# Solucionário Mecanica vetorial para engenheiros - 9 ed - Capitulo 3

(Parte 1 de 3)

CHAPTER 3 CHAPTER 3

PROBLEM 3.1 A foot valve for a pneumatic system is hinged at B. Knowing that a - 28°, determine the moment of the 1 6-N force aboutPoint B by resolving the force into horizontal and verticalcomponents.

>ny c

Noting that the direction ofthe moment ofeach force component about B is counterclockwise,MB =xFy +yFx

PROPRIETARY MATERIAL. © 2010 The McGraw-Hill Companies, Inc. All rights reserved. No pail of this Manual may be displayed,reproduced or distributed in anyform or by any means, without the prior written permission ofthe publisher, or used beyond the limiteddistribution to teachers andeducatorspermittedby McGraw-Hillfor theirindividualcoursepreparation. Ifyouarea student using this Manual,you are using it withoutpermission. 153

PROBLEM 3.2

A foot valve for a pneumatic system is hinged at B. Knowing that a = 28°, determine the moment of the 16-N force aboutPoint B by resolving the force into components along ABC and in a direction perpendicular to ABC.

First resolve the 4-lb force into components P and Q, whereg = (16 N)sin 28°

PROPRIETARY MATERIAL. © 2010 The McGraw-Hill Companies* Inc. All rights reserved. No part of (his Manual may be displayed,reproduced or distributed in anyform or by any means, without the prior written permission ofthe publisher, or used beyond the limiteddistribution to teachers andeducatorspermittedby McGraw-Hillfortheirindividualcoursepreparation. Ifyou areastudentusingthis Manual,you are using it withoutpermission. 154

-100 m- 200 m

A 300-N force is applied at A as shown. Determine (a) the momentofthe 300-N force about D, (b) the smallest force applied at B thatcreates the same moment about D.

(a) O.2.

PC 0>T-**>

Fv =(300N)cos25°

(b) The smallest force Q at B must be perpendicular toDB at 45°£LMD =Q(DB) 41 .700 N m = (2(0.28284 m) Q = 147.4 N L 45° <

PROPRIETARY MATERIAL. © 2010 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed,reproduced or distributed in anyform or by any means, without the prior written permission ofthe publisher, or used beyond the limiteddistribution to teachers andeducatorspermittedbyMcGraw-Hillfortheir individualcoursepreparation. Ifyouareastudent usingthis Manual,you are using it withoutpermission. 155

•-lOOnim-* -200 m *.

125 il

PROBLEM3.4

A 300-N force is applied at A as shown. Determine (a) themoment of the 300-N force about D, (b) the magnitude andsense of the horizontal force applied at C that creates thesame moment about D, (c) the smallest force applied at C that creates the same moment aboutD.

(a) See Problem 3.3 for the figure and analysis leading to the determination ofMd M =41.7N-mHCl^n

0>\7.Siy\ c = at. (b) Since C is horizontal C = Ci r = DC = (0.2 m)i -(0. 125 m)jMD =rxCi = C(0.l25m)k4l.7N-m = (0.l25m)(C)C = 3.60 N (c) The smallest force C must be perpendicular to DC; thus, it forms a with the vertical

PROPRIETARY MATERIAL. €5 20)0 The McGraw-Hill Companies, Inc. AH rights reserved. No part of this Manual may be displayed,reproduced or distributed in anyform or by any means, without the prior written permission ofthe publisher, or used beyond the limited distribution to teachers andeducatorspermittedby McGraw-Hillfor their individual coursepreparation. Ifyou are a student using this Manual,you are using it withoutpermission. 156

PROBLEM 3.5

An 8-1b force P is applied to a shift lever. Determine the moment of V about Bwhen a is equal to. 259 .

SOLUTION First note Px = (8 lb)cos 25°

Noting that the direction ofthe moment ofeach force component about B isclockwise, have

= -(8in.)(3.3809 1b) - (2 in.)(7.2505 lb) = -186.6 lb -in.

i*2. •**.

or Mj =186.6 lb -in.J) ^

PROPRIETARY MATERIAL. © 2010 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed,reproduced or distributed in anyform or by any means, without the prior written permission ofthe publisher, or used beyond the limiteddistribution to teachersandeducatorspermittedby McGraw-Hillfortheir individualcoursepreparation. Ifyouare astudentusing this Manual,you are using it withoutpermission. 157

PROBLEM 3.6 For the shift lever shown, determine the magnitude and the direction of the smallestforce P that has a 21 0-lb • in. clockwise moment about B.

2 in.

SOLUTION For P to be minimum it must be perpendicular to the linejoining Points A. and B. Thus,

2 in. /= 19.98° iand MB =dPn p Z i«.

PROPRIETARY MATERIAL. © 2010 The McGraw-Hill Companies, Inc. All rights reserved. No part ofthis Manual may be displayed,reproduced or distributed in anyform or by any means, without the prior written permission ofthe publisher, or used beyond the limited distribution to teachers andeducatorspermittedby McGraw-Hillfor their individual coursepreparation. Ifyou are a student using this Manual,you are using it withoutpermission. 158

PROBLEM 3.7 An 1 1-lb force P is applied to a shift lever. The moment of P about B is clockwiseand has a magnitude of250 lb • in. Determine the value ofa.

2 in.

By definition where and also

Then

or and

MB =rmPsm.6 = a + (9Q°-tf>)

_i 8 in.<p - tan" 19.9831'2 in.

PROPRIETARY MATERIAL. © 2010 The McGraw-Hill Companies, Inc. All rights reserved. No part ofthis Manual may be displayed,reproduced or distributed in anyform or by any means, without the prior written permission ofthe publisher, or used beyond the limiteddistribution to teachers andeducatorspermittedby McGraw-Hillfortheir individualcoursepreparation. Ifyouareastudentusingthis Manual,you are using it withoutpermission. 159

PROBLEM 3.8

It is known that a vertical force of 200 lb is required to remove the nailat C from the board. As the nail first starts moving, determine (a) themoment about B of the force exerted on the nail, (b) the magnitude of the force P that creates the same moment about B if a ~ .1 0°, (c) the smallest force P that creates the same moment about B.

V—Ar 4in. n SOLUTION

(a) We have MB =raBFN(4 in.)(200 lb)

800 lb -in. or MB =H00\b-m.)<

(c) For P to be minimum, it must be perpendicular to the linejoiningPoints A and B. Thus, P must be directed as shown. Thus

or or

A*W™. cl = fAIB

PROPRIETARY MATERIAL. © 2010 The McGraw-Hill Companies, Inc. All rights reserved. No part ofthis Manual may be displayedreproduced or distributed in anyform or by any means, without the prior written permission ofthe publisher, or used beyond the limited distribution to teachers andeducatorspermittedby McGraw-Hillfor their individual coursepreparation. Ifyou are a student using this Manual,you are using it withoutpermission. 160

PROBLEM 3.9 A winch puller AB is used to straighten a fence post. Knowing that the tension in cable BC is 1040 N andlength d is 1 ,90 m, determine the moment about D of the force exerted by the cable at C by resolving thatforce into horizontal and vertical components applied (a) at Point C, (b) at Point E.

(a) Slope oflineThen

Then

(b) We have or M./J =760N-m>)^

PROPRIETARY MATERIAL. © 2010 The McGraw-Hill Companies, Inc. AM rights reserved. No part ofthis Manual may be displayed,reproduced or. distributed in anyform or by any means, without the prior written permission ofthe publisher, or used beyond the limiteddistribution to teachers andeducatorspermittedby McGraw-Hillfortheir individualcoursepreparation. Ifyou are astudentusing this Manual,you are using it withoutpermission. 161

PROBLEM 3.10 It is known that a force with a moment of 960 N • m about D is required to straighten the fence post CD. Ifd- 2,80 m, determine the tension that must be developed in the cable of winch puller AB to create therequired moment about Point D.

0.2 j.i» SOLUTION o. a?5^i

z-acw OiZ^

Slope of line

Then and We have

24My

T/)B rAH

PROPRIETARY MATERIAL; © 2010 The McGraw-Hill Companies, Inc. All. rights reserved. No part of this Manual may be displayed,reproduced or distributed in anyform or by any means, without the prior written permission ofthe publisher, or used beyond the limited distribution to teachers andeducatorspermittedby McGraw-Hillfor their individual coursepreparation. Ifyou are a student using this Manual,you are using it withoutpermission. 162

PROBLEM 3.1 It is known that a force with a moment of960 N • m about D is required to straighten the fence post CD. Ifthecapacity of winch puller AB is 2400 N, determine the minimum value of distance d to create the specifiedmoment about Point D.

0.2 in

0.875 in o.mtn czom The minimum value ofd can be found based on the equation relating the moment ofthe force TAB about D:MD ={TABmK),(d)

whereNow

 0.875msin #

or 3.7852</2 - 0.40c/ - .8056 = Using the quadratic equation, the minimum values ofdare 0.51719 m and -.41151 m.

PROPRIETARY MATERIAL © 2010 The McGraw-Hill Companies, Inc. All rights reserved. JVo port o/rt/s Manual may be displayed,reproduced or distributed in anyform or by any means, without (he prior written permission ofthe publisher, or used beyond the limiteddistribution to teachersandeducatorspermittedby McGraw-Hill/or theirindividualcoursepreparation. Ifyouareastudentusingthis Manual,you are using it withoutpermission. 163

5.3in. PROBLEM 3.12 12.0 in.

2.33in.mmM 1 >i. i

The tailgate of a car is supported by the hydraulic lift BC. If the lift exerts a 125-lb force directed along its centerline on the balland socket at B, determine the moment ofthe force about A.

SOLUTION First note

Then and

Now where

Then

12.2241 in.

sin 9- 2.3 in. 12.2241 in.

*cb = FCB cos 9\ -FCB \$m9l 1251b

12.2241 in. [(12.0 in.)i- (2.3 in.) j]

M = (15.3 in.)i--(14.3 in.) j ru-nimn* 1251b

\«5»,a> im.

\Z.& >N. 2.3 ttvi.

(116.156 lb -ft)k or M„ = 116.2 lb- ft *)<

PROPRIETARY MATERIAL. © 2010 The McGraw-Hill Companies, Inc. All rights reserved. No part ofthis Manual may be displayed, reproduced or distributed in anyform or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers andeducatorspermittedby McGraw-Hillfor their individualcoursepreparation. Ifyou are a student using this Manual, you are using it without permission.

t§| 1.7.2 hi.

PROBLEM 3.13 The tailgate of a car is supported by the hydraulic lift EC. If the liftexerts a 125-lb force directed along its centerline on the ball andsocket at B, determine the moment of the force about A.

First noteThen cos - 17.2 in.

18.8123 in.

18.8123 in.

Z.O.'S «w.

and

Now where

Then r = (20.5 in.)i- (4.38 in.)j n.z .«»».

4.?.e. >Ni. XKvZ >KJ.

(1538.53 lb -in.)k (128.2 lb -ft)k or M =128.2 lb- ft ^H

IROlRlhTARl MAlhRlAL. © 2010 The McGraw-Hill Companies, Inc. All rights reserved. M> ,«,#/ «///,& MW ,»«> 6* rfwpfereproduced or distributed in anyform or by any means, without the prior written permission ofthe publisher, or used bevond the. limitedd.stnbuUoiMo teachers andeducatorspermittedby McGraw-Hillfortheir individualcourse,preparation. you are using it withoutpermission. 165

120mm

PROBLEM 3.14

A mechanic uses a piece ofpipe AB as a lever when tightening an alternator belt. When he pushes down at A, a force of 485 N is exerted on the alternator at B. Determine the moment of that force about bolt C ifits line of action passes through O.

We have Mc =rwc xFj,

Noting the direction ofthe moment ofeach force component about C is clockwise.

Where and

Mc =xFBy +yFBx

4 j,

PROPRIETARY MATERIAL. © 2010 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers andeducators permittedby McGraw-Hillfor their individual coursepreparation. Ifyou are a student using this Manual, you are using it without permission.

By definition: Now and

PROBLEM 3.15

Form the vector products B x C and B' * C, where B =B\ and use the resultsobtained to prove the identity sinacos/? =-sin(a+ P) +-sin (a - p).

SOLUTION N(>te: B = £(cos/?i + sin/?j)

B'= 5(cos/?i-sin/?j)C - C(cosai +sin aj)|BxC| = flCsin(a-jff)|B'xC| = 5Csin(flf+ y?) B xC = Z?(cos /?i + sin y9j)x C(cos tfi + sin orj)

= BC(cos /?sin « - sin /?cos «)kB'xC - /?(cos /?i -sin /?j)x C(cos ai -f sin #j)- £C(cos yffsin ar+sin /?cos ar)k

Equating the magnitudes of BxC from Equations (I ) and (3) yields:BCs'm(a-p)~ BC(cos ps'm a-sin pcos a)

Similarly, equating the magnitudes of B'xC from Equations (2) and (4) yields;BCsm(a+ p) = BC(cos ps'm a+ sln pcos a) Adding Equations (5) and (6) gives:

sin(a-p) + s'm(a+p) - 2cos /?sin a

PROPRIETARY MATERIAL. © 2010 The McGraw-Hill Companies, Inc. All rights reserved. AV, /«», „/-,/,& AAW ,«, he displayedreproduced or distributed m anyform or by any means, without the prior written permission ofthe publisher, or used beyond the limiteddistribution to teachers andeducatorspermittedbyMcGraw-Hillfortheir individualcoursepreparation. Ifyouareastudentusing this Manualyou are using it withoutpermission. 167

PROBLEM 3.16

A line passes through the Points (20 m, 16 m) and (-1 m, ~4 m). Determine the perpendicular distance rffrom the tine to the origin O of the system ofcoordinates.

Assume that a force F, or magnitude F(N), acts at Point A and isdirected ixomA to B. Then,,

Where

By definition Where

Then

¥~FXm

MQ = \rA xF\ = dFr,=-(lm)i-(4m)j F

E> (Zom, K-rti^

PROPRIETARY MATERIAL. © 2010 The McGraw-Hill Companies, Inc. AH rights reserved. No part ofthis Manna! may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers andeducatorspermittedby McGraw-Hillfor their individual coursepreparation. Ifyou are a student using this Manual, you are using it without permission.

PROBLEM 3.17 The vectors P and Q are two adjacent sides ofa parallelogram. Determine the area ofthe parallelogram when(tf)P--7i + 3j-3kandQ = 2i + 2j + 5k,(6)P = 6i-5j-2kandQ = -2i + 5j~k.

(a) We have where

Then

(b) We have where v4 = |PxQf

Then PxQ i fci 6 -5

PROPRIETARY MATERIAL. © 2010 The McGraw-Hill Companies, Inc. All rights reserved. No part ofthis Manual may be displayed,reproduced or distributed in anyform or by any means, without the prior writ/en permission ofthe publisher, or used beyond the limiteddistribution to teachers andeducatorspermittedbyMcGraw-Hillfortheir individualcoursepreparation. Ifyouareastudent usingthis Manual,you are using it withoutpermission, 169

PROBLEM 3.18

A plane contains the vectors A and B. Determine the unit vector normal to the plane when A and B are equal to, respectively, (a) i + 2j - 5k and 4i - 7j - 5k, (b) 3i - 3j + 2k and -2i + 6j - 4k.

{a) We have where

Then

and

(b) We have where

Then

and

AxB

AxB

|AxB| :li + 2j-5k

|AxB|

AxB

15V(-3)2 +H)2 +(-l)2 -I5>/ri

15(-3f-lJ-lk) or X15VHAxB (-3i-j~k) 4

|AxB| 3i-3j + 2k :-2i + 6j-4k

(12-12)i + (-4+ l2)j + (18-6)k (8j + 12k)

4(2j +3k)413 or X, V (2j + 3k) <

PROPRIETARY MATERIAL. © 2010 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed reproduced or distributed in anyform or by any means, without the prior written permission ofthe publisher, or used beyond the limited distribution to teachers andeducatorspermittedby McGraw-Hillfor their individual coursepreparation. Ifyou are a student using this Manual, you are using it withoutpermission.

(Parte 1 de 3)