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Addison Wesley Longman 1996Addison Wesley Longman Limited Edinburgh Gate Harlow Essex CM202JE England and Associated Companies throughout the world, The right of Marc Bedford and Wallace Fowler to be identified lis authors of this Work has been asserted by them in accordance with the Copyright, Designs and Patents Act 1988, I All rights reserved. No part of this publication may be reproduced, s 'ored in a retrieval system; or transr;nitted in anY form or by any means, electronic, mecbanical, photocopying, recording 'r otherwise, without prior writtep permission of the publisher or a licence permitting restricted copying in the United !~gdom issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1P 9HE. I Photo Credits I Cover: Medford Taylor/Superstock Chapter 2: 2.11, The Harold E. Edgerton 1992 Trust, courtesty Pa m, Press, Inc.; 2.44, courtesy of Intelsa!. I Chapter 4: P4., US. Geological Survey. Chapter 5: 5.4, The Harold Edgerton 1992 Trust, courtesy Palm p: ss, Inc. Chapter 6: 6.44 (a & b), NASA. Chapter 10: 10.20, US Geological Survey. Cover designed by Designers & Partners, Oxford Typset by Techs.t Composition Limited, Salisbury Printed and bound in the United States of America First printed 1996 ISBN 0-20\-40341-2 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Lib Library of Congress Cataloging-in-Publication Data is available

ABOUT THE AUTHORS

III

About the Authors

Anthony Bedford is Professor of Aerospace Engineering and Engineering Mechanics at the University of Texas at Austin. He received his BS degree at the University of Texas at Austin, his MS degree at the California tnstitute of Technology, and his PhD degree at Rice University in 1967. He has industrial experience at the Douglas Aircraft Company and TRW Systems, and has been on the faculty of the University of Texas at Austin since 1968. Dr Bedford's main professional activity has been education and research in engineering mechanics. He is author or co-author of many technical papers on the mechanics of composite materials and mixtures and of two books, Hamilton & Principle in Continuum Mechanics and Introduction to Elastic Wave Propagation. He has developed undergraduate and graduate courses in engineering mechanics and is the recipient of the General Dynamics Teaching Excellence Award. He is a licensed professional engineer and a member of the Acoustical Society of America, the American Society for Engineering Education, the American Academy of Mechanics and the Society of Natural Philosophy. Wallace Fowler is Paul D. and Betty Robertson Meek Professor of Engineering in the Department of Aerospace Engineering and Engineering Mechanics at the University of Texas at Austin. Dr Fowler received his BS, MS and PhD degrees at the University of Texas at Austin, and has been on the faculty since 1966. During 1976 he was on the staff of the United States Air Force Test Pilot School, Edwards Air Force Base, California, and in 1981 1982 he was a visiting professor at the United States Air Force Academy. Since J 991 he has been Director of the Texas Space Grant Consortium. Dr Fowler's areas of teaching and research are dynamics, orbital mechanics and spacecraft mission design. He is author or co-author of many technical papers on trajectory optimization and attitude dynamics, and has also published many papers on the theory and practice of engineering teaching. He has received numerous teaching awards, including the Chancellor's Council Outstanding Tea"hing Award, the General Dynamics Teaching Excellence Award, the Halliburton Education Foundation Award of Excellence and the AlAAASEE Distinguished Aerospace Educator Award. He is a licensed professional engineer, a member of many technical societies, and a fellow of the American Institute of Aeronautics and Astronautics and the American Society for Engineering Education.

iv

PREFACE

Preface

Figure 7.16

For 25 years we ha~'1 taught the two-semester iutroductory course in e engineering mechanic I. Duriug that time, students often told us that they could understand Our c' assroom presentation, but had difficulty understanding the textbook. This com ent led us to examiue what the in:;tructor does in class that differs from the traditional textbook presentatioo, and eventually resulted in this book. Our aPl\roach is to present matetial the way we do in the classroom, using mordl sequences of figur.es and stressing the importance of careful visual analysis I and conceptual understanding. Throughout the book, we keep the student f9remorst iu mind as our audience.I

Goals and TheryesProblem Solving 'fe emphasize the critical importance of good problemsolving skills. In our.llworked examples. we teach students to think about problems before they ~egin their solution. What principles apply? What must be determined, and in 'iha! order? Separate Strategy sections that precede most of the examples ilIusIDjte this preliminary aoalysis. Then we give a careful and complete description (,f the solution, often showing alternative methods. Finally, many examPl s conclude with Di,'cussion sections that point out properties of the solut on, or comment on and compare alternative solution methods, or point out ,ays to check the answers. (See, for instance, Example 3.2, pp. 106-7.) Our 01iective is to teach students how to approach problems and critically judge Ih results. Tn addition, for thos. e stud.ents who tell us that they understand the aterial in class, but don't know how to begin their homework problems te also provide brief Strategy sections in selected homework problems. ,I

I

, t

N

(a) Free-body diagram of the wheel.

(b) Expressing the acceleration of the centreof mass G in tenus of the acceleration of the centre A.

Visualization One Jf the essential elements in successful problem solving is visualization, cspec' y lbe use of free-body diagrams. In the classroom, the instructor can draw a .agram one step at a time, describing each step aod developing the solutio in parallel with the diagram. We have done the same thing iu this book, ing the same sequence of diagrams we use in class and carefully indicating lationships between them. In Example 8.2, pp. 3789, instead of simply sh , . g the free-bOdy diagram.,. we repeat the initial figure with the isolated part h ,ghlighted and everything else showo as a pale ghosted image. In this way we ow the student exactly how to isolate the part that will become the free-body lldiagram. In Example 9.8, p. 456, we use a ghosted image to indicate the motion of a rigid body about an a.xis. This he ips students visualize the true moti~n of the obiect. We use colour to ,elp students distinguish and understand the various using the same colours for particular clements eonelements in figures. sistently - such as blU~~ for force vectors and green for accelerations - we have tried to make the ook easier for students to read and understand. (See, for example, Figure 7! 16 on the left.) In addition, the greater realism of

Bt

PREFACE

V

colom illustrations helps motivate students. (See Figure 3.7. p. 117; Figure 5.13, p. 202; and problem illustrations throughout the book.)

Emphasizing Basic Principles Our primary goal for this book is to teach students fundamental concepts and methods. Instead of presenting dynamics as a sequence of independent methods, we emphasize its coherence by showing how energy and momentum techniques can be derived from Newton's second law. We apply the same approach to a system of particles to obtain the equations describing the dynamics of rigid bodies. Tn describing motions of rigid bodies, we consistently uSe the angnlar velocity vector and the vector equations describing the relative motions of points. Traditionally, dynamics texts have waited until they discuss rigid bodies to show that the sum of the external forces acting on an object is equal to the product of its mass and the acceleration of its centre of mass. We introduce this simple result as soon as we have discussed Newton'S second law, in Chapter 3, because we find our students gain confidence in their solution. They don't need to be concerned about whether a given object can be modelled as a particle; they know they're determining the motion of its centre of mass. To help students identifY important results, key equations are highlighted (see, for example, p. 18), and the concepts discussed in each chapter are reinforced in a chapter-end summary. Thinking Like Engineers Engineering is an exciting discipline, requiring creativity and imagination as well as knowledge and systematic tllinking. In this book we try to show the place of engineering mechanics within the larger context of engineering practice. Engineers in industry and the Accrediting Board for Engineering and 1echnology (ABET) are encouraging instructors to introduce design early in the engineering curriculum. We include simple design and safety ideas in many of om examples and problems without compromising emphasis on fundamental mechanics. Many problems arc expressed in terms of design and safety considerations (for example, Problems 3.101 and 3.102, p. 136); in some cases, students are asked to choose a design parameter from a range of possible values based on stated criteria (for example, Problems 4.118, p. 180; and 4.125, p. 181). Our students have responded very positively to tl,ese motivational elements and have developed an awareness of how these essential ideas are applied in engineering.

Pedagogical FeaturesBased on our own teaching experiences and advice from many colleagues, we have included several features to help students learn and to broaden their perspective on engineering mechanics.

Problem-Solving strategies Worked examples and homework problemsare the heart of a course in engineering mechanics. Throughout the book, we

vi PREFACE I provide descriptions that we use in the examples and which students WIll find helptul m worldng problems. We do not proVide recipes that students are intended follow rigidly, Instead, we describe general lines of thought that apply to IProad classes of problems and give useful advice and helpful warnings of dommon pitfalls, the kind of information we give to students dUling office ~ours, (See, for example, pp. 33, 242, 262 and 311.)

o~ th~ appro~ches

10

engineering practic~ehmging from familiar household items to advanced engineering applicatio I s. In addition, examples labelled. as 'Applications to Engineering' provide lore detailed case studies from different engineering disciplines. These e I pies show how the principles learned in the text are directly applicable to durrent and future engineering, problems. Our goal is to help students see the Vnportance of engineering mechanics in these applications and so gain the 1l10tivation to learn it. (See, for example, pp. 79, 118 and 218.)

Applications Manl of our examples and problems are derived from actual

I

Computer Problems Surveys tell us that most instructors make some use of computers in enginJering mechanics courses, but there is no consensus on how it should be donl. We give the instructor the opportunity to introduce students to computer rpplications in dynamics, including the use of finite differences to integrat equations of motion, without imposing a particular approach. Optional se9:tions called 'Computational Mechanics' contain examples and problems suitable for the use of a progranunable calculator or computer. (See, for e~ample, pp. 128 and 174.) The instructor can choose how students solve bse problems, for example by using a progranuning language, a spreadsheel or a higher-level problem-solving environment. These l sections are independe t and self-contained.

r

Chapter Openings ' We begin each chapter with an illustration showing an application of the idea, in the chapter, often choosing objects that are familiar to students. By seeing pow the concepts in this course relate to the design and function of familiar objects around them, students can begin to appreciate the importance and exciteIPent of engineering as a career. (Sec pp. 98, 230 and 302.) ,

Commitment to Students and InstructorsWe have taken precautt'bns that ensure the accuracy of thiS. book to the b.est of our ability. Reviewers I xamined each stage of the manuscript for errors. We have each solved the lroblems in an effort to be sure that their answers are correct and that the pr~blems are of an appropriate level of difficu!J;y:'Eugene Davis, author of the $olutions Manual, further verified the 'lfiSwers while developing his soluti03s. As a further check, James Whitentoi examined the entire text for errors thl.t crept in during the typesetting procss. Any errors that r"nfin are the responsibility of the auth~rs. We welcome conununication from sl',udents and instructors concerning errprs or areas for improvement. Our mai ing address is Department of Aerospa~e Engineering and Engineering Mec~anics, University of Texas at Austin, 'Austin, Texas 78712, USA. Our electonic mail address is bedford@aw.com. 'II

II'il

PREFACE vii

Printed SupplementsInstructor's Solutions Manual The manual for the instructor contains step-by-step solutions to all problems. Each solution includes the problem statement and the associated art. study Guide This guide reinforces the Strategy-Solution-Discussion process outlined in the text. Selected solutions are provided in great detail, accompanied hy suggested strategies for approaching problems of that type. Transparencies Approximately 100 figures from the text have been prepared in four colours on acetate for use on an overhead projector.

Software SupplementsStudent Edition of Working ModefllJ Working Model (KnowledgeRevolution, inc.) is a simulation and modelling program that allows the student to visualize engineering problems. The program calculates the effects of forces on an object (or objects), animates the results, and provides output data such as force, moment, velocity and acceleration in digital or graphical form. The Student Edition make this powerful program affordable for undergraduate students. It is available in both Windows and Macintosh versions.

Working Model Simulations Approximately 100 problems and examples from the text have been re-created on disk as Working Model simulations. These simulations have been constructed to allow the student to change variables and see the results. The student can explore physical situations in a 'what if' manner and thereby develop deeper conceptual insights than possible through quantitative problem solving alone. Students cao purchase these simulations combined with the text for a nominal additional charge.

PREFACE ix

AcknowledgementsWe are grateful to our teachers, colleagues and students for what we have learned about mechanics and about teaching mechanics. Many colleagues reviewed the manuscript and generously shared their knowledge and experi. ence to improve our book. They areNick Altiem Michigan State University James G. AndrewsUmversity of fowa

Mark Frisina r.fentworth Institute

V. J Lopardo US Naval Academy

Robert W. .l = - 180 mm. (a) Detennine the equation for s as a function of time. (b) What is the velocity of the point?

(b) What are the velocity and acceleration at 1 = O? (c) What are the velocity and acceleration at 1 = 4 s? 2.5 A rocket stans from rest and travels straight up. Its height

above the ground is measured by radar from 't = 0 to t = 4 s and is found to be approximated by the function" '" 1 m.

Or

(a) What is the displacement during this interval of time'? (b) What is the velocity at I = 4 s? (c) What is the acceleration during the first 4 s?

i; ,i

P2.2 P2.52.3 The graph of the velocity 11 of a point as a function of time is a straight line. When 1=2s, v=4m!s, and when 1=4s, v = -IOm!s. eaJ Detennine the acceleration of the point by calculating the slopeof the straight line.

The position of a point dUting the interval oftime from t = 0 to t = 65 is s = (_~t3 + 612 +41)m. (a) What is the displacement of the point during this interval of 2.6

(b) Obtain the equation for v as a function of time and use it to detcnnine the acceleration of the point.

time?(b) What is the maximum velocity during this interval oftitne, and at what time does it occur? (c) What is the acceleration when the velocity is a maximum?

28

CHAPTER 2 MOTION OF A POINT

2.7 The position of a point during the interval oftimc from 1 ~ 0 to 1 = 3 s is s = C12 + 5r - ,') m. (a) What is the maximum velocity during this interval of time, and at what time does it occur?

(b) What

i~

the acceleration when the velocity is a maximum?

2.8 A seismograph measures the horizontal motion of the growld during an earthquake. An engineer analysing the data detennines that for a lOs interval of time beginning at t = 0, the position is approximated by s = 100 cos(2nl) mm. What arc the Ca) maximum velocity and (b) maxhmun acceleration of the ground during the lOs interval?

2.11 uppose you wan. t to approximate the position of a vehicle you are testing by the power series s = A Sf CI' Di', where A, B, C and D arc constants. The vehicle starts from rest at t = 0 and ,/,0. At 1 = = 54m and at t = 8s" = 136m. (a) Dctknnine A. B. C and D. (b) WJJ~t are the approximate velocity and acceleration of the vehiclellat t= 8 s?

ii'

+ +

+

4.,,,

2.12 fhe acceleration of a point is a=20tm/s2 When t = 0, s=40rl, and v=-lOm/s. What are the position and velocity att

= 3 s~i"

2.9 During an assembly operation, a robot's arm moves along a straight hne. During an interval of time from t :::::::: 0 to t = 1 S, its position is given by s = (75r-SOr') mm. Detennine, during this I s intervaL (al the displacement of the ann; (b) the maximum andminimwn values of the velocity; (c) the maximum and minimum values of the acceleration.

t

2.13 = 0,

he acceleration of a point is a = (60, 361') m/s'. When 'il= 0 and v =20 m/s. What are the position and velocity as

functio*s of time?

2.14 uppose that during the preliminary design of a car, you assume !its maximum a.cceleration is' approximately constant. What constant acceleration is necessary if you want the car to be able to acceler~te from rest to a velocity of 88 km/hr in lOs? 'What distanc~ would the car travel during that time?II

' t

2.15

~n entomologist estimates that a flea l,mm in length attains

jumpin~.

a velocity of 1.3 m/s in a distance of one body length when What constant acceleration is necessary to achieve that veloci~r!

P2.92.10 In a test of a prototype car, the driver starts the car from rest at t = 0, accelerates, and then applies the brakes. Engineers measuring the position of the ear find that from t = 0 to t = 18 s it is approximated by s=(1.5t'+0.1i' O.006t4 )m. (a) What is the maximum velocity, and at what time does it occur? (b) What is the maximum acceleration, and at what time does itoccur?

P2.15

P2.10

2.2

STRAIGHT LlNE MOTION

29

2.16 Missiles designed for defence against ballistic missiles achieve accelerations in excess of 100 g's or one hundred times the acceleration of gravity. If a missile has a constant acceleration of 100 g's, how long does it take to go from rest to 96 km/hr'l What is its displacement during that time?

2.19 Tn 1960 R. C. Owens of the Baltiinore Colts blocked a Washington Redskin, field goal attempt by jumping and knocking the ball away in front of the cross bar at a point 3.35 m above the field. If he was 1.90m tall and could reach a.36m above his head, what was his vertical velocity ali he left the ground?

2,20 The velocity of a bobsled is v=3tm/s. When t = 28, it,position is s = 7.5m, What is its position when t = lOs?

P2,202.21 The acceleration of an object is a={10 - 21)m/,'- When 0, s 0 and v = 0, What is its maximum velocity during the interval of time from t = 0 to t = lOs?

=

P2,162.17 Suppose you want to throw some keys to a friend standing on a first-floor balcony, If you release the keys at 1.5 m above the ground, what vertical velocity is necessary for them to reach your friend's hand 6 m above the ground?

2.22 The velocity of an object is v = (200 - 2(') m/s. When l = 3 s, its position is s = 600 II1. What arc the position and acceleration of the objecl at t = 68?

2.23 The aeceicration of a part undergoing :l machining operation is measured and determined to be a = (12 - 61) mm/s'. When t = 0, v = D. For the interval of time from t == 0 to t = 4 s,determine: (a) the maximum velocity; (b) the displacement.

2,18 The Lunar Module descends toward the surface of the moonat 1 m/s when its landing probes) which extend 2 ttl below the landing gear, touch the surface, automatically shutting off the engines. Determine the velocity with which the landing gear contacts the surface. (The acceleration due to gravity at the surface of the moon is 1,62 m/s'.)

2.24 The missile shown in Problem 2.16 starts from rest and accelerates straight up for 3 s at 100 g's, After 3 s, its weight and aerodynamic drag cause it to have a constant deceleration of 4 g's. How long does it take the missile to go from the ground to an altifilde of 15240 m'l2.25 A car is traveling at 48 km/hr when a traffic light 90 m ahead turns amber. The light will remain amber for 5 s before turning red. (a) What constant ac.celeration will cause the car to reach the light at the instant it films red, and what will the velocity of the car be when it reaches the light? (b) If the driver decides not to try to make the light, what constant rate of acceleration will cause the car to corne to a stop just as it reaches the light?

30 mi/hr

vlew...-Driver', - [ .

,~~~--~--~----~~,

P2.IS

I------~~~

295 ft

30

CHAPTER 2 MOTION OF A POINT2.32~uppose that a person unwisely drives at 120 km/hr in anII

2.26 At t = 0, a motorist travelling at 100 km/hr sees a deer standing in the road 100m ahead. After a reaction time of 0.3 s, he applies the brakes and decelerates at a constant rate of 4m/s2 , If the decr takes 5 S ITom t = 0 to react and leave the road, docs tho

motorist miss him?2.27 A high-speed rail transportation system has a top speed of 100m/so For the comfort of the passengers, Ole magnitude of the acceleration and deceleration is limited to 2m/s2 , Determine the minimum time required for a trip of lOOkm. Strategy: A graphical approach can help you solve this problem. Reeall that the change in the position ITom an initial time to to a time t is equal to the arca defined by the graph of tho velocity as a function of time from to to t.

80km/~r zone and passes a police car going at 80km/hr in the same ditection. If the police offiCt"J'Sl begin constant acceleration at the insu{nt they are passed and increase their velocity to 130 km/hr in 4 s, hbw long does it take them to be level with the pursued car?IIII

2.33 If $= 1 rod and dO/dt= 1 rad/s, what is the velocity of P relative ~o O? " Strategj!: You can write the position of P relative to 0 as

s = (2m)00sO

+ (2m) cos 0,

then the derivative of this expression with respect to time to detenni*e the velocity.

t:ai\b

P2.27 P2.33

Thc noarcst star, Proxima Centauri. is 4.22 light years ITom the ealth. Ignoring relative motion between the solar system and Proxima Centauri! suppose that a spacecraft accelerates from the vicinity of the earth at 0.0 I g (0.0 I times the acceleration due to gravity at sea level) until it reaches one-tenth the speed of light, coasts until time to decelerate, then decelerates at 0,01 g tUltii it comes to rest in the vicinity of Proxima Centauri. How long does the trip take? (Light travels at 3 x IO'm/s. A solar year is 365.2422 solar days.) 2.29 A racing car starts ITom rest and accelerates at

2.28

Problem 2.33, if O=lrad, dO/dt=-2rad/s and d'O/dt'I=0, what are the velocity and acceleration of P relative to 0'1 2.34II

'I I~

2.35

If e=1 rad and dO/dt = I rad/s, what is the velocity of P

relative

io 01

a=(1.5 + 0.61) m/s 2 for lOs. The brakes are then applied, and the car ha."I a constant acceleration a = -9 m/s 2 until it comes torest. Determine: (a) the maximum velocity; (b) the total distance travelled; (c) the total time of travel.

2.30 When' = 0, the pooition of a point is s = 6 m and its velocity is v=2m/s. From t = 0 to t = 65, its acceleration is a = (2 + 2r)m/s2 , From t = 6 s until it comes to rest, its acce1eration is a = -4 O1/s2, (a) What is the total time of travel? (b) What tola1 distance docs it move?P2.35 2.31 Zoologists studying the ecology of the Screngcti Plain estimate that the average adult cheetah can run 100 km/hr and the average springbok can run 65 km/hr. If the animals nm along the same straight line, stalt at the same time, and are each assumed to have constant acceleration and reach top speed in 4 s, how close must a cheetah be when the chase begins to catch a springbok in

15 s7

2.2 STRAIGHT-LINE MOTION

31

Acceleration Specified as a Function of Velocity Aerodynamic and hydrodynamic forces can cause an object's acceleratioo to depend on its velocity (Figure 2.11). Suppose that the acceleration is a known function of velocity a(v);-

dv = a(v) . dt

(2.14)

Figure 2.11 Aerodynamic and hydrodynamic forces depend on an object'svelocity. The faster the object moves relative to the fluid, the greater is the force resisting its

motion.

We cannot integrate this equation with respect to time to determine the velocity, because a(v) is not known as a function of time. But we can separate variables, putting terms involving v on one side of the equation and terms involving t on the other side;

dv -=dt a(v)We Can now integrate

(2.15)

ids

'dV dt "' a(v) - to

1t

(2.16)

where .0 is the velocity at time to. In principle, we can solve this equation for the velocity as a function of time, then integrate the relation

-=v dtto determine the position as a function of time.

32 CHAPTER 2 MOTION OF A POINTBy using the chain we can also determine the velocity as a function of the position. Writing thell acceleration asII

ru\:e.

dv dvds..J dv dt = dsdt --r ds vand substituting it into I1huation (2,14), we obtain~v

dv = at,,) ds

Separating variables,

-=ds a(v)and integrating,

vdv

J"

r vdva(v)

we can obtain a relation ~etween the velocity and the position.

Acceleration as a Function of Position Gravitational forces and forces excrte~ by springs can cause an object's acceleration to depend on its position, r~ the acceleration is a known function of position,dv dt= a(s)

Specif)~d

(2,17)

we cannot integrate with ~espect to time to determine the velocity because s is not known as a functio~ of time. Moreover, we cannot separate variables, because the equation cont~ins three variables, v, t and s. However, by using the

chain rule .-=--=--v

dv dt

dvdsdv ds dt ds

we can wnte Equation

(2i 17) as

dv ~v = a(s) dsNow we can separate variables,:1

vdvand integrate:

= a(s)ds'

(2.18)

lvo

r 'vdv = [(1(S) dsSo

(2.19)

2.2

STRAIGHTLlNE MOTION

33

In principle, we can solve this equation for the velocity as a function of the position:v=

ds _ () -v s

dt

(2.20)

Then we can separate variables in this equation and integrate to detennine the position as a function of time:

.

ds -= '" v(s)

l''0

dt

The next two examples show how you can analyse the motion of an object when its acceleration is a fanction of velocity or position. The initial steps are summarized in Table 2.1.

Table 2.1. Dctcnnining the velocity when you know the acceleration as a function of velocity or position.

If you know a - a(v):

Separate variables,

dv

dt - a(v)dv -=dt a(o)or apply the chain rule,

-=--=-v_a(v) dt ds dt dsthen separate vaIiables,

dv

d1)ds

dv

.

--=dsa(v)

vdv

If you know a - a(s):

Apply the chain rule,dv dvds dt = J;didvv

ds

a(s)

then separate variables!vdv

= a(s)ds

34

CHAPTER 2 MOTION OF A POINT

After deploying its dral! parachute, the aeroplane in Figure 2,12 has an accelemtion a = -0,004.' mis, i (a) Determine the time required for the velocity to decrease from 80m/s to lOm/s, (b) What distance do"" the plane cover during that time?

Figure 2.12

STRATEGYIn pan (b), we will u~c the chain rule to express the acceleration in tenns of a derivative with respect!1 to position and integrate to obtain a relation between the velocity and the positiqIl.

SOLUTION(a) The acceleration is"a

=

dv, , = -0,004vdt

We separate variables,dv , =-O,Q04dt

and integrate,

defining!~ =,0!i

0 to be the time at which v = 80m/s:

19oEvaluating the

r ~' = I:', -O,004dtintegral~

and solving for t, we obtain

II

t=250(L~)" 80

2.2 STRAIGHTLlNE MOTIONThe time required for the plane to slow tol' = lOm/s is 21.9s. We show the velocity of the aeroplane as a function of time in Figure 2,13,

35

XI )

70o(

I

li\

Figure 2.13 Graph of the aeroplane's velocity as a function of time,

,

S()4( )

\

E

\

i

3(I

'\

I

2UII I()

"'- -..U 5 IU

iII!

"

,20' 2530

15r, seconds

21.9

(b) We write the acceleration as

a = - = - - = -11 = -0.004,)dt d" dt d"

dv

dvds

dv

2

separate variables,

and integrate. defining s

~

0 to be the position at which v ~ 80m/s:

L80

dp - = [ -O.004dsIi

0

Evaluating the integrals and solving for s, we obtain

s = 250

InC:)

The distance required for the plane to slow to v = lOm/s is 519.9m.

DISCUSSIONNotice that our results predict that the time elapsed and distance travelled continue

to increase without bound as the aeroplane's velocity decreases. The reason is thatthe modelling is incomplete. The equation for the acceleration includes only aerodynamic drag and does not account for other forces, such a~ friction in the

aeroplane's wheels,

36

CHAPTER 2 MOTION OF A POINT

In terms of distance oS' ~m the centre of the earth, the magnitude of the acceleration due to gravity is gR~/~l. where RE is the. radius of the earth. (See thc discussion of gravity in Section 1.3.)1 If a spacecraft is a distance So from the centre of the earth (Figure 2.14), what ou ard velocity Vo must it be given to reach a specified distance ~ . II h ,rom the centre of th earth?

SOLUTIONFigure 2.14The acceleration due t~: gravity is towards the centre of the earth:

gR' a=-~ s '!Applying the chaina

rul~,~vds

=-=t---~- =~v=~-

dtidt

dv

gR~,2

lis dt

as

and separating

variable~,

we obtain

R' vdv=-~{lS'~.2

We integrate this equation using the initial condition, 'I) = v() when s = So, as the lower limits, and the fi~al condition, v : 0 when s = h, as the upper limits:

Evaluating the integralS and solving for 1)(), we obtain the initial velocity Vo necessary for the spacc~raft to reach a distance h:

Vo

=

2gR~ r1 - -h :~()

~

I)

DISCUSSIONWe can make an interesting and important observation from the result of this example. Notice that as:1 the distance h increases, the necessary initial velocity Vo approaches a tinite limi~, This lunit,

Ve~c

=

,

.

::1

h-+c4

1ml Vo;;;::::

tgRf;So

--

is called the escape ve~~city. In the absence of other effects~ an object with this initial velocity will co~tinuc moving outwards indefinitely. The existence of an eO

M

M~O

. 2 sin(M/2) n 1 1m

M

To evaluate this limit, we write it in the form

de dt

= lim 'in(M /2) L\8 n"140

M /2

M

In the limit as M approaches zero, sin(L\O/2)/(M/2) equals I, L\O/L\t equals dO/dt, and the unit vector n is perpendicular to e(t) (Figure 2.19(c)). Therefore the time derivative of e is

-=-n=wn

de dt

dO dt

(2.33)

where n is a unit vector that is perpendicular to e and points in the positive 8 direction (Figure 2.19(d). In the following sections we use this result in deriving expressions for the velocity and acceleration of a point in different coordinate systems.

Figure 2.19 (a) A unit vector e and reference line L o. Cb) Thc change .l. in from t to t + .It. (e) As At goes to zero, n becomes perpendicular to e(t).(d) The time derivative of e.(a)

~--~----------------LO

(b)

-""-0""(1)0

deill

deIt

~=----------------- to(c)

~

__.J.::._______________ ld)

to

52

CHAPTER 2 MOTION OF A POINT

Example 2.7The rotor of a iet engin~ is rotating at 10 000 rpm (revolutions per minute) when the fuel is shut off. The e~suing angular acceleration is O! ~ -0.02(0, where (J) is the angular velocity in !""ddt s.(a) How long does it take the rotor to slow to 11)00 rpm? (b) How many revolut~bns does the rotor turn while decelerating(0

1000 rpm?

STRATEGYTo analyse the angular'motion of the rotor, we define a line L that is fixed to the rotor and perpendicularlto its axis (Figure 2.20). Then we examine the motion of Lrelative to the referenceilline Lo. The angular position, velocity and acceleration of L define the angular moti'pll of the rotor.

Figure 2.20 Introducing a line L and reference line Loto specify the angular position of therotor.

SOLUTIONThe conversion from qjm to cadIs isI rpm = 1 revolutlOn/mm x= (n/30) rad/sIi

(2nrad) x"---O (' min) ,.1 revo utlonI)

S

2.3 CURVILINEAR MOTION

53

~1

P7.74

7.72 The homogeneous) slender bar has mass m and length I. Use integration to determine the mass moment of inertia of the bar about the axis L.

7.75 The brass washer is ofunifonn thickness and tnaSS m. (a) Detennine its mass moments of inertia about the x and z axes. (b) Let Ri = 0 and compare your results with the values given in Appendix C for a thin circular plate. (c) Let R, -+ Ro and compare your results with the solutions of Problem 7.73.)'

P7.72 P7.75

APPENDIX: MOMENTS OF INERTIA

357

7.76 The homogeneous; thin plate is of uniform thickness and weighs lOON. Detennine its mas:s moment of inertia about the y axis.

7.80

Two homogeneous, slender bars, each of mass m and length I, are welded together to the T-shaped oi1iect. Use the parallel-axis theorem to determine the mass moment of inertia

",nn

of the object about the axis through point 0 that is perpendicular to

the bars,

P7.76

7.77 Dctennine the mass moment of inertia of the plate inProblem 7.76 about the x axis.about LI is 10 kg.m 2. \Vbat is its lnass moment of inertia aboutL 2? (The three axe!': lie in the same plane.)

P7.80

7.78 The mass of the object is 10 kg, Its maSS momenl of inertia

7.81 Usc the paral1cl~axis theorem to dctctminc the mass moment of incrtia of the T~shapcd object in Problem 7.S0 aboul the axis through the centre of mass of the object that is perpendicular to the two bars.7.82 The mass of the homogeneous, slender bar is 20 kg. Detennine its mass moment of lneltia about the z axis.

jL

06 m --~+--- 06 Tn L,

P7.78

-1 ~! ~ ~ I/J '~'~"'..1m

.1'

.,

'1,

7.79 An engineer gathering data tor the design of a manoeuvringunit detennincs that the astronaut'~; centre of mass is at x = LO 1 ill, V = 0, 16 m and that his mass moment of inertia about the z axis is 'I05,6kg,m', His mass is 8L6kg, What is his mass moment of inertia about the i axis through his centre of mass?

-J-

-'".

!......

- . - I ..

~m------- I", the angle /3 < O. In Figure 9.30(c), we show an imaginary cone of half-angle /3, called the body cone, whose axis is coincident with the z axis. The body cone is in contact with a fixed cone, called the space cone, whose axis is coincident with the Z axis. If the body cone rolls on the curved surface of the space cone as the z axis precesses abont the Z axis (Figure 9.30(d)), the points of the body cone lying on the straight line in Figure 9.30(b) have zero velocity relative to the XYZ system. That means that the motion of the body cone is identical to the motion of the object. Yon can visualize the object's motion by visualizing the motion of the body cone as it rolls around the outer surface of the space cone. This motion is called direct precession. If f", < I,., the angle P> e. In this case you must visualize the interior surface of the body cone rolling on the fixed space cone (Figure 9.30(e. This motion is called retrograde precession.

Arbitrary ObjectsIn this section we show how the equations of angular motion can be expressed in terms of the Eulerian angles for an arbitrary object. The initial steps are similar to our treatment of an axially symmetric object, but in this case we assume that the xyz coordinate system is body-fixed.

Figure 9.30(c), (d) The body and space cones. The body cone roJJs 011 the stationary space cone. (e) When IX > 0, the interior surface of the body cone rolls on the 8tationary space cone.

z

(a)

(b)

z

Space cone ,

zSpaceCOliC'

z

Spacey

Bodycone ___ .~

---yEorly

cone(e)

/

/~(ell

Body

.

.-'"'-.......x

cone

/~o/-~xX(0)

X

9,5 EULERIAN ANGLES

469

Definitions We begin with a reference position in the x yz and XYZ systems are superimposed (Figure 9.31(a)). First, we rotate the xyz system through the precession angle", about the Z axis (Figure 9.31 (b)) and denote it by x'y'z' in this intermediate orientation. Then we rotate the xyz system through the nutation angle 0 about the x' axis (Figurc 9.31 (c)), denoting it by x''y''z'', We obtain the final orientation oflhe xyz system by rotating it through the angle ep about the z" axis (Figure 9,31(d)). Notice that we have used one more rotation of the xyz system than in the case of an axially symmetric object.

}X,X

,/ and the angular velocity !/J. (d), (e) The components of the angular velocities t/I sin () and (} in the xyz system. (1) The angular velocities w>;, wyand W z .

angles to obtain the equations of angular motion. Figure 9.32(a) shows the rotation 1/1 from the reference orientation of the xyz system to lhe intermediate orientation x'y'z'. We represent the angular velocity of the body.fixed coordinate system due to the rate of change of 1/1 by the vector 1/1 pointing in the z' direction. Figure 9.32(b) shows the next rotation that takes the body-fixed coordinate system to the intermediate orientation x"y"z". We represent the angular velocity due to the rate of change of 0 by the vector iJ pointing in the x" direction. ln this figure we also show the components of the anl,,'1I!ar velocity vector I~ in the y" and z" directions. Figure 9.32(c) shows the third rotation 1J that takes the body-fixed coordinate system to its final orientation defined by the three Eulerian angles. We represent the angular velocity due to the rate of change of 4J by the vector ~ pointing in the z direction. To determine wx , wyand OJ, in terms of the Eulerian angle

iJ sin 4

If you know the Eulerian ,mgles and their first and second time derivatives, you can solve these equations for the components of the total moment Or, if you know the total moment, the Eulerian angles, and the first time derivatives of the Eulerian angles, you can determine the second time derivatives of the Eulerian angles, You can use these equations to detennine the Eulerian angles as functions of time whc,n you know the total moment, but numerical integration is usually necessary.

472

CHAPTER 9 THREE-DIMENSIONAL KINEMATICS AND DYNAMICS OF RIGID BODIES

In the following example we analyse the motion of an object in steady precession. By aligning the coordinate system as shown in Figure 9.28, you can use Equation (9.39) to relate the total mOffle'" about the x axis to Ihe nutation angle 0, precession rate and spill rate ~.

;p,

The thin circular disc of radius R and mass m in Figure 9.33 rolls along a horizontal circular path of radius r. The angle 0 between the disc's axis and the vertical remainsconstant. Determine the magnitude v of the velocity of the centre of the disc as a

function of the angle O.

Figure 9.33

,.- ------ .....

/STRATEGYWe can obtain the vell,Jcity of the centre of the disc by assuming that the disc is in steady precession and determining the conditions necessary for the equations of motion to be satisfied.

SOLUTIONy

I

---

--

In Figure (a) we align the z axis with the disc's spin axis and assume that the x axis remains parallel to the surface on which the disc rolls, The angle (J is the nut1.tion angle. The centre of mass moves in a circular path of radius fG = r - R cos Therefore the precession rate, the rate at which the x axis rotates in the horizontal plane, is

e.

II(a) Aligning the z axis with the spin axis. The x axis is horizontal

From Equations (9.33). the components of the disc's angular velocity are

myill,

= '" sinO =-sin H Yc = + "'cosO = '" +-cosO YG.. . v

.

v

Ii J

I

9,5 EULERINI ANGLES where ~ is the spin rate. To detenlline

473

;p, we use the condition that the velocity of

the point of the disc in contact with the surface is zero. In tenn!ll of the velocity ofthe centre the velocity of the point of contact is

k

O=vi+wx(-Rj)=vi+ 0 ~sinlirr;

1> +~coslir(l

o. vR

-R

o

Expanding the determinant and solving ror

I~''-~(bl

CHAPTER SUMMARY

477

Euler EquationsThe equations governing three-dimensional motion of a rigid body include Newton's second law and equations of angular motion. For a rigid body Totating about a fixed point 0 (Figure (a, the equations of angular motion are expressed in terms of the components of the total moment about 0:

Equation (9.25)

where n is the angular velocity of the coordinate system. If the coordinate system is body-fixed, n = w. III the case of general three-dimensional motion (Figure (b, the equations of angular motion are identical except that they are expressed in terms of the components of the total moment about the centre of mass, The rigid body's angular acceleration is related to the derivatives of the components of (() by~=-=-I+-J+

dm dt

dw.,. dt

dwy , dt

dw, " k+ .. xw dt

Equation (9.24)

If the coordinate system does not rotate or is body-fixed, the terms dwx/dt, dw,,/dt and dw,jdt m'e the components of the angular acceleration.

Mome,nts and Producl!i of InertiaIn terms of a given coordinate system xyz, the inertia matrix of an object is defined by [EquatIOn (9.11)]-fryfyy

-"] _.'",-fyz

I"

-1Lm

m

xydm

-1(x"

m

xzdm

zxdm

1 -L

(x2 +Z2)dmzydm

LYZdm

L+

y')dm

where x, y and z are the coordinates of the differential element of mass dm. The terms lxx, I", and T" are the moments of inertia about the x, y and z axes, and 'XY' Tj7 and 1M are the products of inertia,

478

CHAPTER 9 THREE-DIMENSIONAL KINEMATICS AND DYNAMICS OF RIGID BODIES

If x'y'z' is a coordinate system with its origin at the centre of mass of an object and xyz is a parallel system, the parallel-axis theorems state that

Ixx = Ix'x' + (d; I,y '" Jy.y.T zz

+ d;)m

+ (d; + d;)mEquation (9.16)

= I,.,. + (d; + d;)m

where (dx, dy , dz) are the coordinates of the centre of mass in the xyz coordinate system. The moment of inertia about an axis through the origin parallel to a unit vector e is given by~qu.tlon

(9.17)

For any object and origin 0, at least one coordinate system exists for which the products of inertia are zero. The coordinate axes are called principal axes, and the moments of inertia are called the principal moments of inertia. If the inertia matrix is known in terms of a coordinate system x'y'z', the principal moments of inertia are roots of the cubic equation[3 _

+ I).). + T,.,. )[2 + (.lx'xl/),.y, + ly",'IZ't' + iZ'Z'I.l;lx' . .(Ix'x'

_12, , xy

,2, , _[2, ,)[yz zx

For each principal momcnt of inertia

r,

the vector V with componentsI}.,Equation (~.20)

1/,.fly..

= (l,.y. - 1)(1,.,.

T)

1x 'y(1," - I)

+ Ixzly '"

z

y

is parallel to the corresponding principal axis.~--y

Eulerian Angles: Axisymmetric ObjectsIn the case of an object with an axis of rotational symmetry, the orientation of the xyz system relative to the reference XYZ system is specified by the precession angle ofr and the nutation angle (Figure (c)). The rotation of the object relative to the xyz system is specified by the spin angle 1/>.

x(e)

x

e

CHAPTER SUMMARY

479

The components of the rigid body's angular velocity relative to the XYZ system are given byWx

= iI (Oy = ~ sinew, =

Equation (9.33)

+ ~cosi1

The equations of angular motion expressed in terms of these Eulerian angles are [Equations (9.36) (9.38)]: 'LM,'L M,

= luO + (fa -lu)~2 sinOcosO +1,"~ sin a = I a( ~ sm 0 + 2ljJiJ cos 0) = I,,(~ + 1,& cos f) i,,(

1,iI + IjJU cos 0)

EM,

~b sin II)

In steady precession of an axisymmetric spinning object, the spin rate , the nutation angle (I and the pre'ccsslon rate ~ are assumed to be constant. Withthese assumptions, the equations of angular motion reduce to [Equations (9.39}-(9.41)]

EM, =0

Eulerian Angles: Arbitrolry ObjectsIn the case of an arbitrary object, the orientation of the body-Axed xyz system relative to the reference XYZ system is specified by the precession angle 1jJ, the nutation angle 0 and the spin angle q, (Figure (d)). The components of the rigid body's angular velocity relative to the XYZ system are given by

w" = ~ sin 0 sin rp

+ iJ cos q,IJ sin q,Equation (9.44)

w,

= IjJ sin cos q, w, = ~ cos () + ,i,

If xyz is a set of principal axes, the equations of angular motion in terms of Eulerian angles are given by Equations (9.46).

480

CHAPTER 9

THREE-DIMENSIONAL KINEMATICS AND DYNAMICS OF RIGID BODIES

9.92 The slender bar oflength I and mass m is pinned to the Lshaped bar at O. The L-shaped bar rotates about the vertical ..xis with a constant angular velocity CUo- Detemline the value of Wo necessary for the bar to remain at a constant angle /3 relative to the

verticaL

9.94 The thin plate ofmnss m spins about a vertical axis with the plane of the plate peI]lendicular to the floor, The comer of the plate at 0 rests in an indentation so that it remains at the same point on the floor. The plate rotates with constant angular velocity (J)o and the angle fJ is eonslanL (a) Show that the angular velocity WI) is related to the angle j3 by

hW6gsin'

2 cos j3 - sin 13

fJ - 2 sin fJ cos fJ - cos2 P

(b) The equation you obtained in part (al indicates that WI) = 0 when 2 cos /J - sin fJ = o. What is the interpretation of this result'!

P9.92

9.93 A slender bar oflength I and mass m rigidly attached to the centre of a thin circular disc of radius R and mass m. TIlecomposite object undergoes a motion in which the bar rotates in the horizontal plane with constanl angular velocity (1)0 about the

centre of mass of the composite object and the disc rolls on the floor. Show that 'va =

2JiiR.

o

P9.94

9.95 [n Problem 9.94, detennine the range ofvalucs of the angle {J for which the plate will remain in the steady motion described.

1---1--1P9,93

REVIEW Pr.'OBLEMS9.96 AR11 Be has a mass of 12 kg, and its moments and products of inertia in terms of the coordinate system shown are 2 iu ::::: 0,03 kg.m 2 , ~vy::::: lu = 4 kg.m , Ixy = I)'z = 1.1;; = 0, At the instant shown, arm All is rotatil.lg in the horizontal plane with a constant angular velocity of 1 rad/s in the counterclockwise dircc~ tion viewed from above. Relative to arm AB, arm Be is rotating about the z axis with a constant angular velocity of 2 rad/s. Determine the force and couple exerted on aIm Be at B.

481

9.99 The mass of the homogeneous thin plal(~ is 1 kg. For a coordinate ~ystcm with its origin at O. detennioe the principal moments of inertia and the dircotions of unit vecl!>rs parallel to the corresponding principal axes.\

b~~~=1.''.

/'

T 160III III

! ! -.'.=~.~

T'Illm

:I

'

160

I

/~40(;~;~~:~.'

0'

','

I

i

...J~~~_.~. ~,'

P9.99

P9.969.97

SL!-ppose that you throw a football in a wobbly spiral with a nUlation angle of 25". The f()otball'~ moments of inertia are lu = I," = 0.003 kg.m' and In = 0.001 kg.m 2 If the spin rate is = 4~-revolutions per second what is the magnitude of the precession rate (the rate at which it wobbles)'!

9.100 The aeroplane's principal moments of inc,rtia in kg,m 2 are Ix> = BOOO, I"" = 48 000 and I" = 50 000. (a) The aeroplane begins in the reference position :;hown and mano~mvres into the orientation rjJ = 0 = = 45 '. Draw a sketch s:howing its orientation relative to the X YZ system. (b) If the aeroplane is in the onentation described in part (a), the rates of change of the Eulerian angles arc !/I == 0, i:' = 0.2 rad/s and == O.2rad/s) and their second! time derivatives are :.:-;cro, what arc' the components of the total moment about the aeroplane's centre of mass?

P9.1009.101 What are the x,y at)d z components of the angular acceleralion of the aeroplane descL'ibed in Problem 9.100?

P9.97

9.98 Sketch the body and space cones for the motion of the rootball in Problem 9.97.

9.102 If the orientation of the aeroplane in Problem 9.100 is = 45', () = 60" '" 45', the rates of change of the Eulerian angles are Ifr = 0, Ii 0.2 rad/s and Ip = 0.1 rad/s, and the com ponents of the total moment about the centre of maRR arc LM, =400N.m, LM, = 1200N . m and LM, = 0, what arc the

VI

=

x, y and z components of the aeroplane's angular acceleration?

: :'

ngineers use 'shake tables' to simulate the vibrations of buildings and other structures during earthquakes and investigate methods for minimizing structural damage. The tables can be programmed to simulate the magnitudes and time histonc8 of the ground vibrations measured during actual earthquakes. In this chapter we analyse the vibrations of simple

E

mechanical systems.

Vibrations

V

IBRATIONS have been of concern in engineering at Icast since the beginning of the industrial revolution.

The oscilllatory motions of rotating and reciprocating engines

subject their parts to large loads that must be considered in their design. Operators and passengers of vehicles powe:red by these engines must be isolated from their vibrations. JBeginning with the development of e1eetromechanical devic()s capable of cn:ating and measuring mechanical vibrations, such as loudspeakers and microphones, engineering applications of vibrations have included the various areas of acoustics., from architectural acoustics to earthquake detection and analysis. In this chapter we consider vibrating systems with one degree of freedom; that is, the position, or configuration, of each system is speeified by a single variable. Many actual vibrating systems either have only one degree of freedom or their motions can be modelled by a one-degree-of-freE:dom system in particular circumstances. We discuss fundamental eoncepts, including amplitude, frequency, period, damping and resonance, that are also used in the analysis of systems with multiple degrees offreedom.

483

484

CHAPTER 10 VIBRATIONS

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

10. 1 Conservative Systems

We begin by presenting different examples of one-degree-of-freedam systems subjected to conservative forces, demonstrating that their motions are described by the sarne differential equation. We then examine solutions of this equation and use them to describe the vibrations of one-degree-of-freedom conservative systems.

ExamplesThe spring-mass oscillator (figure 10.1 Cal) is the simplest example of a onedegree-of-freedom vibrating system. A single coordinate x measuring the displacement of the mass relative to a reference point is sufficient to specify the position of the system. We draw the free-body diagram of the mass in figure 10. I (b), neglecting metion and assuming ihat the spring is unstretched when x = O. Applying Newlon's second law, we can write the equation describing horizontal motion of the mass as

(a)

~!", "',,', '::~g,1N

....................................

ri

'"

(10.1)

(b)

We can also obtain tbis equation by using a different method that you will find very useful. The On ly force that docs work on the mass, the force exerted by the spring, is conservative, which means that the sum of the kinetic and potential energies is constant:

I -m -

2

(dx)2 +-1 OJ, the system is said to be supercritieally damped. The general solutiol1 isx

= Ce-Ia-h)t + De-(d+h)t

Equation (10.24)

where C and D are constarllts and h is defined byEquation (10.23)

If d = w. the system is said to bc critically damped. TI,e general solution isEquation (10.25)

where C and D are constants.

526

CHAPTER 10 VIBRATIONS

Forced VibrationsThe forced vibrations of many damped, one-degree-of-fi'eedom systems are governed by the equation dx 2 d 2x +2d-+w x=a(t) dt dt

-2

Equation (10.26)

where aCt) is the forcing function, The general solution of Equation (10,26) consists of the homogeneous and particular solutions:

The homogeneons solution Xh is the general solution ofEquation (10.26) with the right side set equal to zero, and the particular solution xp is a solution that satisfies Equation (10.26),

Oscillatory Forcing Function

If aCt) is an oscillatory function ofthe fonn

aCt)

= ao sin wot + bu cos wut

where aD, bo and Wo arc constants, the palticu1ar solution is (Equation 10.30)

.tp

=[