Engineering Dynamics: AComprehensive Introduction—Errata

N. Jeremy Kasdin & Derek A. Paley

Last updated November 10, 2017

PRINCETON UNIVERSITY PRESS

PRINCETON AND OXFORD

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Chapter 2:

i) [First printing only] In tutorial 2.4, the solution for the criticallydamped case (starting at the bottom of page 33) solves for λ1, λ2 =−ζω0. This is propagated through to the solution (Eq. (2.26)), whichmakes it appear as if ζ is a variable in the solution. This is misleadingas ζ can only equal 1 in this case. The two equation should be rewrittenwithout ζ (3 instances).

ii) [First printing only] In problem 2.16, the downward accelerationshould be g − c

m y2.

iii) In problem 2.16, the problem statement should be changed to add theassumption that the rock is released from rest. The phrase ‘withoutthe use of a computer’ should be interpreted as ‘without the use ofnumerical integration.’

Chapter 3:

i) [First printing only] In Section 3.4.1, the final equation on Page 68defining the transformation matrix should be in terms of the scalarcomponent magnitudes, not the unit vectors. That is, the elements ofthe two matrices should not be bold. It should thus read:[

b1b2

]B

=

[cos θ sin θ− sin θ cos θ

] [a1a2

]A

ii) [First printing only] In Section 3.8.4 on page 93 describing the vec-tor Taylor series the vector (and components) were mistakenly writtenat t+a rather than t. Thus, the sentence before Eq. (3.68), that equa-tion, and the following two equations should be written:

First, write the vector r(t) as components in a frameA = (O,a1,a2,a3):

r(t) = r1(t)a1 + r2(t)a2 + r3(t)a3. (1)

Then take the Taylor series of the scalar magnitude of each component.For i = 1, 2, 3, we have

ri(t) = ri(a) +d

dtri

∣∣∣∣t=a

(t− a) +1

2!

d2

dt2ri

∣∣∣∣t=a

(t− a)2 + . . .

Expanding and rearranging each coefficient in Eq. (3.68) yields

r(t) = r1(a)a1 + r2(a)a2 + r3(a)a3

+d

dtr1

∣∣∣∣t=a

(t− a)a1 +d

dtr2

∣∣∣∣t=a

(t− a)a2 +d

dtr3

∣∣∣∣t=a

(t− a)a3

+1

2!

d2

dt2r1

∣∣∣∣t=a

(t− a)2a1 +1

2!

d2

dt2r2

∣∣∣∣t=a

(t− a)2a2 +1

2!

d2

dt2r3

∣∣∣∣t=a

(t− a)2a3

+ . . .

2

iii) In the caption of Figure 3.32 on page 97, the units of µ should be m−1

(two times).

iv) [First printing only] In Problem 3.2, F3 = (5− 4√

6)ey N.

v) [First printing only] Figure 3.45 is missing a downward gravityarrow labelled g.

Chapter 4:

i) [First printing only] In Example 4.1 on pgs. 115-116, the expressionsfor x(t) and x(t) should be written relative to the initial time t2,

x(t) =−FP (t1, t2)

mPω0sin [ω0(t− t2)]

x(t) =−FP (t1, t2)

mPcos [ω0(t− t2)]

ii) [First printing only] On page 138, the first unnumbered equationshould not have a minus sign.

iii) [First printing only] In Tutorial 4.3, the second-to-last line on page140 should say “The magnitude of the specific angular impulse...” Also,the two moment expressions on page 141 should be divided by thesatellite mass to correspond to the change in specific angular momen-tum, i.e., the left-hand-side of the two unnumbered equations shouldbe ‖MA/O(t1, t2)‖/mP and ‖MB/O(t1, t2)‖/mP , respectively.

iv) In Problem 4.5, the groove is semi-circular when viewed from above.Also, ignore the coefficient of friction corresponding to the horizontalcomponent of the normal force required to make the puck travel aroundthe groove: i.e., assume the only source of friction arises from thevertical component of the normal force.

v) [First printing only] In Problem 4.11 on page 146, the feedbackequation for the base motion should be for the acceleration u of thebase and not µ, i.e., the equation should read

u = −g sin θ + kθ + bθ

cos θ

vi) In Problem 4.13, for clarity, the problem statement, and part (a),should both say ‘linear impulse’.

Chapter 5:

i) [First printing only] Top of page 155 should not be gray box, up tothe start of Example 5.3.

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ii) [First printing only] On page 163, in Example 5.8, the middle un-numbered equation has a typo; the sign is wrong on the last term. Itshould read:

Tb1 − µcNb1 +Nb2 −mP g sin θb1 −mP g cos θb2 = 0

iii) [First printing only] On page 183, in Figure 5.18, right side, thelength of the string should be longer so that the height of the whiteparticle is the same as the height of the black particle.

iv) [First printing only] On page 183, the ellipse in Figure 5.19 shouldbe taller so that it only intersects the circle at point P, i.e., it com-pletely encloses the circular orbit of radius ρ.

v) [First printing only] In Problem 5.14, page 184, the value given forC in part (b) is actually the square root of the correct value. C shouldequal 0.33, not -0.58.

Chapter 6:

i) On page 193, the first line after Equation (6.7), the right-side of theinline equation should use Fj,i not Fj,i.

ii) [First printing only] Equation 6.10 on page 193 should use F(ext)

(t1, t2)

instead of F(ext)

. The following line should read “where F(ext)

(t1, t2)is the total external impulse acting on the system from t1 to t2.”

iii) On page 196, the final line of text before Equation 6.14 is incorrect.It should say: “A consequence of this definition is that the sum of themass-weighted positions of all of the particles in the collection relativeto the center of mass is zero.”

iv) The first equation in Example 6.7 (page 202) is missing the total massterm (mG). It should read:

mG

Id2

dt2rG/O = F

(ext)G

v) [First printing only] The second line of Equation 6.23 on page 203should be

=−k(m1 +m2)

m1m2

(‖r2/1‖ − l0

)r2/1.

vi) [First printing only] Figure 6.14(b) on page 214 is incorrect. Itshould be altered as below.

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Figure 1 Corrected figure 6.14(b).

vii) [First printing only] On page 235, the unnumbered equation in the

second bullet of the Key Ideas should use F(ext)

(t1, t2) instead of F(ext)

.

The following line should read “where F(ext)

(t1, t2) is the total externalimpulse from t1 to t2.”

viii) [First printing only] In Problem 6.7 on p. 239, the lower summation

index should be i not j, so that it reads∑N

i=1.

ix) In Problem 6.15 on p. 242, the given values for parameters x0 and l0do not match the problem description (that initially the springs arecompressed). If the are reversed, so that x0 = 5m and l0 = 10mthe problem is much more physical, although the solution remainsunchanged.

x) [First printing only] In Problem 6.23 on p. 244, the value for theexhaust velocity should be 2.75 km/s, not 1 km/s.

xi) [First printing only] In Problem 6.24 on p. 244, the value for thespring rest length should be l0 = 1 m, and the damper constant shouldbe b = 25 N·s/m. The second-to-last sentence should read “...in therocket, starting with the spring at its rest length.”

xii) [First printing only] In Problem 6.25 on p. 244, the second and thirdsentences should read “Assume the model rocket weighs 2 kg and thefuel weighs 3 kg. Assume a mass flow rate of 0.1 kg/s and an exhaustvelocity of 500 m/s.”

Chapter 7:

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i) In Example 7.1 on pp. 249-250, both the figure (in terms of the place-ment of the center of mass) and the final equation of motion are onlycorrect if we assume mP = mQ. In the case where they are unequal,the equation of motion becomes:

θ =g

l (mP +mQ)

(mP cos

(θ +

π

3

)−mQ sin

(θ +

π

6

))ii) [First printing only] In Example 7.4 on p. 254 there is a typo in

the third equation (definition of the angular momentum of the centerof mass about the origin). The unnumbered equation reads “IhG/O =

rG×...” but should be “IhG/O = rG/O×...”.

iii) [First printing only] In Example 7.8 on p. 265 there is a typo inthe expression for TG/O. The square should be on yG not outside theparenthesis, i.e.,

TG/O =1

2mG‖IvG/O‖2 =

1

2mG(x2G + y2G).

iv) [First printing only] In Example 7.11 on p. 273, the unit vectors ofthe polar frame have been defined as er and eθ so the b1 and b2 areincorrect. The equations for rP/G and IvP/G should be written

rP/G =−r2er

IvP/G =− r2er −

r

2θeθ.

Likewise, in the sentence that follows, the unit vector b1 should bereplaced by er.

v) [First printing only] In Example 7.11 on p. 274, the final conditionsin the last paragraph are not correct. The rotation rate at infinityshould be θ(∞) = 2hG/r

20. The expression for the nonconservative

work should be

W (nc,int) =mh2Gr20− m

4

(r(0)2 +

4h2Gr(0)2

)− 1

2k(r(0)− r0).

vi) [First printing only] On p. 274, penultimate paragraph there shouldbe a leading Eqs. before (7.11) and (7.12).

vii) [First printing only] On p. 275, Figure 7.10(b), the depiction of θ isnot correct. It should be the angle measured counter-clockwise fromthe e1 axis to the er axis. The figure should be corrected as annotatedhere.

6 ANGULAR MOMENTUM AND ENERGY OF A MULTIPARTICLE SYSTEM 275

I B

O

G

G

re1

rG/O

r2/O

r1/O

r1/2

m1

m2

e2

er

eθ

θ

e3

(a) (b)

Figure 7.10 Two bodies of masses m1 and m2 in orbit about their common center ofmass in the plane orthogonal to e3. (a) Inertial frame I = (O, e1, e2, e3). (b) Polar frameB = (G, er, eθ , e3).

also perhaps different from our intuitive image of the earth orbiting the sun. In fact,the earth and sun orbit their common center of mass; because the sun is so muchlarger than the earth, the center of mass of the two bodies is actually inside the sun.Consequently, the sun’s motion is very small.3 (The same is true of the earth’s motionwhen it is orbited by a small satellite.) Nevertheless, it is very common and usefulto find a description of the earth’s motion relative to the sun (or, to be more general,body 1 relative to body 2) given by r1/2(t).

Using the inertial frame I = (O, e1, e2, e3), we write the dynamics of body 1relative to body 2. We start with the definition of the center of mass,

rG/O = m1

m1 + m2! "# $!=µ1

r1/O + m2

m1 + m2! "# $!=µ2

r2/O,

where we have introduced the dimensionless masses µ1!= m1/(m1 + m2) and µ2

!=m2/(m1 + m2). Note µ1 + µ2 = 1 and m1 + m2

!= mG. Rearranging and using thevector triad r2/O = r2/1 + r1/O , we find

µ1r1/O = rG/O − µ2r2/O = rG/O − µ2(r2/1 + r1/O)

= rG/O + µ2r1/2 − µ2r1/O,

which means

r1/O = rG/O + µ2r1/2.

3 Scientists searching for a planet about another star infer the presence of a planet by measuring the (small)motion of the star.

Kasdin third pages 2010/12/9 6:33 p. 275 (chap07) Princeton Editorial Assoc., PCA ZzTEX 13.9

I

Figure 2 Annotated correction to Figure 7.10(b)

viii) [First printing only] On p. 277 the final equation for the angularmomentum relative to the center of mass is incorrect. The correctequation is

IhG =m1m2

m1 +m2r2θe3,

ix) [First printing only] On p. 277 in the description following thesecond unnumbered equation for IhG, the specific angular momentumfrom Tutorial 4.2 should be hO rather than hP .

x) [First printing only] On p. 287, everywhere in the last bullet shoulduse EO instead of E (four times).

xi) [First printing only] On p. 289, Problem 7.5, the last sentence beforethe hint should read “...when the baton falls, assuming it starts nearlyupright.”

xii) [First printing only] On p. 290, Problem 7.7, the text of the problemshould read “initial speed v0” rather than “initial velocity v0”. Also,in part (a), it should read “final speed vf” rather than “final velocityvf”. Part (b) is correct as written.

Chapter 8:

i) [First printing only] On p. 298, in Example 8.2, the equation at thebottom of the page should be

IvO′/O = −ωBe3 × (−Re2) = −RωBe1Likewise, both of the b unit vectors in the preceeding paragraph shouldbe e unit vectors.

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ii) [First printing only] On p. 315, In the second equation, the velocityshould be with respect to frame B rather than I. The equation shouldthus be:

BaP/L = − flmP

rP/L − 2ωrb3 × BvP/L − ω2rb3 × (b3 × rP/L).

iii) Original Kindle Edition Only Problem 8.1(a), IvP/O′ should beIvP/O.

iv) [First printing only] On p. 330, in Problem 8.3, the radius rB fromthe problem statement is missing from Figure 8.24. It is the radius ofthe back gear, which has center O′′. Also, the rotation of the crank isdrawn as going clockwise and so the rotation rate should be -2 rpm.

v) [First printing only] p. 331, Problem 8.4, the problem does notconstrain the initial orientation of the astronaut’s velocity vector withrespect to the station’s velocity vector. To remove ambiguity, theproblem should state that when the astronaut is at O′ (when r = 0),she starts moving in the same direction as IvO′/O.

vi) p. 331, Problem 8.4, the solution will include v0 also.

vii) [First printing only] p. 331, Problem 8.7. Delete the word “mass-less” from the first line.

viii) [First printing only] p. 332, Problem 8.8, the first letters of itemizedproblem statements (a) and (b) should be capitalized.

ix) [First printing only] p. 333, Problem 8.10, the first line should read“Consider a disk of radius R = 1 m that is...”

Chapter 9:

i) On page 352, the second line should read, “its signed magnitude hG,which can be positive or negative depending on the sign of the angularvelocity. We do so...”

ii) [First printing only] On p. 365, in the middle of the page, the phrase“...fixed, implying that IaP/O = 0 in Eq. (9.30)” should read “...fixed.

Although IaP/O 6= 0, it is perpendicular to the ramp, implying rQ/G×IaQ/O = 0 in Eq. (9.30), where Q is replaced by P here.”

iii) [First printing only] On p. 384, the sentence before Eq. (9.61) shouldread “...two unknowns θ and IωB.”

iv) [First printing only] On p. 389, all of the expressions for the momentabout the center of mass, MG, are missing the unit vector in the b3

direction, which should appear at the end of each equation (4 times).

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v) [First printing only] On p. 399, Problem 9.5. In figure, the distancel should go to the center of the bar, not the edge. Also, the arrow onθ should be reversed.

vi) [First printing only] On p. 401, Problem 9.7, the area moment ofinertia should be J = a4/12.

vii) [First printing only] On p. 405, Problem 9.19 the equation of motionfor the motor has an incorrect sign on the motor torque. It should read:

IGθ = −bθ + ki.

Also, in text for part (b) it should read “the inductance L to 0.75 H”rather than “the inductance h to 0.75 H”.

Chapter 10:

i) [First printing only] On p. 418, the second-to-last equation shouldstart “β − θ2...”

ii) [First printing only] On p. 422 the third unnumbered equation from

the top should have√x2 + y2 in the denominator instead of x2 + y2

(two times).

iii) [First printing only] On p. 430, the paragraph following Eq. (10.37),in the second-to-last sentence, Cij = bi · ej should be ICBij = bj · ei

iv) [First printing only] On p. 439, in the first row of Eq. (10.50) andin the first of the following unnumbered set of three equations, csc θshould be sec θ (two times).

v) [First printing only] On p. 448, the second and third unnumberedequations, and the line between them, should read as follows:

ICBij = ei · bj i, j = 1, 2, 3.

Using the fact that bj = ej prior to the rotation, we can substitutefor bj from Eq. (10.60) to obtain

ICBij = ei · (ej cos θ − (ej × k) sin θ + (ej · k)k(1− cos θ)) .

vi) [First printing only] On p. 448, the last unnumbered equation

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should read

ICB11 = cos θ + k21(1− cos θ)ICB12 =−k3 sin θ + k1k2(1− cos θ)ICB13 = k2 sin θ + k1k3(1− cos θ)ICB21 = k3 sin θ + k1k2(1− cos θ)ICB22 = cos θ + k22(1− cos θ)ICB23 =−k1 sin θ + k2k3(1− cos θ)ICB31 =−k2 sin θ + k3k1(1− cos θ)ICB32 = k1 sin θ + k3k2(1− cos θ)ICB33 = cos θ + k23(1− cos θ).

vii) [First printing only] On p. 449, Eqs. (10.63)–(10.66) should read

θ= 2arccos

√

1 + ICB11 + ICB22 + ICB33

2

k1 =

ICB32 − ICB232 sin θ

k2 =ICB13 − ICB31

2 sin θ

k3 =ICB21 − ICB12

2 sin θ.

viii) [First printing only] On p. 462, Problem 10.7, change “when itreaches the bottom of the funnel” to “when it reaches a height of h/2.(Note that a solution exists only for sufficiently small s0.)”

ix) [First printing only] On p. 463, Problem 10.12 is worded incorrectly.The proper wording is “...Sensors on the plane measure the positionand velocity of the released vehicle relative to the C-130, rP/O′ andBvP/O′ . The C-130 telemeters...”

Chapter 11:

i) [First printing only] On p. 467, in the second-to-last sentence ofExample 11.1, “Because m = p

∫B dV ...” should be “Because m =

ρ∫B dV ...”.

ii) [First printing only] On p. 471, the first equation at the top of thepage has two sign errors. It should read:

0 = −2mP l2θφ cosφ−mP l

2θ sinφ+ hφ

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iii) [First printing only] On p. 471, Eq. (11.5) has a sign error. It shouldread

θ + 2θφ cotφ− hφ

mP l2 sinφ= 0

iv) On p. 488, Example 11.9 - while there is no explicit error in thisexample, it should be pointed out that the constant of motion definedby Eqs. 11.31 and 11.33 can only be constant if Ω is not assumed tobe constant (this is not a stated assumption in the example, but isone that students may tend to make based on previous examples inthe chapter). So, if integrating the equations of motion give by 11.30and 11.32, and treating Ω as constant, you will get a slightly incorrectmotion, and the constant will not be conserved. In order to conservethe constant, you must integrate the complete system, including theexpression for Ω that can be derived from 11.31:

Ω = −ψ sin (θ)− ψθ cos (θ)

There also exists a second constant of motion, obtained by consideringthe same angular momentum balance, but using I frame components,in which case a zero exists in the right-hand side in the e3 component,and the quantity:

I1ψ cos2 (θ) + I2

(Ω + ψ sin (θ)

)sin (θ)

can be shown to be conserved. Just as for the original constant, thisquantity is also only conserved when Ω is allowed to vary.

v) [First printing only] On p. 492, the first line after Eq. (11.43) shouldbe ”As long as ω2

n > 0 . . . ”

vi) [First printing only] On p. 494, Eq. (11.45), csc θ should be sec θ.

vii) [First printing only] On p. 496, the unnumbered equation before(11.53) should have dm instead of i (four times).

viii) On p. 497, Eq. (11.54) contains an error. The equation is for rotationabout a point Q, and so the moment term on the right-hand sideshould be about Q: [MQ]B.

ix) [First printing only] On p. 501, Example 11.13, in the first line of thelong equation at the bottom of the page, the leading I2 in the secondrow of the first matrix should be deleted, so it reads d

dt(I2ψ sin θ+ h).

x) On p. 502, Example 11.13, just as with the comment above for Exam-ple 11.9, the constant of motion in Eq. 11.63 will only be constant if his allowed to be time-varying, and if the numerical integration includesa term for h derived from this equation.

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xi) [First printing only] On p. 509, Example 11.16, first equation onpage should be

rO/G2= − l2

2c1 − l1b1 −

h

2a1

xii) On page 516, in the first line below the two unnumbered equationsbelow Equation (11.80), the first inline equation should be ω1(t0) =−ω0.

xiii) On page 516, in Equation (11.81), the first term should be negativeand the second term positive, so it reads

IhG = −IOω0 cosωntb1 + IOω0 sinωntb2 + Jωsb3

xiv) On page 518, in the first line below the third unnumbered equation,the inline equation should be ψ = hG/IO.

xv) [First printing only] On p. 529, in Problem 11.11, Figure 11.30,arrow on rotary pendulum for angle ψ should point the other way.

xvi) On p. 530, in Problem 11.13(d), the given spring constant results inphysically non-intuitive and non-illustrative system motion. A stifferspring (at least 10 N/m) produces much better results.

xvii) [First printing only] On p. 531, in Problem 11.17, Figure 11.34makes it look as if the cord is attached to an arbitrary point along therod, whereas it should be attached at the end of the rod (point A).

xviii) [First printing only] On p. 533, in Problem 11.19, the value givenfor D is actually the value of D/IT , so that the problem should read“...D/IT = 0.5...”

xix) On p. 533, in Problem 11.19, the initial condition for ω2 should begiven as ω2(0) = 2 rad/s.

Chapter 12:

i) [First printing only] On p. 540, in the first unnumbered equation, allof the “cosωdt” terms should be replaced by “cos(ωdt)” (three times)and all of the “sinωdt” terms should be replaced by “sin(ωdt)” (threetimes).

ii) [First printing only] On p. 540, in the second line of the sec-ond unnumbered equation, the “cos 2ωdt” term should be replaced by“cos(2ωdt)” and the “sinωdt” term should be replaced by “sin(2ωdt)”.

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iii) [First printing only] On p. 541, the last equation on p. 541 shouldbe

x(t) =g

ω20

+ e−ζω0t

(√2glω2

0 − gζω20

√1− ζ2

sin(ωdt)−g

ω20

cos(ωdt)

).

iv) [First printing only] On p. 553, the first of the two unnumberedequations before Eq. (12.19) should be “y1 = y2”.

v) [First printing only] On p. 558, the final unnumbered equation andthe preceding sentence should be “...from the initial conditions y(0)and y(0):

y(0) = c1v1 + c2v2

y(0) =λ1c1v1 + λ2c2v2.”

Chapter 13:

i) [First printing only] On p. 583, bottom of the page, boxed equationshould read

fk(r1/O, r2/O, . . . , rN/O, t) = 0, k = 1, . . . ,K.

ii) [First printing only] On p. 591, the right side of the first unnum-bered equation should be

NC∑i=1

N∑j=1

F(act)j ·

I∂rj/O

∂qi︸ ︷︷ ︸,Qi

δqi,

Appendix A:

i) [First printing only] On p. 625, the sentence following DefinitionA.2 should read “Figure A.2 shows examples of continuous functionswith continuous and discontinuous derivatives.”

ii) [First printing only] On pgs. 628-630, Examples A.1, A.2, and A.3should be in gray background like all other examples.

Appendix C:

i) [First printing only] On p. 647, Example C.1 should have graybackground like all other examples.

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ii) [First printing only] On p. 658, fifth line of text, “TSPAN” shouldbe “TSPAN”.

iii) [First printing only] On p. 658, seven lines from the bottom, low-ercase “l” should be uppercase “L” (two times).