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Unformatted text preview: C/1 INTRODUCTION Appendix G contains an abbreviated sum
mary and reminder of selected topics in basic
mathematics which ﬁnd frequent use in me
chanics. The relationships are cited without
proof, The student of mechanics will have fre
quent occasion to use many of these relations,
and he or she will be handicapped if they are
not well in hand. Other topics not listed will
also be needed from time to time. OF MATH WTIS As the reader reviews and applies math
ematics, he or she should bear in mind that
mechanics is an applied science descriptive of
real bodies and actual motions. Therefore, the
geometric and physical interpretation of the
applicable mathematics should be kept clearly
in mind during the development of theory and
the formulation and solution of problems. C/2 PLANE GEOME TRY 1. When two intersect ing lines are, respec
tively, perpendicular
to two other lines, the angles formed by each pair are equal. 91 = 02 —
. Similar triangles T
x h — y
_ 2 — h
b h T
A
b . Any triangle 1
Area = %bh h 4. Circle Circumference = 27v
Area = erZ Arc length 5 = r0
Sector area = %r2() . Every triangle inscribed within a semicircle is a right triangle,
91 01 + 02 = Tr/Z
6. Angles of a triangle 9
2
01 + 62 + ()3 2 180°
04 : 91 + ()2
91 03 04 688 SELECTED TOPICS OF MATHEMATICS Appendix C T/3 SOLID GEOMETRY 3. Rightcircular cone
1. Sphere
4 3 Volume = ﬁnrzh L
Volume = inf ‘ Lateral area = an h
Surface area : (1an L = /,.2 + h2 i
L r 4 2. Spherical wedge F 1 4. Any pyramid or cone T
0
Volume 2 §r30 ﬂ Volume = éBh h where B : area of base J
7/4 ALGEBRA
1. Quadratic equation 4. Cubic equation
ax2+bx+c=0 x3:Ax+B
x = —_b i W, b2 2 4ac for real roots Let p : A)3, £71: B/2.'
Case I: q2 — p" negatlve (three roots real and
2. Logarithms distinct)
b"=y,x=logby cosu=q/(p/E),0<u<180°
Natural logarithms x1 = 2/13 cos (u/ 3)
b 2 e : 2.718 282 x2 : 2/13 cos (u/3 + 120°) e"2y,x=logpy=lny log (ab) 2 log a + log 1) x3 2 2”: COS (ll/3 + 240 ) 10g (a/b) 2 log a — log 1) Case 11:
log (1/n) = flog n log a" = n log a log 1 = 0 x1 = (q + Wﬂﬂi + (q _ WPB logw x = 0.4343 In x q2 , p3 positive (one root real, two
roots imaginary) Case III: q2 — p3 = 0 (three roots real, two 3. Determinants roots equal) 2nd order
x1 2 2(11/3, 362 = x3 I _q1/3
(11 bl = a b a b .
a2 b2 1 A z 1 For general cubic equation
3rd order x3 + ax2 + bx + c = 0
a1 b1 cl Substitutex = x0 — (1/3 and get x03 =Ax0 + B.
a2 b2 C2 = + a1b2c3 + a2b301 + (13bIC2 Then proceed as above to ﬁnd values of x0 from
as b3 C3 —a3b201 — 02b103  (11b302 which 36 = x0 — (1/3. “ Article C/6' TRIGONOMETRY 689 C/5 ANALYTIC GEOMETRY
1. Straight line 3. Parabola ‘— ————————— x
y = a + mx
2. Circle 4. Ellipse
y y y
l  I
l 
l

l ———x
——x :_
I
b 2 2
i x—2+ i; = 1
a
x2 + 3'2 = r2 L—‘a———i——— x
(x—a)2+ (y b)2=r2 5. Hyperbola C/6 TRIGONOMETRY 1. Deﬁnitions 1 II J III IV
sin 0 : a/c csc () = c/a sin U 7i —[ + 7 ~
cos (I : b/c sec () : c/b (I —*
._ ‘ _ + i + tan () , (1/1) not (I — b/a COS _L
2. Signs in the four quadrants tan 0 + , 4 7 r—
(+) I II (+) csc (I + + v 4] ,
a 9 0 o _ _
«JA(+)(_)X1¥ (_)ﬂ:l¥ $0?) SEC U + J— +
t (1 l— , + ,
III (_) (_) IV CO 690 SELECTED TOPICS OFMATILEMATICS .. ,, WeeeeeApe C . Miscellaneous relations 4. Law of Sines
sin2 0 + cos2 0 = 1 g _ sinA
1+tan20=sec20 b_sinB 1 + cot2 (9 = csc2 6 0 sin E = ‘/%(1 — cos 6) c a 0 cos—2 \/%(1 +cos 0) D _
2 b ’—
sin 20 = 2 sin 9 cos 6 5 Law of cosines
cos 26 = cos2 9 — sin2 0 '
sin(aib)=sinacosbicosasinb c2=a2+b2—2abcosC
cos(a:b)=cosacosb¢sinasinb cz=a2+b2+2abcosD
1/7 VECTOR OPERATIONS _————> Q
P
1. Notation. Vector quantities are printed in boldface type, and scalar
quantities appear in lightface italic type. Thus, the vector quantity
V has a scalar magnitude V. In longhand work vector quantities R
should always be consistently indicated by a symbol such as Z or Q
V to distinguish them from scalar quantities. p
2. Addition . Direction cosines l, m, n are the cosines of the angles between V Triangle addition P + Q = R Parallelogram addition P + Q = R
Commutative law P + Q = Q + P
Associative law P + (Q + R) = (P + Q) + R, p . Subtraction PQ Q
P—Q=P+V® Unit vectors i, j, k
V : in + Vyj + Vzk
where IV! = V : t/VI2 + sz + V22 and the x, y, zaxes. Thus,
1: Vx/V m = Vie/V n = VZ/V
so that V = V(li + mj + nk) and l2+m2+n221 Article C/ 7 VECTOR OPERATIONS 691 . Dot or scalar product
P  Q = PQ cos 0 This product may be viewed as the magnitude of P multiplied by P
the component Q cos (I of Q in the direction of P, or as the magnitude
of Q multiplied by the component P cos () of P in the direction of Q. Commutative law P  Q = Q  P % From the deﬁnition of the dot product
ii=jj =kk= 1
ij =ji=ik=ki=jk=k'j=0 p. Q 2 (Pxi + P_‘.j + sz)(Qxi + Qyj + (9210
:PxQx+Py'Qy+PZQZ PP =sz +Py2 +1322 It follows from the deﬁnition of the dot product that two vec
tors P and Q are perpendicular when their dot product vanishes,
P ' Q = 0. The angle (1 between two vectors P1 and P2 may be found from
their dot product expression P1 P2 = P1P2 cos (1, which gives P1~P2 P1IP2I + PIVPZV + PIZPZZ cos (1  P1P2 , HR; = [112 + mlmZ + ”1’12 where l, m, n stand for the respective direction cosines of
the vectors. It is also observed that two vectors are perpendicular
to each other when their direction cosines obey the relation [112 + mlmz + 71.1712 2 0.
Distributive law P(Q + R) = PQ + PR Q\——~—/ P . Cross 0r vector product. The cross product P X Q of the two vectors
P and Q is deﬁned as a vector with a magnitude P x Q PxQ =PQ sin() and a direction speciﬁed by the righthand rule as shown. Reversing Q
the vector order and using the righthand rule give Q x P = aPXQ. Cl/ 0 P
Distributive law P x (Q + R) = P x Q + P X R From the deﬁnition of the cross product, using a right—handed \
l
\
\
coordinate system, we get i QxP=—PxQ ixj=k jxk=i kxizj jxi=—k kxj2—i ixk=—j
ixi=jxj=k><k=0 692 SELECTED TOPICS OF MATHEMATICS Appendix C With the aid of these identities and the distributive law, the vector
product may be written P x Q = (Pxi + Pyj + sz) x (Q,i + QVj + sz)
: (Psz _ PzQy)i + (PZQ): — PerU + (PrQ) — Per)k The cross product may also be expressed by the determinant i j k
PxQ= PX P}, PZ
Qx Q.v Q2 8. Additional relations Triple scalar product (P X Q)  R = R(P X Q). The dot and cross
may be interchanged as long as the order of the vectors is main
tained. Parentheses are unnecessary since P x (Q  R) is meaningless
because a vector P cannot be crossed into a scalar Q  R. Thus, the
expression may be written PxQR=PQXR The triple scalar product has the determinant expansion P, P,. P:
PXQ'R: Qx Q)‘ Q:
Rx R). R2 Triple vector product (P x Q) X R = —R X (P X Q)
= R x (Q x P) Here the parentheses must be used since an expression P x Q x R
would be ambiguous because it would not identify the vector to be crossed. It may be shown that the triple vector product is equivalent
to (PXQ)><R=RPQ—R~QP
or Px(QxR)=PRQ—PQR The ﬁrst term in the ﬁrst expression, for example, is the dot product
R  P, a scalar, multiplied by the vector Q. 9. Derivatives of vectors obey the same rules as they do for scalars. %:P:Pxi+1§\j+nk
d(Pu):Pii+Pu dt d(PQ)_  _ dt —PQ+PQ
—d(Pd:Q):PxQ+PxQ Article C79 DERIVATIVES 693
W 10. Integration of vectors. If V is a function of x, y, and z and an
element of volume is dr = dx dy dz, the integral of V over the
volume may be written as the vector sum of the three integrals of
its components. Thus, der=iJder+ijydr+kazdr C/8 SERIES
(Expression in brackets following series indicates range of conver
gence.) _ 1 __ _ 2 ‘
(1ix)n:1:nx+n(nsz2:n(_n_13)'(n___zx3+.n [x2<1]
' — x3 x_5 x_7 ... 2<x
Slnx—x—§+5!—7!+ [x ] x2 x4 x6 2
cosx=1—ﬁ+4—!~a+n ”(30]
‘ ex _ e’X x3 x5 x7 2
smhxz 2 =x+§+5+ﬁ+~ [x<oc] e’ + e“ x2 x4 x6 2
COthz 2 21+E+E+a+'u [x<oc]
fix) =%)+ 2 ancosTZZ—er 2: busin$ n=1 11:1 [ l
where an : % J, f(x) cos €95 dx, 1),, = % f f(x) sin % dx
7/ —z [Fourier expansion for —l < x < l] C/9 DERIVATIVES «e
ﬂ_ "*1 d(uv)_ @+ Q U _ dx dx
dx ~nx ’ dx _udx de’ dx _ 02 lim sin Ax : sin dx : tan dx = dx
Ax~>0 lim cos Ax = cos dx = 1 AXHO
dsinx dcosx . dtanx .
= cosx = ~smx = seczx
dx ’ dx ’ dx
d sinh x d cosh x . d tanh x 2
—— = cosh x, — = smh x, _—— : sech x
dx dx dx m 694 SELECTED TOPICS OF MATHEMATICS Appendix C ‘1 0 INTEGRALS ‘ XI! 1
I x d} _ n +1
‘(lx
J— : Inx
x
I / 2 /' .1}
I Va + bx (1x 4 % V'm + bx)
, 2 ‘ /___3
. xv/a + bx (1x : 15b2 (3bx — Zak, (a + bx)
. a ,r 2 v a 9 f—‘ﬂ—T
J xwa + bx (1x 2 105b" (80* — IZabx + 15b~x')V/(a + bx)‘
J‘ dx , ZV/a + bx
V/a + bx b
‘V/a+xd ‘ —’11+x”b—x+(a+b)sin'1 a+x
JV/bx x— V \/ a+b
1
Jde =—;Ia+bx—aln(a+bx)l
a + bx b“
J’xdx i<a+bx)1”(a+bx_ a )
(a + bx)” _ b3 2 — n 1 — n,
/— /' b
{ dx . : #— tan 1x» ab 01‘ I, 1 tanh”1xV a
 a + bx2 v/ab a \/ —ab (1
xdx 1
2 —1 ( + b 2)
[a + bx2 2b n a x
J V/x" i (1"2 dx = “xv/x2 i a2 i a2 In (x + \/x2 i (12)]
JV/az — x2 dx : é (xv/oz2 , x2 + a2 sin"1:—:)
fag/a2 — x2 dx = —;‘;\/(a2 — x2)3
2
foV/az — x2 dx = £ /(a2 — x2)3 + 1(xv/a2 — x2 + a2 sin’1£>
4 8 a
[JCS/0.2 — x2 dx = — ;—')(x2 + Eda/(a2 — x2)3
r—+  b 1 *1 . b + 20x ‘
[:w_—+:l:ln(va+bx+cx2+xv/c+ r) or ,—sm1(—~2=._~
\/a + bx + cxZ V/C 2V/c v’ —c V/b  4ac f—dx—:1n(x+\/x2:a2) Article CH 0 INTEGRALS 695
0 sin x (1x: #005 x cosxdx—sinx . lx
:sm —
—‘’——x a
x_d_x ,, .)
: V'x“ , a“
I _ iv'azfxz
Jxv’ 3—50 dx : JW/(xz :03)3
. 2 4
1 , .ia , 1 a «.
JXZV/  :a 3 (1x : 4—1 V/(xl : (1‘)“ + gag x1 t a“) — E In (1c V/3cZ t (13) 1 1+sinx
lvsinx sin 2x sin x dx— ‘ f 2 4
J_C sin 2x
2 4 COS“ x dx ~ sing x
2 sinx cos xdx — sinh x dx — cosh x
cosh x dx — sinh x tanh x dx — 1n cosh x xe“"d : ax  1)
:e’“(a sin px — p cos we)
a2 + p2 e"‘(a cos px + p sin px)
a2 + p2 e‘” cos px dx # J
J
J
J
J
J
J
Jinxdx; xlnxix
J
J
J
J 696 SELECTED TOPICS OF MATHEMATICS Appendix C
m . ‘ eux . . 2
e’” 3an x dx : 2 (a 51113 x sm 2x + —)
4 + a (1
e111 2
e‘” cosgxdx: 2(acos3x+s1r12x +—
4 + a a
etlr a
e“ 5m x cos x dx : ., — sm 2x ,_ cos 2x)
4 + a“ 2
. , cos x . .
sm3xdx= , 3 (2+s1n3 x)
, sin x .
c053 x dx = (2 + cos3 x) cos5x dx : sin x — § sin“ x + % sin5x sinx~xcosx R R
g a
m {3 R R Q Q. R R
H cosx+xsinx
xzsinxdx=2xsinx—(x3—2)c0sx xgcosxdx:2xcosx+(x2—2)sinx dy 2 3/2
[1 3 (an) ] ER%%\R%%ER I)” Z L33“
Radius of dx2
curvature (1,: 2 3‘2
,2 + _.
: (10) ]
[Jr/r : _—»'2—‘—
dr dzr
2 __ _ _
r + 2 (d0) r d62 Article C/l 1 N EWTON’S METHOD FOR SOLVING INTRACTABLE EQUATIONS 697
W C/11 NEWTON ’8 METHOD FOR SOLVING
IN TRACTABLE EQUATIONS Frequently, the application of the fundamental principles of
mechanics leads to an algebraic or transcendental equation which
is not solvable (or easily solvable) in closed form. In such cases, an
iterative technique, such as Newton’s method, can be a powerful
tool for obtaining a good estimate to the root or roots of the equation. Let us place the equation to be solved in the form f(x) : 0. The
apart of the accompanying ﬁgure depicts an arbitrary function ﬁx)
for values of x in the vicinity of the desired root x,” Note that x, is
merely the value of x at which the function crosses the xaxis. Suppose
that we have available (perhaps via a handdrawn plot) a rough
estimate x1 of this root. Provided that x1 does not closely correspond
to a maximum or minimum value of the function f (x), we may obtain
a better estimate of the root x,‘ by projecting the tangent to f (x) at [(x) Tangent to
ﬁx) at x = x1 l (a) (b) (C)
x1 such that it intersects the xaxis at x2. From the geometry of the
ﬁgure, we may write tan (i : f'(x1) 2 where f ‘(x 1) denotes the derivative of f (x) with respect to x evaluated
at x = x]. Solving the above equation for x; results in The term —f(xl)/f'(x1) is the correction to the initial root estimate
x1. Once x2 is calculated, we may repeat the process to obtain x;;,
and so forth. Thus, we generalize the above equation to ﬂack)
x _ ~— x, , _,
I‘ l A / (xi)
where
xi. 1 2 the (k + 1)th estimate to the desired root x,‘ 698 SELECTED TOPICS OF MATHEMATICS Appendix C xk = the kth estimate to the desired root x,
ﬂack) : the function f(x) evaluated at x : xi. f’lxk) = the function derivative evaluated at x = x}, This equation is repetitively applied until flxk . 1) is suﬁiciently close
to zero and xk1 E xk. The student should verify that the equation is
valid for all possible sign combinations of xi, [(30,], and Wm). Several cautionary notes are in order: 1. Clearly, f’(x,,) must not be zero or close to zero. This would mean, as restricted above, that x], exactly or approximately
corresponds to a minimum or maximum of fix). If the
slope ﬂag.) is zero, then the slope projection never intersects
the xaxis. If the slope ﬂag.) is small, then the correction to
xi. may be so large that x,” is a worse root estimate than
xk. For this reason, experienced engineers usually limit the
size of the correction term; that is, if the absolute value of
f(x,\,)/f’(x,,) is larger than a preselected maximum value. the
maximum value is used. . If there are several roots of the equation ﬁx) — 0, we must be in the vicinity of the desired root x,. in order that the
algorithm actually converges to that root. The bpart of the
ﬁgure depicts the condition that the initial estimate x1 will
result in convergence to x5, rather than an]. . Oscillation from one side of the root to the other can occur if, for example, the function is antisymmetric about a root
which is an inﬂection point. The use of onehalf of the
correction will usually prevent this behavior, which is depicted
in the c—part of the accompanying ﬁgure. Example: Beginning with an initial estimate ofxl : 5, estimate the
single root of the equation 9" — 10 cos x e 100 : 0. The table below summarizes the application of Newton’s method to the given equation. The iterative process was terminated when
the absolute value of the correction —f(x;,)/f’(x,,) became less than 10 6.
, , ﬁx.)
k xk f‘xk) flxk) x1e»! , xi 7 ’ fwd“
1 5.000 000 45.576 537 138.823 916 770.328 305
2 4.671 695 7.285 610 96.887 065 70.075 197
3 4.596 498 0.292 886 89.203 650 70.003 283
4 4.593 215 0.000 527 88.882 536 770.000 006
5 4.593 209 72(10 8‘) 88.881 956 2.25 (10 1") Article 0/12 SELECTED TECHNIQUES FOR NUMERICAL INTEGRATION 699 0/12 SELECTED TECHNIQUES FOR NUMERICAL INTEGRATION 1. Area determination. Consider the problem of determining
the shaded area under the curve y = f(x) from x = a to x = b as
depicted in the apart of the ﬁgure and suppose that analytical
integration is not feasible. The function may be known in tabular
form from experimental measurements or it may be known in
analytical form. The function is taken to be continuous within the
interval a < x < b. We may divide the area into n vertical strips,
each of width Ax = (b — a)/ n, and then add the areas of all strips
to obtain A = f y dx. A representative strip of area A l is shown with
darker shading in the ﬁgure. Three useful numerical approximations
are cited. In each case the greater the number of strips, the more
accurate becomes the approximation geometrically As a general
rule, one can begin with a relatively small number of strips and
increase the number until the resulting changes in the area ap
proximation no longer improve the desired accuracy. ((2) Rectangular
Ai =ymAx yi+1 .Ym AZIydeEymAx ___l I. Rectangular [Figure (b)] The areas of the strips are taken
to be rectangles, as shown by the representative strip whose height
ym is chosen visually so that the small crosshatched areas are as
nearly equal as possible. Thus, we form the sum Eym of the effective
heights and multiply by Ax. For a function known in analytical
form, a value for ym equal to that of the function at the midpoint
x, + Ax/ 2 may be calculated and used in the summation. 700 SELECTED TOPICS OF MATHEMATICS Appendix C
W II. Trapezoidal [Figure (c)] The areas of the strips are taken
to be trapezoids, as shown by the representative strip. The area A,
is the average height (yi + yM)/ 2 times Ax. Adding the areas gives
the area approximation as tabulated. For the example with the
curvature shown, clearly the approximation will be on the low side.
For the reverse curvature, the approximation will be on the high
side. Trapezoidal Ai:yi+gi+ldx A =fydx§(=y§9+y1 +312 + +yn~1 +%)Ax yi+1 Parabolic 1
AA = 3 01+ 4yi+1+yi2mx yi” A :1. ydx E éty0+4y1+2y2+4y3+2y4 + +2yn72+4yn71+ynMx III. Parabolic [Figure (d)] The area between the chord and
the curve (neglected in the trapezoidal solution) may be accounted
for by approximating the function by a parabola passing through
the points deﬁned by three successive values of y. This area may be
calculated from the geometry of the parabola and added to the
trapezoidal area of the pair of strips to give the area AA of the pair
as cited. Adding all of the AA’s produces the tabulation shown, which
is known as Simpson’s rule. To use Simpson’s rule, the number n of
strips must be even. Example: Determine the area under the curve y = x ‘/1 + x2 from
x = 0 to x = 2. (An integrable function is chosen here so that the
three approximations can be compared with the exact value, which isA = fgx J1 + x2 dx = a1 + 3653/23 = %(5/5 — 1) = 3.393 447.) Article C/12 SELECTED TECHNIQUES FOR NUMERICAL INTEGRATION 701
m AREA APPROXIMATIONS NUMBER OF ——~——————————————————
SUBINTERVALS RECTANGULAR TRAPEZOIDAL PARABOLIC 4 3.361 704 3.456 731 3.392 214 10 3.388 399 3.403 536 3.393 420 50 3.393 245 3.393 850 3.393 447 100 3.393 396 3.393 547 3.393 447 1000 3.393 446 3.393 448 3.393 447 2500 3.393 447 3.393 447 3.393 447 Note that the worst approximation error is less than 2 percent, even
with only four strips. 2. Integration of firstorder ordinary differential equations.
The application of the fundamental principles of mechanics fre quently results in differential relationships. Let us consider the ﬁrst d . . .
order form 3”: = /(t), where the function f(t) may not be readily
integrable or may be known only in tabular form. We may numerically
integrate by means of a simple slopeprojection technique, known as Euler integration, which is illustrated in the ﬁgure. dy /
S] = I t)
ope ( y(t)
= Accumulated
I algorithmic
l error
 
Slope = f0] l l
,  a l :“
~ I l [ya I
l I yz l  —>etc
WI  I 
 I I I
 l  l
l L  I t
1’1 t2 t3 t4 Beginning at t1, at which the value yl is known, we project the
slope over a horizontal subinterval or step (t; — t1) and see that
yg : yl + f(t1)(t2 — t1). At t2, the process may be repeated beginning
at yg, and so forth until the desired value of t is reached. Hence, the
general expression is 30:41: yk + f(tkl(tk1—tk) If y versus t were linear, i.e., if f(t) were constant, the method
would be exact, and there would be no need for a numerical approach
in that case. Changes in the slope over the subinterval introduce 702 SELECTED TOPICS 0F MATHEMATICS Appendix C W error. For the case shown in the ﬁgure, the estimate yg is clearly
less than the true value of the function y(t) at t2. More accurate
integration techniques (such as Runge—Kutta methods) take into
account changes in the slope over the subinterval and thus provide
better results. As with the areadetermination techniques, experience is helpful
in the selection of a subinterval or step size when dealing with
analytical functions. As a rough rule, one begins with a relatively
large step size and then steadily decreases the step size until the
corresponding changes in the integrated result are much smaller
than the desired accuracy. A step size which is too small, however,
can result in increased error due to a very large number of computer
operations. This type of error is generally known as “roundoff error”,
while the error which results from a large step size is known as
algorithm error. Example: For the differential equation % = 5t with the initial condition y = 2 when t = 0, determine the value of y for t = 4.
Application of the Euler integration technique yields the fol
lowing results: NUMBER OF
SUBINTERVALS STEP SIZE y at t = 4 PERCENT ERROR
10 0.4 38 9.5
100 0.04 41.6 0.95
500 0.008 41.92 0.19
1000 0.004 41.96 0.10 This simple example may be integrated analytically. The result is
y = 42 (exactly). ...
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