepointB? e) Istheoriginco-planarwiththepoints mg
A,B,andC? {O
2.81 Vector Calculations and Geometry. problem2.85:
*
The 5N fo*rce F1 is along the line OA. The 2.83 PointsA,B,andCinthefiguredefinea (Filename:pfigure2.3.rp2)
7NforceF2isalongthelineOB. plane.
a) FindaunitvectorinthedirectionOB. a) Findaunitnormalvectortotheplane.
(cid:3) (cid:3)
b) FindaunitvectorinthedirectionOA. b) Find the distance from this infinite 2.86 Findthenetforceonthepulleyduetothe
(cid:3) (cid:3)
planetothepointD. belttensionsshowninthefigure.