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ENGINEERING PROBLEMS
ILLUSTRATING MATHEMATICS
The quality -of the materials used in the manufacture
of this book is governed by continued postwar shortages.
Engineering Problems
Illustrating Mathematics
A Project of the Mathematics Division of the Society
for the Promotion of Engineering Education
JOHN W. CELL
Chairman of Committee
Associate Professor of Mathematics in the College
of Engineering, North Carolina State College
FIRST EDITION
FIFTH IMPRESSION
McGRAW-HILL BOOK COMPANY, INC.
NEW YORK AND LONDON
1943
ENGINEERING PROBLEMS ILLUSTRATING MATHEMATICS
COPYRIGHT, 1943, BY THE
EDITORIAL COMMITTEE
J. Wj CELL, W. C. BRENKE,
G. E. MOORE, C. C. TORRANCE
PRINTED IN THE UNITED STATES OP AMERICA
All rights reserved. This book, or
parts thereof, may not be reproduced
in any form without permission of
the Editorial Committee.
CONTENTS
PAQB
FOREWORD: A PROJECT OF THE MATHEMATICS DIVISION OF THE SOCIETY
FOR THE PROMOTION OF ENGINEERING EDUCATION v
PREFACE ix
INTRODUCTION BY ROBERT E. DOHERTY xi
PART I
COLLEGE ALGEBRA
RADICALS AND EXPONENTS 1
QUADRATIC EQUATIONS 2
COMPLEX NUMBERS 5
LOGARITHMS 5
SIMULTANEOUS EQUATIONS 7
VARIATION 8
BINOMIAL THEOREM 9
PROGRESSIONS 10
THEORY OF EQUATIONS 12
DETERMINANTS AND SIMULTANEOUS LINEAR EQUATIONS 15
PART II
TRIGONOMETRY
FUNDAMENTAL IDENTITIES 22
RIGHT TRIANGLES 23
OBLIQUE TRIANGLES 25
MULTIPLE ANGLE IDENTITIES 27
GRAPHS OF THE TRIGONOMETRIC FUNCTIONS. LINE VALUES 28
TRIGONOMETRIC EQUATIONS 31
COMPLEX NUMBERS. DEMOIVRE'S THEOREM 33
PART III
ANALYTIC GEOMETRY
ELEMENTARY FORMULAS 37
STRAIGHT LINES 38
CURVE SKETCHING 43
GRAPHS OF THE CURVES: y = ax n 46
CIRCLES 49
PARABOLAS 51
ELLIPSES 54
vii
viii CONTENTS
PAGE
HYPERBOLAS 55
ROTATION OF AXES 57
GRAPHS OF SINE WAVES 58
EXPONENTIAL AND LOGARITHMIC GRAPHS 61
HYPERBOLIC FUNCTIONS 63
POLAR COORDINATES 64
PARAMETRIC EQUATIONS 66
EMPIRICAL CURVES 71
ELEMENTARY FORMULAS IN SOLID ANALYTIC GEOMETRY 75
SURFACES IN SOLID ANALYTIC GEOMETRY 79
PART IV
DIFFERENTIAL CALCULUS
USES AND INTERPRETATIONS OP DERIVATIVES 83
TANGENT AND NORMAL 86
CURVE SKETCHING. MAXIMUM, MINIMUM, AND FLEX POINTS .... 90
MAXIMUM AND MINIMUM PROBLEMS 93
VELOCITY AND ACCELERATION. RECTILINEAR MOTION 100
VELOCITY AND ACCELERATION. CURVILINEAR MOTION 102
DIFFERENTIALS 105
CURVATURE 106
INDETERMINATE FORMS 107
RELATED RATES 110
PARTIAL DIFFERENTIATION 110
PART V
INTEGRAL CALCULUS
CONSTANT OF INTEGRATION 114
EVALUATION OF DEFINITE INTEGRALS 122
DEFINITE INTEGRALS. SIMPSON'S RULE AND TRAPEZOIDAL RULE. . . 125
AREAS. AVERAGE ORDINATE 127
LENGTH OF ARC. AREAS OF SURFACES OF REVOLUTION 131
VOLUMES 133
FIRST AND SECOND MOMENTS 135
TOTAL FORCE 140
WORK 141
OTHER APPLICATIONS OF INTEGRAL CALCULUS 143
INFINITE SERIES 149
HYPERBOLIC FUNCTIONS 153
INDEX 157
ANSWERS 161
PREFACE
The problems in this collection are designed to give to the freshman
and sophomore engineering student some understanding of the uses of
mathematics in junior and senior engineering courses, and hence of the
necessity of a thorough foundation in mathematics. These problems
should help to answer the eternal question, "Why study this topic?"
The early training of an engineering student should contain an
appreciation of how engineering problems are translated into mathe-
matics problems. The Report on the Aims and Scope of Engineering
Curricula (in The Journal of Engineering Education , vol. 30, March,
1940, pp. 563-564) states that
"The scientific-technological studies should be directed toward:
"1. Mastery of the fundamental scientific principles and a command
of basic knowledge underlying the branch of engineering which the
student is pursuing. This implies:
" a. grasp of the meaning of physical and mathematical laws, and
knowledge of how they are evolved and of the limitations in
their use;
"b. knowledge of materials, machines, and structures.
"2. Thorough understanding of the engineering method and elemen-
tary competence in its application. This requires:
" a. comprehension of the interacting elements in situations which
are to be analyzed;
"6. ability to think straight in the application of fundamental
principles to new problems; ..."
Another quotation from an address by Dr. Marston Morse (President
of the American Mathematical Society and Chairman of the War
Preparedness Committee), which he delivered to the National Council
of Teachers of Mathematics (December, 1940), will further emphasize
the purposes of this collection:
"Teachers of mathematics should present to their pupils technological
and industrial applications whenever possible. This can and should
be done without abandoning the concept of mathematics as a general
tool."
Most of these problems are suited to direct assignment to the superior
student, and many of them can be assigned even to an average student.
ix
x PREFACE
Some problems, marked with asterisks, would more properly be used in
outline form in the classroom or for bulletin board display.
Some engineering terminology is necessary in stating engineering
problems. The Committee has tried to keep the use of unfamiliar terms
to a minimum and has resorted at times to somewhat crude translations
into nontechnical language. However, many engineering freshmen have
some technical vocabulary derived from a high-school physics course or
from some hobby.
Two groups of important topics in mathematics receive little or no
attention in these problems. One group consists of those topics, such
as variation in college algebra and mensuration in trigonometry, for
which most mathematics texts contain sufficient application problems.
The other group consists of essentially mathematical concepts that form
the groundwork for subsequent material of application.
These remarks by no means imply that mathematics courses should
be solely utilitarian. It is important, furthermore, that these problems
be used as supplementary material and for their avowed purposes. The
necessary drill problems should continue to be of the traditional variety.
This collection will be misused if there is any attempt to teach junior or
senior engineering concepts in the freshman or sophomore mathematics
classroom.
JOHN W. CELL.
NORTH CAROLINA STATE COLLEGE,
May, 1943.
ENGINEERING PROBLEMS
ILLUSTRATING MATHEMATICS
PART I
COLLEGE ALGEBRA
RADICALS AND EXPONENTS
< _ |YP'Y A /PA*"*'"]*
1 ' If = , >how that [(-) -(-) J
v , j 1 \ k-L. i* ^is simplification arises in a
derivation for the theory of flow in a nozzle in thermodynamics.
2. The gas in an engine expands from a pressure pi and volume Vi to a
7?2 fv\\ n
pressure p* and volume v 2 according to the equation = ^ J . (Boyle's
law is the special case of this equation when n = 1.) Solve for Vi in
terms of #2, pi, PZ, and n.
3. The study of a vacuum tube involves the following equations:
I = kJSK, P = El, G = I/E, and R = #//, where / is current in
amperes, E is voltage in volts, R is resistance in ohms, P is power in
watts, and G is the conductance; & is a constant. Determine formulas
for each of the quantities E, /, P, G, and R, respectively, in terms of
each of the other quantities. (There will be 20 such equations in
all.)
4. Simplify the following expressions :
(a) g| [o(ac + 6) + (ax - 26) Q) (ax
1 /1\ (
^ - b (2) (
2 1 /1\ ax -
Note to Teachers: Additional problems of this character may be obtained
by differentiating the answers in any standard table of integrals. Similar
problems may be obtained for use in trigonometric simplification.
1
2 ENGINEERING PROBLEMS
QUADRATIC EQUATIONS
5. The design of a square timber column of yellow pine, of length
20 ft., to support a load of 30,000 lb., is accomplished by the use of an
" approximate straight-line column formula":
~ = 1,000 -
In this problem P = 30,000 lb., a = w 2 (where w is the width of the
1000 Jbs.
, A' i.
,
L_ 4 '_J
r*~*\
300 Ibs./ft. ]
^
- t
1000 Ibs.
4000 Ibs.
FIG. 1.
square column in inches), L length of column = (20) (12) in., and
r = w/A/12 (a property of the shape of the cross section). Substitute
these quantities in the given equation and solve for w correct to the
nearest second decimal.
6. Figure 1 shows a beam that supports its own weight of 300 lb.
per ft. and a concentrated load of 1,000 ib. at a
point 4 ft. from the left end. There is a point x
ft. from the left end where there is no compression
or tension (push or pull) in the beam, x must be
larger than 4 and less than 16 and is a root of the
equation
3,000s - 1,000(3 - 4) - 150z 2 = 0.
Find x.
7. Figure 2 shows a right circular cylinder of
height H and radius r, mounted on a smooth
hemisphere of radius r. The combined body will
FlG * 2 * not tip if the height H is less than the positive
value of H which satisfies the following equation :
( \
(r)
H
Determine this critical value for H.
8. A study of the flow of a gas through a nozzle requires the solution
of the equation
COLLEGE ALGEBRA 5
COMPLEX NUMBERS
17. In electrical engineering it is customary to use the complex
numbers Ej I, and Z to designate, respectively, the voltage, current?
and impedance. If E IZ, find the third quantity (in simplified
form) when given,
(a) E = 100 + j4Q volts, / = 4 + j3 amp.; (j = V^
(6) 1 = 2 + J3iamp., Z = 30 - jl2 ohms.
(c) E = 120 volts, Z = 20 - J20 ohms.
18. Two loads Z\ = 3 + ^4 ohms and Z 2 = 8 ,76 ohms are con-
nected in parallel. Determine an equivalent single load Z from the
relation 1/Z = 1/Zi + 1/Z 2 .
19. If F = 1/Z = G - jB and if Z = R + jX, determine relations
for the real quantities G and B in terms of the real quantities R and X.
and B are called, respectively, the conductance and the susceptance
in the electric circuit.
20. A study of damped forced vibrations requires the solution for x
of the equation
mw 2 x + jwcx + kx P.
m, w t c y k, and P are real constants and j = \/ 1. Show that
mw* + k jwc)
X =
-ww 2 + &) 2 +
Also show that the numerical value of this complex number solution is
given by
P
x =
[(-raw 2 + /c) 2 +
LOGARITHMS
21. A gas expands from a temperature Ti = 60 + 460 = 520 (Ti
is in degrees absolute on the Fahrenheit scale) and pressure p\ = 89.7 Ib.
per sq. in. until the pressure is p 2 = 39.7 Ib. per sq. in. If the relation
between temperature and pressure is
where n = 1.40, determine T 2 by aid of a four-place logarithm table.
22. Given that As = wc v log e (T r 2 /Ti) (an equation that you will
study in thermodynamics), compute As if w = 4, c v = 0.1373, T 2 = 940,
and Ti = 500.
23. The work done in compressing a gas at constant temperature
from a pressure po Ib. per sq. ft. and volume VQ cu. ft. to a pressure
6 ENGINEERING PROBLEMS
pi Ib. per sq. ft. and volume v\ cu. ft. is given by
W = potto log, () ft.4b.
Vt/o/
Determine TF if po = 20 Ib. per sq. in., pi = 35 Ib. per sq. in., and
VQ = 3 cu. ft. vi can be determined from the relation p\vi = p vo.
24. The average molecular weight M of a petroleum fraction may be
approximately determined from its atmospheric boiling point, tC.,
from the equation
logic M = 2.51 logio (t + 393) - 4.7523.
a. Restate this equation in exponential form; i.e., solve for t in terms
of M.
b. Compute M, correct to the nearest integer, for t = 160C.
25. It has long been customary to state the brightness of stars in
magnitudes, where M = 2.5 logio (I/I\) is the magnitude of a star
whose light intensity is /, where /i is the light intensity of a star of the
first magnitude. Note that the less brilliant the star the higher is
its magnitude.
Determine the ratio of the light intensities I/I\ for a star of the fourth
magnitude (M == 4).
1 TyT
26. Show that the equation In i = j^ + In r>, where In x log e x,
E
can be written in the exponential form i = ^ e~ t/RC . This problem is
typical of many derivations in electrical engineering. In the resulting
equation i is the current in amperes in a series circuit t sec. after the
switch is closed, E is the voltage of the battery in the circuit, R is the
resistance in ohms, and C is the capacitance of the condenser in farads.
27. A boat is brought to rest by means of a rope, which is wound
three times around a capstan. The boat exerts a pull on the rope of
2,000 Ib. Determine the pull on the other end of the rope (F measured
in pounds) if
2,000 = Fe< 3 >< 2ir ><- 25 >
The equation used in this problem is T = Fe^ a where a is the angle in
radians subtended by the contact between the rope and post and p is
the coefficient of friction between the rope and the post.
28. The pressure P in pounds per square inch required to force water
through a pipe L ft. long and d in. in diameter at a speed of v ft. per sec.
is given by
P = 0.00161 ~p
a. Solve for v in terms of P, L, and d.
COLLEGE ALGEBRA 7
b. Determine P correct to the nearest three significant figures if
v 8 ft. per sec., L = 2 miles, and d = 6 in.
SIMULTANEOUS EQUATIONS
29. The study of the expansion of a gas in an engine utilizes the
equations piV\ n pzVi n , p\V\/T\ = p^V^/T^ where n is a positive
constant, p is pressure, V is volume, and T is absolute temperature.
Use these two given equations to obtain the following equations :
a
30. In electrical engineering occur the two simultaneous equations
- -
= R } S ( j^ 2 _ j ) = D. (K cannot be 1)
Eliminate S and solve for K in terms of D and R. Also show that the
two answers for K are reciprocals.
P TT 2 E
31. Euler's column formula (from strength of materials) is = yrry 2 -
(1 \'>/T)
P S
The Gordon-Rankine formula is = ^ ; '-//T-CV Determine an
a 1 + <t>(l/r) 2
expression for <p in terms of l/r so that the value of P/a will be the same
from the two equations.
32. A beam (see Fig. 5) weighs w Ib. per ft. and has five supports
which are L ft. apart. To determine how much of the Joad each support
L
I wlbs./ft. I
A A A A A
FIG. 5.
bears, it is necessary to solve the following set of equations for the M 's.
Determine the AT s.
M l + 4M 2 + M s = - ^,
M, + 4M 3 + M 4 = - ~>
Mi and Ms = 0.
33. A waste mixed acid left over from nitrating is composed of 60.12
per cent H 2 SO 4 , 20.23 per cent HN0 3 , and 19.65 per cent HA It is
8 ENGINEERING PROBLEMS
required to make a mixture of 1,000 Ib. containing 60 per cent H 2 S0 4y
22.5 per cent HNO 3 , and 17.5 per cent H 2 O. A 97.5 per cent H 2 SO 4
(containing 2.5 per cent H 2 0) and a 90.5 per cent HNO 3 (containing 9.5
per cent H 2 0) are available. How many pounds of each of these two
acids and of the waste acid must be used to make up the required mixture
if no additional water is to be used? Solution: Let W, S, and N denote,
respectively, the number of pounds of waste, sulphuric acid, and nitric
acid to be used.
Then l.OOOOTF + 1.00005 + 1.00007V = 1,000.
0.60121F + 0.9750AS = 600. (Sulphuric acid)
0.2023 W + 0.9050N = 225. (Nitric acid)
34. The alternating currents in the circuit, shown in Fig. 6 may be
found by solving the circuit
. .1 .f^ equations
-A/VY: 1
J,
= 0,
12
where E is the impressed voltage,
/i and 7 2 are the currents flowing
y 6 in the two meshes, and the Z's are
impedances. Obtain literal ex-
pressions for the currents and evaluate numerically if E = 100 volts,
Zi = I + j2 ohms, Z 2 = 11 + J2 ohms, Z\ = 1 +J10 ohms, where
j = V-i.
36. A particle moves along a straight line such that its speed v is given
in numerical value by
v = co(r 2 - x 2 )^,
where co is in radians per second, r in feet is the maximum distance the
particle moves from a certain center point, and v in feet per second is
the speed at a distance x in. from the center point, (x is numerically less
than r.) If v = 90 in. per sec. when x = 4 in., and v = 80 in. per sec.
when x = 6 in., determine the positive values for co and r. This problem
occurs in the study of harmonic motion.
VARIATION
36. The quantity Q of water that will flow each second over a particu-
lar type of weir varies directly as the product of the breadth B of the weir
and the three-halves power of the head H. If Q 160 cu. ft. per sec.
when B = 6 ft. and H = 4 ft., determine the formula. Then determine
Q when H = 9 ft. and Q = 5 ft.
37. The lift L of the wings of an airplane varies directly as the product
of the area A of the wings and the square of the speed V of the plane. If
COLLEGE ALGEBRA 9
the wing area is A = 100 sq. ft. and the lift is L = 1,480 Ib. when the
speed is V = 65 miles per hour, find the formula (use A in feet, L in
pounds, and V in miles per hour). If the airplane is exactly doubled in
all linear dimensions, what will happen to the weight, to the wing area,
and to the lift?
BINOMIAL THEOREM
38. A derivation in fluid mechanics for the quantity of flow of water
through a vertical rectangular orifice requires the expansion by the
binomial theorem of the two binomial quantities in the following equa-
tion and simplification to obtain the second equation. Perform the
intermediate steps.
39. The following thermodynamical equation occurs in the study of
flow of a gas out of a nozzle :
( r /VA (*-i)/*-] /H
V., = 223.7 j cr, L! -() Jf .
a. Evaluate (p>2/p\) (k ~^ /k by the binomial theorem if
(1) (V*) = 0.940 and k = 1.25.
\PI/
(2) \^J = 0.980 and k = 1.40.
b. Write (p2/pi) (k ~ ly/k in the form
p l p, 2 \(k-l)/h
and expand to three terms by aid of the binomial theorem. Then
simplify 1 (pz/pi) (h ~ 1)/h to
k I pi
JL fa - P*\* , 1
2k\ pi ) ^ ' ' ' J'
40. In 1879 the American engineers Fteley and Stearns proposed the
formula
Q = 3.31BH* (l + ^) % + 0.
0.007B
for the quantity of water that would flow each second over a certain type
of weir. Expand the binomial to two terms and simplify the equation.
Then determine K = Q/(BRK).
10 ENGINEERING PROBLEMS
41. A study of a certain type of meter in fluid mechanics starts with
the equation
Show that this equation may be written successively in the following
forms :
p 2 r Wl (k - 1) Fi 2 i */<*-i>
p~i * L pik ~2g +
-1)
TV \
20 + ' /
20
42. (Taken from a text in ceramics.) If 6 is small, show that an
approximate formula for a = (100)[1 \Xl (&/T56)] is a = 6/3.
43. (Taken from a text on principles of flight of airplanes.) Expand
the square root in Q g = W[l -\/\ (Q/W)] to three terms and
simplify. W is the maximum gross weight of plane and fuel, Q is the
maximum fuel load, and Q { , is the permissible quantity of fuel that may
be consumed on the outward flight toward an objective.
44. Engineering courses, such as thermodynamics or fluid mechanics,
frequently require the computation of such quantities as (1.045) 1 - 41 ;
(0.976) 1 - 28 ; (0.917) - 938 ; and (0.994)- 1 - 26 . Although these can be com-
puted directly by aid of a log log slide rule, the use of the binomial
theorem is just as rapid. Moreover, the accuracy by use of the slide
rule is limited, whereas that by use of the binomial theorem is not.
Compute the values of each of the preceding quantities, each correct to
the nearest third decimal.
PROGRESSIONS
45. An automobile makes a trip of K miles from A to B with an
average speed of Si miles per hour. It makes the return trip with an
average speed of Sz miles per hour. Show that the average speed for the
total distance (i.e., that speed which, when multiplied by the total
elapsed time, gives the total distance traveled) is the harmonic mean of
Si and $2.
46. A plane travels one-half of a given distance d in miles at a speed
of Si miles per hour, and the remaining half distance at a speed of $ 2
miles per hour. Show that the average speed for the entire distance is
the harmonic mean of Si and $ 2 . Half of this average speed is called the
" radius of action per hour" of the plane; i.e., it is the outbound distance
that a plane can travel and return from in 1 hr. The " radius of action "
COLLEGE ALGEBRA
11
of a plane would be the " radius of action per hour" multiplied by the
number of hours in flight.
47. Find the "radius of action" (see Prob. 46) of a plane for a 4-hr,
flight when the outgoing ground speed is 80 miles per hour and the
incoming speed is 120 miles per hour. Also find the critical time of
turning.
48. A plane encounters a changing wind in flying a round trip. On
the first half of the outbound flight the head wind is such that the ground
speed is only 75 miles per hour, and on the second half it is 100 miles per
hour. For the first half of the return flight the speed is 125 miles per
FIG. 7.
hour; for the last half of the return flight the tail wind enables the pilot to
make 150 miles per hour.
a. Find the average speed 8 for the entire flight.
6. Compare the total distance traveled in 4 hr. under the conditions
stated in the problem with the distance it would travel if the plane
traveled for 1 hr. at each of the speeds 75, 100, 125, 150 miles per hour.
49. The current flowing in an electric circuit has the following proper-
ties: When time t 0, the current i = 4 amp.; when t 0.1 sec.,
i = 4e~- 2 amp.; when t = 0.2 sec., i 4e~ QA amp.; etc. The values
of the currents at intervals of 0.1 sec. form a geometric progression.
a. What is the expression for i when t = 1 sec.? When t 3 sec.?
b. Determine a general expression for current as a function of time.
60. A ball rolls down an incline 5.47 ft. the first second, 16.41 ft. the
second second, and in each succeeding second 10.94 ft. more than in
the preceding second. Determine the distance it rolls in the tenth
second and the total distance it rolls during the first 10 sec. How far
does it roll during the first t sec.?
51. A damped free vibration is shown in Fig. 7 (where displacement
is plotted as a function of time). Such a vibration could be a weight
vibrating on the end of a spring. The numerical values of the maximum
displacements are A. Ae~ cir / u> , Ae- 2 '*/ 40 , ....
12 ENGINEERING PROBLEMS
a. Show that these values form a geometric progression.
6. Show that the logarithms to the base e of these numbers form an
arithmetic progression.
The numerical value of the common difference is^called the " log-
arithmic decrement." This term appears quite frequently in connection
with vibration problems. What is its value in this problem?
52. Gravel for a highway is to be hauled from a gravel pit 200 yd.
from the highway and at a point opposite one end of a strip to be
graveled. Each truck hauling gravel travels this 200 yd. plus the
distance already graveled. If one load of gravel suffices for ea.ch yard
of highway, determine the number of miles traveled by the highway
trucks in graveling 2 miles of highway. How much distance for x miles
of highway?
THEORY OF EQUATIONS
53. If a solid sphere of radius R and made of material with a specific
gravity s(s < 1) is placed in water, it will float. The depth (h in.) of
the part of the sphere under water is such that the weight of the solid
sphere is equal to the weight of the water displaced. Given, from solid
geometry, that the volume of a spherical segment of height h is
TT/> 2
V = ^ (3R - h),
show that h is a solution of the Aquation -5- (3R A) =
If R = 4 in. and s = 0.70, compute the value of h correct to the
nearest 0.01 in.
54. The two-mesh electric circuit shown in Fig. 8 has the following
data:
Ri = 10 ohms, Ri2 = 5 ohms, #2 = 3 ohms,
Li = 0.01 henry, d = 0.000,001 farad,
C 2 = 0.000,000,5 farad.
To determine the currents after the switch is closed it is necessary to
solve the following algebraic equation:
COLLEGE ALGEBRA
13
R,
.
w
which, for the given data, becomes
0.08w 3 + 20,095w 2 + 38,000,000w + 2,000,000,000,000 = 0.
Show that this equation has one real root w = a. and find it correct to
the nearest three significant figures. Then divide by w - a and neglect
the remainder. Use the resulting quadratic to determine the .two
complex roots.
Lft.
Curve of beam
FIG. 9.*
55. Figure 9 shows a beam that is built in at one end and simply
supported at the other end. To determine the horizontal distance to
the lowest point on the curve, it is necessary to solve the equation
8z 3 9L.-C 2 + L 3 = 0. The figure shows that the desired value of x
is approximately 0.4L. Determine this value of x, correct to the nearest
three significant figures. Notice that the equation has one rational
root.
j Curve of beam
FIG. 10.
56. To find the maximum deflection y for the beam shown in Fig. 10
it is necessary first to solve the equation
-lOOz 3 + l,950z 2 - 38,650 =
for its root between x = and x = 7. Find this root correct to the
nearest three significant figures.
14 ENGINEERING PROBLEMS
57. The formula d 5 = Ad + B is used to find the diameter d in. of a
pipe that will discharge a given quantity of water per second. A and
B are known constants. If A = 15.5 and J5= 400, compute the value
of d correct to three significant figures by the following method :
To obtain a first estimate for the value of d } neglect the term Ad and
solve d* = 400 for d. Substitute this value in Ad + B and then deter-
mine a second estimate for d from d 5 equated to this computed value.
Continue this process until the value for d does not change in the third
significant figure.
This is a method of practical approximation that is especially suited
to the use of a slide rule. This method frequently can be used on a
cubic equation if one first transforms the equation (by a proper reduc-
tion in the size of the roots) so that the quadratic term is missing.
68. The amount of water (q cu. ft. per sec.) that flows over a trape-
zoidal spillway is given by
where g = 32.2 ft. per sec. per sec., b is the width of the bottom of the
trapezoid, and the width at the top is b + 2z. The depth of the water
is H ft.
If z = #/4, b = 6 ft., q = W)0 cu. ft. per sec., determine the proper
value of // correct to the nearest three significant figures. For a "cut-
and-try" method, show that the equation can be written in the form
HK = 93.46/(30 + H).
59. The equation q = 0.622 \/2g (b - 0.2A)(/^) is used to determine
the quantity of water (q cu. ft. per sec.) that flows each second over a
rectangular weir of width b ft. when the head (height of water source
above that flowing over the weir) is h ft. If q = 24.4 cu. ft. per sec.
and b = 3.00 ft., compute the value of h correct to three significant
figures. You may use either the methods explained in your text on
college algebra or the following cut-and-try method. For a first
approximation, solve for h in q (0.622) (20) 1 /"(&)(/^~). Substitute
this value for h in the expression b 0.2/& and again solve for h. Con-
tinue this sequence of steps until the computed value for h does not
change in the third significant figure.
60. A hollow iron sphere has an external radius of 2 ft. What wall
thickness should the sphere have so that the sphere will just float in
water? Iron weighs approximately 7.8 times as much as water. Sug-
gestion: The weight of a solid sphere of iron of radius 2 ft. less the weight
of a solid sphere of radius 2 t ft. = weight of a solid sphere of water
of radius 2 ft.
61. A design problem in chemical engineering led to the following
question: A right circular cone has an altitude of 10 in. and radius of
base 6 in. It is required to inscribe a right circular cylinder whose
COLLEGE ALGEBRA 15
volume will be one-third the volume of the cone. What are the dimen-
sions of the cylinder?
62. (Taken from a text on vibrations.) Determine the exact values
of the roots of the equation
' 47/3 _ 10 ^ 2 + Q F _ i = 0>
63. The following equations are solved in a text on vibration problems
in engineering. Obtain all the solutions, each correct to the nearest
three significant figures:
(a) x 3 - 407z 2 + 34,492z - 296,230 = 0.
(6) p - 6/fy 4 + 100 2 p 2 - 40 s = 0.
(c) p 4 - 106,000p 2 4- 2,060,000,000 - 0.
(d) 10- 21 co 6 - 3.76(10- 14 )co 4 + 1.93(10- 7 )co 2 - 0.175 = 0.
Remarks: Two of the preceding problems suggest cut-and-try methods for
the determination of irrational roots. In many engineering problems an
accuracy of three significant figures is quite sufficient, and the required roots
can be obtained rapidly by aid of a slide rule. Certain types of problems
lead to algebraic equations that can be solved rapidly by special methods.
For example:
If the quartic equation with four complex roots
ttow 4 + a\w 3 + 2^ 2 -|- a*w + cu =
arises from an electric-circuit problem in which the resistances are small or
from a vibration problem in which damping is small, then one can neglect
the terms of odd degree and determine satisfactory estimates for the imagi-
nary parts of the complex roots. The real parts may then be determined by
relations between the coefficients and roots.
It is worth noting that, although the Horner method itself does not appear
very much in engineering literature, many of the ideas which the student
learns in the chapter on the theory of equations do appear in other
circumstances.
The standard method for computing complex roots is Graeffe's root-
squaring method, which is explained in advanced mathematics and engineer-
ing texts.
DETERMINANTS AND SIMULTANEOUS LINEAR EQUATIONS
64. In an analysis of a class A radio amplifier a derivation of a
so-called "load line" requires that the three equations
ib = h ip, e b = E b + e p , e p = i p R L
be combined to obtain the equation
e b = E b + R L (h - i b ).
Perform the required algebra to obtain this last equation.
16
ENGINEERING PROBLEMS
65. By applying Ohm's and KirchhofT s laws to the three-mesh elec-
tric circuit, which contains resistances (in ohms) as shown in Fig. 11,
the following set of equations is obtained :
47i - 2/ 2 - / 3 = Ei t
/i + 14/ 2 - 3/3 - Ei,
/! ~ 3/2 + 6/3 = J03-
fRi = 1 ohm, #2 = 9 ohms \
#3=2 ohms, #12 = 2 ohms J
\Rit = 1 ohm, #23 = 3 ohms /
a. Determine the currents /i, /2, /s (in amperes) flowing in each mesh
of the circuit if EI = 10 volts, E 2 = 5 volts, and E 3 = 2 volts.
b. Compute /i if E\ = # 3 = and $2 = $. Also compute /2 if
$2 = /?3 = and /?i = #. Show that the results are the same in
both cases.
FIG. ll.
c. Solve the three equations simultaneously for the I's in terms of the
E's.
Remarks: The literal equations for the given network are as follows:
(#1 4" #12 4~ #13)/1 #12/2 #13/3 EI,
#12/1 4" (#2 4~ #12 ~H #2s)/2 #23/3 ~ ^2j
#13/1 #23/2 + (#3 4" #13 "T" #2s)/3 == ^3-
The answer to question c can be interpreted as saying that the same cur-
rents /i, / 2 , /s would flow in a three-mesh electric circuit as shown in Fig. 11
with the following data: #i 2 = #13 = #23 = 0, #1 = #2 = #3 = 1 ohm,
E'i' - 0.3^! + 0.06^ 2 + 0.087^3 volts, E 2 ' = O.OG^i + 0.092^ 2 + 0.056# 3 ,
ES = 0.08J5J! 4- 0.056# 2 4- 0.208^ 3 .
66. In a five-mesh electric circuit similar to the one in the preceding
problem, it is shown in electrical engineering courses that the currents
/i, / 2 , /a, /4, and /5 satisfy equations of the following form:
aJi + a 2 / 2 4- 3/3 4- aJ 4 + aJ 5 = EI,
2/l + 6 2 /2 4- ^>3/3 + &4/4 4- &5/5 = ^2,
a 3 /i 4- &3/2 4- c 3 / 3 4- c 4 / 4 + c 5 / 5 = E 3 ,
aJi + 64/2 4- c 4 /3 4- ^4/4 4- 4/5 = Et,
a 5 /i 4- 65/2 + c 5 /3 4-
COLLEGE ALGEBRA
17
a. Indicate the solution for I\ and for / 3 by the use of fifth-order
determinants. Do not expand.
b. By the theory of determinants, show that the solution for I\
when #3 E and all the other #'s are zero is the same as the solution
for /a when E\ = E and all the other #'s are zero.
c. Show that the solution for I\ in terms of the E's can be obtained
as follows: Let In denote the value of /i when #1 = 1 volt and all the
other E's are zero. Let 1 12 denote the value of /i when # 2 1 volt
and all the other E's are zero. Let Iu, Iu, In have similar definitions.
Then the required solution for /i in terms of the E's is given by
/I = #1/11 + #2/12 + #3/13 +
#4/14 -f- #5/15
R
Remark: This problem suggests two
theorems about electric circuits, which
will be proved in senior courses in
electrical engineering.
67. The Wheatstone bridge shown
in Fig. 12 is a circuit for measuring
a resistance #4 in terms of three
known resistances #1, #2, and #3. G
is a galvanometer of resistance R g . E is the impressed battery voltage.
The currents satisfy the equations:
/!-/- / 2 = 0, / + I g - h = 0, / - 7. - /i = 0,
#1/1 4~ #2/2 #3/3 #4/4 0,
#1/1 + #,/<; - #3/3 = 0, #1/1 + #2/2 = #
a. Show that if #i# 4 = # 2 # 3 then I g = 0.
b. Obtain an expression for I g li E = 10 volts, #1 = 100 ohms,
# 2 = 1,000 ohms, 72 3 = 201 ohms, # 4 = 2,000 ohms, and R g = 1,000
ohms.
68. The current supplied by a
"three-phase alternator" to an
"unbalanced load" may be ob-
^c tained by solving the simultaneous
equations :
(Zi -\- Z a )I a (Zi -f- Z b )I b #60,
/ rj \ rr \ T f rr \ rr \ T TTT
\Ji ~T ^b)J-b \i "I Jc)lc -C'cfc,
/ a + I b + I c = 0,
FIG. 13.
in which Z is the internal impedance of the alternator; Z , Zi and Z c
are the load impedances; and E ab and E cb are the generated voltages
between two pairs of terminals, as shown in Fig. 13.
18
ENGINEERING PROBLEMS
Solve for the currents I a , /<>, I c first in literal form and then in numerical
form for the following data: Zt = 1 + jlO ohms, Z a = 20 + J15 ohms,
Z b = 40 + J30 ohms, Z c = 60 +j_0 ohms, E ba - 866 + #00 volts,
#<* = -j 1,000 volts, and j = V-l.
69. A beam weighs 1,000 Ib. per ft. and has six supports as shown in
Fig. 14. To determine how much of the load each support bears, it is
1000 Ibs./ft.
10'
12'
10'
FIG. 14.
Give the
necessary to solve the following set of equations for the M's.
final results correct to the nearest three significant figures.
11M 2 + 31/3 = -170,500,
621/2 + 26A/ 3 + 7M 4 = -559,000,
7M 5 + 24M 4 + 5M 5 = -468,000,
5M 4 + 16M 5 = -189,000.
70. A ladder leans against a wall as shown in Fig. 15. There is some
friction between the ladder and the floor and between the ladder and
the wall. A problem in mechanics would be to determine the value of the
force P that would cause the ladder to be on the verge of moving up
the wall.
From Fig. 16, one learns in mechanics how to write down the following
simultaneous equations. Solve them for P.
p -G - N =
4F = N,
0, -12F +
2G = M.
- 300 = 0,
71. Three 1-oz. balls are strung on a string of length 4 ft. and tied as
indicated in Fig. 17. The string is pulled taut and tied to stationary
vertical boards. The string is then pulled vertically upward a short
COLLEGE ALGEBRA
19
distance and released so that the three balls will oscillate in a vertical
direction. Let y\ be the vertical distance from the horizontal to the
first ball, 2/2 for the second ball, and 7/ 3 for the third ball (Fig. 18). By
methods of mechanics one can obtain the following three equations,
which are valid at a particular time after the balls are released :
(3 - 4F)7/! + 2y 2 + 2/3 = 0, 7/ t + 2(1 - F)y 2 + y* = 0,
2/i + 27/2 + (3 - 4F)y, = 0,
where F depends on the weight of the balls, their distances apart, how
tight the string was originally stretched, etc. For most values of F,
these equations will have only the simultaneous solution:
2/i = 2/2 = 2/3 = 0,
which means that after a short time the balls would come to rest. But
1' 1' , 1'
1'
FIG. 17.
FIG. 18.
for three special values for F, these equations have other solutions than
the preceding one.
Determine these three values for F and then find the solutions y\,
7/2, 7/3 for each of these values for F and on the assumption that y\ \
unit. Then plot the string and balls for each of these three solutions.
Remark: Actually, the vertical displacements vary with time according to
the form y yk sin wt.
*72. Suppose that a body falls in a vacuum from an original height
s ft. and with an initial speed of v ft. per sec. The distance (s ft.)
that the body will fall during the first t sec. will then conceivably depend
on t, s , ^o, the acceleration due to gravity g ft. per sec. per sec., and the
weight of the body W Ib. Expressed mathematically, we can say that
s is some function of these quantities; i.e.,
s = F(t,VQ,8o,g,W).
20 ENGINEERING PROBLEMS
Now suppose that this function can be expressed as the sum of terms
each of which is a product of powers of the separate variables. Then
each term will be of the form:
Since the left-hand member of the original equation, s, has the dimension
of feet, each of the terms on the right-hand side must have the dimension
of feet. Thus, substituting the dimensions for each factor, we obtain
where W == force, L s== length, and T = time.
For this to be an identity we must have
b + c + d = 1, a - b - 2d = 0, e = 0.
Problem: Solve the first two equations for a and d in terms of b and c.
Then show that the following sets of five numbers are solutions of the
preceding three equations:
abode
20010
11000
00100
Notice that e = means that the distance is independent of the
weight of the body.
The conclusion that one could then make is that every term in the
expansion of the given function is a combination of the three terms
t 2 g, tv , and s , each of which has the dimension of feet. The simplest
combination of these would be to try s = A(t 2 g) + B(tv Q ) + C(s ).
There is no justifiable reason for this, save as a trial. If you took this
formula to the laboratory to see if it could be true and, if so, determined
the values for A, B, and C, you would obtain s = (gt*/2) + v Q t + s .
73. Determine x, y, and z in terms of n so that the following will be
an identity:
W - ( i^(W\*(W\(
~ (L} (j?) (LT) \T
This problem arises in fluid mechanics, in much the same manner as
Prob. 72, as a result of a study of pipe friction.
74. The theoretical analysis of a damped vibration absorber on a
machine requires the simultaneous solution of the following pair of
equations (j = V r T) :
(-Mw 2 + K + k + jwc)xi - (* + juc)x 2 = P,
(* + jwc)xi + (-mco 2 + k + jwc)x2 = 0.
COLLEGE ALGEBRA
21
Indicate the solutions for x\ and z 2 by means of determinants. Notice
that the solutions are both in complex-number form. Determine the
absolute value for the complex number solution for x\. Thus, if
Xi = a + jbj you are to determine the value of (a 2 + 6 2 )^ 2 .
Remarks: Determinants are used in engineering in the following ways:
1. An important aid in proofs of theorems.
2. A method of obtaining "formulas" for the simultaneous solutions.
These formulas are convenient if the coefficients are complex numbers.
3. A convenient method for manipulating operational expressions such as
are met in the solution of simultaneous li^ar differential equations and in
problems such as number 74.
76. An algebraic equation is said to be "stable" if its real roots are
all negative and its complex roots all have negative real parts. It can
be shown that the quartic equation
a # 4 + ai# 3 + azx 2 + dzx + a\ 0, a positive,
is stable if the values of each of the following determinants are positive :
14
ai ool
CLi do
ai ao
a 3 o 2 |
as a>2 QI
>
as a 2 ai ao
#4 as
a 4 as a 2
a 4
a. Test for stability in the following equation taken from a text on
vibrations: z 4 + 8z 3 + 10z 2 + 5x + 7 = 0.
b. Test for stability: x* + 4x* + Go; 2 + 5x + 2 = 0. Then show
that there are two rational roots, determine all four roots, and verify
the definition of stability from the roots themselves.
PART II
TRIGONOMETRY
FUNDAMENTAL IDENTITIES
76. In the magazine Electrical Engineering (vol. 52, 1933, p. 724) is
given a short cut for finding the value of \/a 2 + & 2 - The method is
as follows: Assume that a > b. Rewrite the given expression in the
form a \/l + (&/0 2 - Determine the acute angle 6 if tan = b/a.
The value of the given expression can be found by dividing a by cos
or by multiplying a by sec 6. Prove that this is a correct method.
77. Let two wattmeter readings for a particular electric circuit be
w\ and ic/'2. Then the so-called "power factor" for the problem can be
determined from these wattmeter readings by aid of the equation
?8 - MI) _ V3 d
W 2 + Wi S
Show that cos = 1/[1 + 3(d/s) 2 ]^.
78. 'The follower on a cam at a time t sec. has for its abscissa
x = 5 sin 2-7Tco in.
and for its ordinate y = 4 cos 27ro> in. The quantity o> is a constant.
Show that a; 2 /25 -f- ?/ 2 /16 always has the value +1, irrespective of
the time.
79. It is necessary to make a table of values of the function
(16 -
for x = 0, 0.1, 0.2, . . . , 1.3. Show that this computation may be
accomplished readily by replacing 3x by 4 sin and then simplifying
the given expression to M tan 6. Make up the table of required values
correct to four decimals.
RIGHT TRIANGLES
80. A weight of 50 Ib. is suspended at the free end of a horizontal bar,
which is pivoted at the left end and is held in position by a cord that
makes an angle of 20 with the horizontal (Fig. 19). Determine the
tension in the cord (T Ib.) and the tension in the bar (P Ib.) from the
triangle of forces (Fig. 20).
22
TRIGONOMETRY
23
81. Three forces of 60, 90, and 100 Ib. act at angles 45, 150, and
210, respectively, with the horizontal (Fig. 21).
a. Determine the x and y components of each force.
I
90>
50 Ibs.
b. Determine the sum of all three x components and the sum of all
three y components.
c. Determine the resultant of these total x and y components. Give
the result both in magnitude and
direction.
82. Three forces of 61.2, 93.8, and
106 Ib. act, respectively, at angles
51.2, 145.1, and 281.3 with the
horizontal.
a. Determine the horizontal and
vertical components of each force, each correct to slide-rule accuracy.
6. Determine the sum of the horizontal components and the sum of the
vertical components.
c. Determine the resultant (in magnitude and direction) of the total
horizontal and total vertical components.
P
FIG. 20.
90 Ibs.
60 Ibs.
100 Ibs.
FIG. 21.
83. Because of " centrifugal force" a man on a bicycle going around
a curve will tend to lean at an angle 0, which depends on his weight, his
speed, and the radius of the circular curve around which he is moving.
If the combined weight of the man and bicycle is 200 Ib. and if he is
going around a curve of 60 ft. radius at a speed of 15 ft. per sec., then
24
ENGINEERING PROBLEMS
the centrifugal force is 23.3 Ib. Use Fig. 22 to determine the angle
at which he should lean.
23 - 3 lbs - 84. An airplane weighs 9,000 Ib. and is climbing
at an angle of 5.00. Compute the components
of the weight of the airplane in the line of flight
and perpendicular to this line. Give results to
three significant figures.
85. Two balls hang by cords as shown in Fig.
23. The length of each cord is 2 meters and the
weight of each ball is 0.5 gram. A charge of
5(10" 6 ) coulombs is placed on each ball. A ques-
tion asked in a course in electrical engineering
fundamentals was to determine the angle 6
between the cords, assuming that the cords are
straight. The solution of the problem shows that
Plane of
bicycle
200 lbs.
FIG. 22.
the angle must be such that
tan o =
Start with this equation and show, by aid of Fig. 23, that d must be a
solution of the equation
4d 6 + 529d 2 = 2,116.
Then solve this equation correct to the nearest two
significant figures.
86. Two poles are 42 ft. apart. One is 30 ft. tall
and the other is 24 ft. tall. A 48-ft. cable is fastened
to the top of each pole and the cable supports a
weight of 400 Ib., which hangs from it by a trolley.
When the trolley comes to rest, the figure is as shown
in Fig. 24 (assuming that the two pieces of the cable
are straight).
The sum of the x components (in Fig. 25) must be zero and the alge-
braic sum of the y components must also be zero.
FIG. 23.
TRIGONOMETRY
25
Hence,
and
P cos = P cos <p,
P sin + P sin <p = 400.
From the ge6metry of Fig. 24: n + p = 48,
n cos + p cos
n sin p sin
= 42,
= 30 24 = 6.
Verify these five equations and solve them simultaneously for the
acute angles and <p, the lengths n and p ft., and the tension in the
cable P Ib.
87. A cord whose length is 2L ft. is fastened at A and B (Fig. 26) in
the same horizontal line; the two points A and B are 2a ft. apart. A
Wlba.
FIG. 20.
FIG. 27.
smooth ring on the cord is attached to a weight of W Ib. (see Fig. 27).
Show that the tension in the cord (a < L) is
T Ib. = HTr
TFL
2(L 2 - a 2 ) 1 ^
88. A condenser having "capacitive reactance" of 10 ohms, a
resistance of 12 ohms, and an air
coil with an "inductive reactance "
of 16 ohms are all connected in
series. Determine the "imped-
ance" (Z ohms) and the power
factor (cos 0) for the circuit. Also
determine the angle 6 correct to
the nearest minute (see Fig. 28).
16-10
= 6 ohms
12 ohms
FIG. 28.
If this series circuit is connected to a 60-cycle, E 100 volt, a.c.
source, and the current / = 100/Z (amperes), calculate / and
P = El cos watts,
each correct to the nearest three significant figures.
OBLIQUE TRIANGLES
Note: Trigonometry texts abound in good examples of the use of oblique
triangles in engineering. The following few problems are thought to be
somewhat different and are included for that reason.
26
ENGINEERING PROBLEMS
89. An airplane flies from point A on a track due east (90 track).
The wind is 30 miles per hour from 240 track and the air speed of the
airplane is 105 miles per hour.
a. In Fig. 29, find the angle 0.
b. Find the heading of the airplane (the angle measured from the
north which the forward axis of
the airplane makes with that direc-
tion) by adding B to 90 in this
particular problem.
c. Determine the ground speed
of the airplane VQ. The ground
speed is the vector sum of the speed
of the airplane relative to the air
plus the speed of the wind relative
FIG. 29.
240
to the ground.
90. What are the answers to the
three questions in Prob. 89 if the airplane flies due west?
91. In Prob. 89, a second airplane leaves point P located 90 miles
southeast (bearing of 225 track) from A at the same time as the airplane
at A and has an air speed of 150 miles per hour. The wind conditions
remain the same as in Prob. 89.
a. If 6 is the angle between the air-speed vector of airplane P and
the horizontal and if t is the time required for airplane P to intercept
airplane A show that
150* cos B + 30Z cos 30 = 90 cos 45 + 130J,
150* sin 6 + 30* sin 30 = 90 sin 45.
. Solve for t and B, and finally for the direction of the track of air-
plane P.
92. In Fig. 30 w represents a constant wind speed making an angle B
with the track of an airplane along which the ground speeds Si and /S 2
(outbound and incoming speeds) are attained by an airplane of constant
air speed v.
a. Show, from the figure, that
Si = w cos B + v cos <p,
82 v cos (p w cos 0,
w sin B = v sin <p.
b. Show that the product 8182 is a constant (note that w and v are
assumed to be constant).
c. Show that
2 -\A 2 - w* sin 2
TRIGONOMETRY
27
d. Show that for a wind of any given speed the " radius of action"
(see Prob. 46)
R =
is least for a head-tail wind (6 = 0) and greatest for a wind at right
angles to the track (0 = 90).
FIG. 30.
MULTIPLE ANGLE IDENTITIES
93. A sphere weighing W Ib. rests between two smooth planes, as
shown in Fig. 31. Figure 32 shows the weight of the sphere and the
forces that the two planes exert upon the sphere. Since the algebraic
R
FIG. 31.
sum of the horizontal components of the forces must be zero and the
algebraic sum of the vertical forces must likewise be zero, we obtain
*
R sin <p P sin 8 = 0,
R cos <p + P cos = W.
Solve these two equations simultaneously for R and P in terms of
W, 0, and <p and show that your results can be put in the form
W sin <p
sin (9 + <f>)'
R
WsinB
sin (& + <?
28 ENGINEERING PROBLEMS
94. The period of vibration of a pendulum is given by the approximate
formula
which is much more accurate than the one customarily given in an
elementary course in physics, namely, T = 2ir(L/g)M. L is the length
of the pendulum and g = 32.2 ft. per sec. per sec. 6 is the angle that the
pendulum makes with the vertical at the instant it is released.
a. Evaluate the part in parentheses for = 2, 30, 60.
6. Show that the quantity in parentheses can be written in the follow-
ing form: 1 + 3 % 12 - K28 cos % 12 cos 26.
95. The voltage in an electric circuit is
e = 40 sin 1207r + 5 sin 3607r volts.
The current is
i = 4 sin 120rr + 2 sin 360rr amp.
Determine an expression for the power p = e - i watts and leave your
final result in a form free of powers and products of trigonometric
functions.
96. Two voltages :
ei = 40 sin (l2tort + |) volts,
e 2 = 60 sin (l2(hrt - |) volts,
are simultaneously impressed in series on an electric circuit. Combine
these into a single voltage by performing the operation e = e\ + e%.
Give your final result in the form E sin (1207r + 0).
97. If the voltage in an electric circuit is
e = E m sin a.
and the current is
i = J m sin (a + 0),
show that the power, p = e-i watts, can be expressed in the form
P = (^^] [cos - cos (2a + 0)].
Remark: This derivation is to be found in every text on a.c. circuits.
98. The value of the voltage e in volts due to "amplitude modulation"
is given by
e = 100(1 + 0.7 cos 4,000 - 0.3 cos 8,0000 sin 4,000,000*
where t is in seconds. Show that this can be rewritten in a form free
TRIGONOMETRY
29
of products of trigonometric functions, i.e., as the sum of simple sine
functions. Then determine the amplitude, period, and frequency for
each of the resulting terms.
*99. Figure 33 shows a connecting rod, crank-arm mechanism from
an engine.
FIG. 33.
a. Show that
x = r cos + (L 2 - r 2 sin 2 0)V*
(r 2 \ ^
1 - 5 sin2 0) '
b. Expand the binomial to four terms by aid of the binomial theorem
and obtain
r 2 sin 2 r 4 sin 4 r 6 sin 6
x = r cos + L
2L
8L 3
16L 5
c. Transform this expression so that there are no powers of trig-
onometric functions present and show that the result is
" ( L 4L 64L 3
+ cos 20
cos 40
256L 5
r 4 13r 6
r cos
r 4 3r 6
64L 3 + 256L 5
d. Simplify the preceding expression if L/r = 5.
e. What would this last result be if you used only the first two terms
of the binomial expansion?
Remark: The result in (e) is commonly used in engineering problems,
since the coefficients of the higher harmonics arc small (L/r ^ 5). How-
ever, there are times when it is necessary to know something about the
higher harmonics, and you obtained some of them in this problem.
30 ENGINEERING PROBLEMS
GRAPHS OF THE TRIGONOMETRIC FUNCTIONS. LINE VALUES 1
100. If a voltage is described as having a sinusoidal wave form, a
maximum value of 140 volts, and an angular velocity of 377 radians per
second (60 cycles per second), and if it is agreed to determine time from
a point of zero voltage where the slope of the tangent line to the curve
is positive, the mathematical equation for the alternating voltage as a
function of time is
e = 140 sin 377Z volts.
Compute the voltages at t = 0.001 sec. and t = 0.002 sec. Sketch the
voltage wave for several cycles.
101. A voltage in an electric circuit is e 140 sin 377Z volts and
the current is i = 7.07 sin 377 1 amp. If the power is
p = e i = 500(1 - cos 7540,
sketch on the same graph the voltage wave, current wave, and power
wave. *
102. At a certain instant the voltage in an electric circuit is repre-
sented by the vector e volts as shown in Fig. 34. At that same instant
t=4 amperes
= radians
e=100 volts
Fia. 34.
the current i amp. is likewise shown as a vector. Let both vectors rotate
together in a counterclockwise direction at 60 cycles per second or 1207T
radians per second. Sketch a graph showing time as abscissa and the
vertical projections of these two vectors as ordinates. Show the graph
for several cycles. Notice on your resulting graph that the current
always leads the voltage by 60 or Tr/3 radian. Also show that the
equations of the two curves are e = 100 sin 120rr, i = 4 sin (120rr + 60).
103. The piston of an engine transmits a force of F = 100 sin 30rr Ib.
Plot a graph of force as a function of time.
104. The approximate tide curve for Cape Cod Bay is
i. 1 f 2wt
h = h m sin l-jr
1 For other problems on graphs of the trigonometric functions, see Part III.
TRIGONOMETRY
31
where h m
tsec.
6 ft. and T = 44,715 sec. Plot h as" a function of time
TRIGONOMETRIC EQUATIONS
105. The relation between the
size of feed (radius 6), the space
between the rolls (2a), the radius
of the rolls (r), and the angle of
"nip" (N) is
fN\ r + a
cos
(f)-
r + b
a. Derive this equation from
Fig. 35.
b. What is the relation for N in
terms of r, a, 6? For r in terms of
N, a, 6?
106. Figure 36 shows four circles
that possess the indicated tangency
properties. The radii of three
circles are known: OC = 5 in.,
AB = 3 in., and ED = 2 in. Determine the coordinates of the center
P and the radius r of the circle BCD, this fourth circle being tangent to
each of the three given circles.
FIG. 35.
FIG. 36.
Suggestion: Angle is common to the two triangles OAP and OEP, and the
dimensions of the sides for both of these triangles can be determined in
terms of r.
32 ENGINEERING PROBLEMS
107. When a block of weight W Ib. is pulled up an inclined plane by
a horizontally directed force (P Ib.), the angle 6 which the plane makes
with the horizontal will make the efficiency a maximum if
sin 2(0 + <f>) = sin 20.
Mechanical efficiency is defined as the ratio of .the
useful work performed to the total energy expended
(see Fig. 38).
Tan <p is a measure of the friction between the
block and the plane. Solve for the smallest acute
angle 6 if tan <p = 0.347 (the proper value if the
block is made of cast iron and the plane of steel).
F 7 108. A ski jumper starts down a hill from the
point marked A (Fig. 37). The cross section of the
hill is a circle of radius R. It can be shown, by methods of physics
and mechanics, that the radius to the point at which he will leave the
surface of the hill (neglecting friction, which is small) will make an angle
6 with the horizontal where
sin = 2(1 - sin 0).
Determine this angle.
109. A body weighing W Ib. rests
on a rough plane inclined at an angle ^s^\ W
with 'the horizontal (Fig. 38). To
determine the force P Ib. that will just
cause the body to begin to slide up the hill, one applies methods of
mechanics to obtain the following equations:
P cos - W sin = F, N - W cos = P sin 8, F = N tan <p,
where tan <p is a measure of the friction between the body and the plane
and F is the frictional force. Solve these three equations simultaneously
for P and obtain P = W tan (6 + <p).
110. The equation
tan a 2 _ u\
tan ai ~~ u<i
is used in electrical engineering to determine the change in direction
when magnetic lines pass from one medium to another. A special case
yields
tan air = 1,000 tan ow.
Compute a &ir in degrees correct to the nearest minute when a ir0 n = 0,
0.1, 1, 15, 30, 60.
111. Sneirs law from physics is
sin (p\ HZ
sin <f>2 n\
TRIGONOMETRY
33
where n\ and n^ are the indices of refraction for two mediums through
which light is passing, <pi and <?* are the corresponding angles. Tabulate
the values for vWer, correct to the
nearest minute, corresponding to the
following values for <p a ' 1T if
1.33 sin ^ wa ter == 1.000,292 sin <p a ir,
^ air = 0, 1, 10, 30, 45, 60, 90.
COMPLEX NUMBERS
DEMOIVRE'S THEOREM
112. At a certain instant the voltage
in an electric circuit may be repre-
sented by a vector as shown in Fig.
40 and may be represented algebraically in each of the following forms :
Trigonometric form:
E = 100(cos 45 + j sin 45), j = V^l,
Polar form:
E = 100 Ijtfj
Exponential form:
E = lOOc^/ 4 , 2.718,
Rectangular form:
E = 70.7 + j 70.7.
At the same instant the vector representing the current (7 amp.) is as
shown in the figure.
a. Represent the current / in each
of the above forms.
b. If the instantaneous power is
the product of the imaginary parts
of the voltage and current vectors,
find its value.
c. If the average power is the pro-
duct of the lengths of the voltage
and current vectors multiplied by
the cosine of the angle between these
two vectors, find its value.
113. Suppose the voltage in an electric circuit is e = E sin ut volts
and the current is i I sin (o> + 0) amp.
a. Show that these two quantities can be written as the imaginary
parts (or vertical projections) of the following vectors:
+
E =
I = /[cos (o)t + 6) + j sin
34 ENGINEERING PROBLEMS
b. Represent these two vectors on a neat sketch. (Take ut to be
about 30 or Tr/6 radians and 6 to be about ?r/4 radians.)
c. Give the values of these vector quantities in each of the other possi-
ble forms for complex quantities.
d. If the voltage and current are written in the forms
E = Ei + jE 2 , I = 7i + jh,
the " average power " in the electric circuit is given by
P ==: E\II ~
What does this become if one uses the trigonometric forms for the two
quantities?
Remark: Electrical engineering analysis of a.c. circuits may be made
by aid of sine waves, line values, or complex quantities. The sine-wave
method utilizes the graph of the vertical or horizontal projections of these
complex quantities (vectors) in terms of time, whereas the complex quanti-
ties show the circuit situation at a particular time, this time being repre-
sented as the angle for the voltage vector. Normally, the analysis uses the
angle for the voltage vector as some integral multiple of 2?r.
For a reasonable understanding at the junior level in electrical engineering,
the student must have at his command an understanding of line values, sine
waves, and complex-quantity manipulation.
114. An electric circuit has in series a resistance R ohms, an inductance
L henrys, a capacitance C farads, and a voltage e = E sin ait volts. The
"impedance" function for this circuit is
a. Compute Z and write your result in three other forms if
(1) R = 10 ohms, L = 0, 1/C = 0, co = 1207T radians per second.
(2) R = 30 ohms, L = 0.1 henry, 1/C = 0, co = 120rr radians per
second.
(3) R = 0,L = 0.0425 henry, 1/C = 0, = 120rr radians per second,
coL = 16.0. (Take 90 for the angle.)
(4) R = 0, L = 0, C = 0.000,001,06 farad, co = 120?r radians per
second, 1/wC = 2,500. (Take = -90.)
b. If E = I E and E = 100 + j'O, determine 7 for each of the prob-
lems in (a).
115. If 7 = 10/30, V b = 30/-60 , and V c - 15/145, and if
aV b + a*7 c ),
Vd = a7.i,
where a = 1/120, determine the values of F i, 7n, V c \.
TRIGONOMETRY 35
116. If F.I + F2 + FaO = Fa,
a 2 Fai + aF 2 + F o0 = F*,
and aFai + a 2 F o2 + F o = F c ,
where a = 1 /1 20, solve for F a o, F^, and F i each in terms of F, F&>
and'F c . Notice that 1 + a + a 2 = 0.
117. Solve simultaneously the following three equations :
(19.4/68^)7! - (9.70/68)J 2 = 0,
-(9.70/68_)7! + (19.4/68^)7 2 - (3.04 /80.5)/ 3 = 0,
- (3.04 /80.5)7 a + (8.08 /60)/ 3 = 4,000 /-60,
which arise from an analysis of a three-mesh electric network. The
results for /i, 7 2 , 7 3 are the currents flowing in the three branches.
118. The following mathematical manipulations are to be found in
textbooks on a.c. circuits:
a. Perform the following indicated operations:
(1) (5+j7) + (3-'j2) - (4-J3).
(2) (5.00 + j8.66)(7.07 - J7.07).
(3) (3 - J4)(10 /W)(cos 30 + j sin 30)(4e^/ 2 ).
(4) (60 + j'80)^ (give that root which has the smaller positive angle).
(5) log (10 /60) (give the angle in radian measure).
(6) The value of e' w * if t = 0.002 sec. and co = 377 radians per second.
(7) (4/60)V(2/--30 ) 2 .
6. Find the rectangular, trigonometric, polar, and exponential expres-
sions for a vector whose magnitude is 10 units and whose position is
(1) 60 ahead of the reference (positive horizontal) axis.
, (2) 120 behind the reference axis.
(3) 90 ahead of the reference axis.
(4) On the reference axis.
(5) 180 ahead of the reference axis.
c. Determine the values of R and if
(120 + j'O) + 4ft /-60 = 225 /-0.
d. Plot Ac^'and Ae~' ut m vector form if co = lOOrr radians per second
when
(1) t = 0.0025 sec.
(2) * = 0.005 sec.
(3) * = 0.00666 sec.
e. Plot the vertical projection of At*" 1 as a function of co for one
complete cycle.
/. flot (AeW + Ae~i at )/2 as a function of ut for one complete cycle.
36 ENGINEERING PROBLEMS
119. Determine the "attenuation" a and the "phase shift" for a
certain type of electric filter if
(a) a + ft = 2 log, [V(0.03155/180) 4- Vl + (0.03155/180)].
Note: Use the smallest positive angles for the square roots.
(b) a +J0= 2 log* (A +jg),
where A - > B - + 1,
Zi = 29.6 /86.1, and Z 2 = 10.61 /-90.
Remarks: The preceding problems are all stated in terms of electric net-
works. However, much the same sort of problem could be stated for
problems in mechanical vibrations.
PART III
ANALYTIC GEOMETRY
ELEMENTARY FORMULAS
120. Figure 41 shows a bridge truss. List the coordinates of A, B,
etc. Then find the slopes and inclinations of the members DE and EF.
121. A weight of 3 Ib. is placed at A (1,2) and one of 5 Ib. at 5(7,4).
Determine the coordinates of the centroid of the system (that point
which could be used as the fulcrum for a
lever with ends at A and B) by finding the
coordinates of a point P on AB such that
AP/PB = %.
122. The modulus of elasticity, much
used in strength of materials, is defined by
E = s/e, where s is the unit stress (load
per unit area) and e is the unit strain
(stretch per Original unit length). A rod
of steel stretches e 0.0005 in. per in.
when subjected to a stress of s = 15,000
Ib. per sq. in. Determine E. Also de-
termine the slope of the straight line joining the origin to the point
with coordinates (0.0005, 15,000).
Remark: The stress-strain graph for steel is shown in Fig. 42. The
modulus of elasticity, by definition, is the slope of the straight part of the
graph. Hooke's law, which will be studied in physics, states that stress is
proportional to strain, another definition for E.
37
FIG. 42.
38 ENGINEERING PROBLEMS
123. The graph of centigrade temperature as a function of Fahrenheit
temperature is a straight line. The student knows that 0C. corre-
sponds to 32F. and 100C. to 212F. Sketch the graph relating
centigrade (vertical) to Fahrenheit and determine the slope of the
resulting straight line. State the meaning of the slope value in good
English as it relates to the temperatures.
Note on Locus Derivations: There are no illustrative engineering applica-
tions of this topic in these problems. However, the following observations
are important. The solution of a locus derivation problem involves the
following steps:
First Step: Sketch a figure and label the given data.
Second Step: Select a general point on the locus, preferably one that seems
graphically to satisfy the statement of the problem.
Third Step: Make a geometric statement that must hold for this general
point on the basis of the statement of the problem.
Fourth Step: Translate this geometric statement to algebraic form with
the aid of the coordinates of the general point.
Fifth Step: Simplify.
Sixth Step: Check.
Compare these steps with the four steps that constitute the engineering
method for the solution of problems:
First Step: Sketch figures, such as free-body diagrams; label all relevant
points, lines, etc.
Second Step: Determine what fundamental engineering principle applies
(Newton's laws of motion, Ohm's law, basic theorems from plane geometry,
etc.); apply and obtain a mathematical problem.
Third Step: Solve the mathematical problem.
Fourth Step: Discuss the engineering implications of the mathematical
results, the limitations that were originally imposed, and their consequences.
Sometimes it is necessary to make more stringent assumptions at the
second step. This may lead to a more difficult mathematical problem to
solve in the third step. Such a situation arises when the results obtained
in the fourth step are insufficiently accurate.
A committee of the Society for the Promotion of Engineering Education,
in a report on the Aims and Scope of Engineering Curricula (The Journal of
Engineering Education, March, 1940, p. 563), states that an engineering
education should be directed, among other things, to a thorough under-
standing of this engineering method and elementary competence in its
application.
Perhaps the student will see, by a comparison of the locus derivation
process and the steps in the engineering method, that he is beginning in his
freshman year to develop this required understanding and competence.
STRAIGHT LINES
124. Figure 43 shows a straight-line speed-time diagram. The times
required for the three different motions are denoted by ti, fa, fa and the
total time by T. The corresponding numerical values of the accelera-
ANALYTIC GEOMETRY
39
tions shown in the accompanying legend are ai, a?, as, and these are the
numerical values of the slopes of the three straight lines.
a. Determine the coordinates of the points P, Q, and R.
b. Determine the total area enclosed by the polygon OPQRO, where
is the origin.
c. Since T = t\ + fa + fa and a\ti ad* a z fa = (why?), elim-
inate fa and 3 between these two equations and the equation for A
(the area) and simplify. .
Q
FIG. 43. Slope OP = ai, slope PQ = -a 2 , slope QR
125. The natural length of a spring is 8 in. and a force F of 40 Ib. is
required for each inch it is increased in length. Show that the equation
F = 40 (L 8) Ib. states these facts. Sketch the graph of the equation.
Then determine the area between this straight line, the L axis, and
L = 9 in. to L = 12 in., and give the proper units for the result.
126. A train (weight 200 tons without locomotive) starts to move
with a constant acceleration. If the resistance to motion due to friction
and air resistance is always 0.005 times the weight of the train, the pull
in the drawbar between the locomotive and train is given by
F = F 1 + Ma Ib.
where Fi = (0.005) (200) (2,000) = 2,000 Ib. = resistance due to friction
and air resistance, M = (200)(2,000)/32.2 = 12,400 slugs = "mass"
of train, and a is the acceleration.
Note: Slug = jr -
& ft. per sec. per sec.
Sketch for acceleration a from to 10 ft. per sec. per sec. What is
the force in the drawbar when the acceleration is zero and what geo-
metrical significance does this value have?
127. Suppose ib and e& are the variables in the fourth equation in
Prob. 64. What geometrical significance can you assign to the constants
E b , I b , and # L ?
40
ENGINEERING PROBLEMS
128. The horizontal beam shown in Fig. 44 weighs 100 Ib. per ft. of
length and supports a pile of sand distributed as shown.
a. Obtain an equation for the weight of the beam itself for the first x
ft. from the left support.
FIG. 44.
6. What is the equation of the straight line that forms the top of the
sand (the ordinate is 400 Ib. per ft. when x = and is zero when x = 10
ft.)?
c. What is the area of the trapezoid, shown in the figure, whose base
is x ft., and hence what is the weight of the sand above the first x ft.
of the beam?
d. What is the equation for the total weight of beam and sand for
the first x ft.?
2000 Ibs. 4000 Ibs. 6000 Ibs.
3' 3' j I' j 5'
6600
Odw 4300 J 4000
ill
>
3' 3'
I
-6100 [
1 "
FIG. 45.
129. A beam is 12 ft. long as shown in Fig. 45. It supports three
concentrated loads as shown. In strength of materials one learns
what the term "shear" means.
a. In this problem you are to write the equations for the four shear
lines (straight lines shown in the lower part of Fig. 45).
ANALYTIC GEOMETRY 41
b. Also determine the total area bounded by these shear lines and the
x axis (the area is positive if above the x axis and negative if below).
c. Determine the area between the first shear line and the x axis from
x = to x = X, where X is between and 3 ; between 3 and 6 ; between
6 and 7; between 7 and 12.
130. Let R denote the electrical resistance in ohms of a copper wire
1 mm. in diameter and 1 meter long at a temperature of TC. If
R = 0.0203 ohm when T = and R = 0.0286 ohm when T = 100C.,
and if the relationship between R and T is linear, obtain the equation
for R in terms of T arid state your resulting equation in good English.
131. Sketch a number of graphs for E = IR (Ohm's law in electrical
engineering) using 7 as the independent variable and E as the dependent
variable. Show graphs for JR = 0, 1, 2, 3, and 4 ohms. Notice that
this equation has physical meaning only for positive values of E and /.
132. The actual over-all fuel consumption of alcohol-gasoline com-
pared to unit volume of ordinary gasoline as found by road tests gives
practically a straight line when plotted on the basis of calorific values
of alcohol and gasoline. If R is consumption. and A is added alcohol:
72 = 1.00 + 0.0060 A. Sketch and give the physical meanings for the
slope and R intercept.
133. Diihrmg's rule gives an empirical relationship between the
absolute boiling points of two substances at two different pressures.
If T A and TB are the boiling points of two materials (A and B) at one
pressure and TA' and 7V at a second pressure, then
T A - T A ' = k(T B - T B '),
where k is a constant depending on the two materials. Discuss the
graphical significance of this rule.
134. The pressure on a certain piston is related to the volume between
the piston and cylinder head by the equation
p = 900 F + 3,000 Ib. per sq. ft.
The work done in compressing the volume from 1 to 0.5 cu. ft. is equal
to the area between the given curve, the V axis, from V = 0.5 to V = 1.
Determine this area.
135. The specific heat of mercury c at a temperature of TC. is given
by c = 0.03346 - G.000,009,27 7 calorics per degree centigrade (at con-
stant pressure). Sketch and determine the area between the straight
line, the horizontal T axis, and T = to T = 50. This area is equiva-
lent to the heat required to raise the temperature of 1 gram of mercury
from to 50C.
136. A cylinder is 12 ft. long and 10 in. in diameter and is lying on its
curved side. One end is kept at a temperature of 10C., and the other
42
ENGINEERING PROBLEMS
end at 100C. If the curved part is perfectly heat-insulated and if the
temperature at any point inside the cylinder is a linear function of its
distance from one end, determine the equation for temperature in terms
of the distance from the 10 end.
137. For an ideal gas the volume is a linear function of the temperature
for a given constant gas pressure. If the equation is v = ^o(l + ozT),
where v is the volume at abs. and a = His, sketch the graph for
V/VQ in terms of T.
138. Figure 46 shows the graph for a water cycle. H is the heat
quantity in calories per unit mass and T is temperature in degrees
centigrade.
Melting
E
C
Point
H
T
A
- 20
- 20
B
C
+ 70
D
170
+ 100
E
529
100
F
625
300
FIG. 46.
a. Determine the equation
of each of the straight-line
segments.
1 liquid" line, the T axis, from
b. Determine the area between the
T = to T = 100.
139. The vertical load (s Ib. per sq. ft.) at a distance of x ft. from the
left edge of the base of the dam shown\ in Fig. 47 (the load is caused
by the weight of the concrete and by
water pressure) is given by
s = 1,650 + 70.9(x - 10).
Identify and sketch.
Remark: It might happen, though it docs
not occur in this problem, that s would be
negative for certain values of x within the
base of the dam.. This would mean, phy-
sically, that the water was tending to over-
turn the dam and that in this interval the
dam was tending to pull its earth support upward.
Determine the total load on the bottom of the dam by multiplying
the area under the straight line from x to x = 20 by the length of
the dam, which is 40 ft.
ANALYTIC GEOMETRY 43
CURVE SKETCHING
140. An insulation wall is made up of a thickness of one material
with a thermal conductivity k\ and an equal thickness of a second
material of thermal conductivity k z . The thermal conductivity of the
total thickness k is then given by
a. Sketch the graph using k/k 2 for the dependent variable and k/ki
for the independent variable.
b. Sketch the graph using k z /k for the dependent variable and k\/k
for the independent variable.
The conductivity k of a material is the amount of heat in British thermal
units that will flow in 1 hr. through a layer of the material 1 sq. ft. in area
when the temperature difference between the surfaces of the layer is 1F.
per in. of thickness.
141. If x is the number of cubic meters of oxygen used to burn com-
pletely a given amount of carbon and if y is the number of cubic meters
of hydrogen required for the reaction, then
x + 0.5i/ = 0.353,
and g) - 1 - 10.662 (I - l).
Plot the graphs and solve these two equations simultaneously.
142. If t is the thickness of a thick cast-iron cylinder, r is the interior
radius, p is the allowable unit pressure, and s is the allowable unit stress,
then
t = r (5 T- -- 1 ), (for the external pressure)
\Z/p ~T~ 5 /
t = r ( - _ 1 J, (for the internal pressure)
Sketch a graph of each equation using s/p as the independent variable
and t/r as the dependent variable.
Remark: The variables in this problem, as suggested for the graph, are
dimensionless variables. It is common engineering practice to discuss
equations in terms of dimensionless-variable combinations.
143. A formula used in the design of beams that have been reinforced
with steel is
1
44 ENGINEERING PROBLEMS
where p is the percentage of reinforcement area with respect to the total
cross-sectional area, f s is the tensile unit stress for steel, f c is the com-
pressive unit stress for concrete, and n is the ratio of the modulus of
elasticity for steel to that for concrete = E 8 t e ei/E concre t e = 12 approxi-
mately. Sketch a graph of p as a function of f,/f c .
144. A weight of W Ib. hangs on a spring that stretches k ft.
when a 1-lb. weight is attached. An oscillatory force P sin cot Ib. is
applied to the weight. At any time after the oscillatory force is applied,
the deflection (y ft.) of the weight is given
approximately by
i
1
Va sin
I W \ tion is given by
where a = P/k and co 2 = kg/W (g = 32.2 ft.
per sec. per sec.).
The maximum value (amplitude) of this mo-
y
____ __
a~ I - (/) 2 '
a. Sketch a graph for this maximum value.
Use the dimensionless variable y/a as the dependent variable and co/w ri
as the independent variable.
b. "Resonance" occurs for ail positive values of co/co that would
cause division by zero (which is never allowed). What are these values?
To what in geometrical terms does resonance correspond?
Remark: o> n is the natural frequency of the weight-spring system (without
the oscillatory force). Resonance occurs when the frequency of the force is
equal to the natural frequency of the system.
145. In Prob. 144 let the support oscillate according to the equation
Y ao sin o)t ft. and omit the oscillatory force. Then the maximum
value for the deflection (y ft.) of the weight is given by
y (w/con) 2
do 1 (C0/C0n) 2
Notice that a is the amplitude of motion of the top of the spring and y
is the relative motion between the weight and the top of the spring.
Sketch with the same variables as before and give the "resonance"
values.
146. A simply supported beam (Fig. 49) is 12 ft. long and supports a
concentrated load of P Ib. at a distance of 8 ft. from the left end. The
equation of the curve that the beam would assume is given as follows:
ANALYTIC GEOMETRY
For a: from to 8:
y x* 64s
k ~ 18 9 '
and for x from 8 to 12:
45
k 18 9
A; is a large positive constant.
.y
9.' >
P
*, J
" 4 ~ >|
K
" 5 1
Curve of beam
FIG. 49
Sketch the graph for y/k in terms of x for the entire beam span. Use
the method of addition of ordinates to sketch the right-hand portion.
147. The formula for the Brinell number, used in a hardness test for
metals, is
B = ~ (10
where d is the diameter or width (in millimeters) of the impression which
a ball makes on the test specimen when dropped through a specified
distance.
Plot a graph of B as a function of d with special emphasis on the
interval from d = I to d = 10 mm. Verify the following data:
d
2.40
3.00
4.00
5.00
B
653
415
229
143
148. An approximate equation for the relation between steam decom-
position and producer gas is
Y =
0.267(/c -
where <p is the ratio of the amount of water vapor to the amount of
injected steam, k is the ratio of the amounts of nitrogen to liberated
hydrogen, and Y is a function of these two quantities.
46 ENGINEERING PROBLEMS
Sketch a series of curves by the method of addition of ordinates for
Y in terms of <p for k = 2, 4, 6, 8, and 10. Let <p vary from to 1 .
149. Sketch a graph of
~ = 0.14 - o.os (i - y - o.o
for z/r from to 1. This equation occurs in fluid mechanics.
1 312Q0 14 )
160. Sketch K = ' Q 332^4 for i = 212 - K is the heat-transfer
coefficient, t is the preheat temperature, and T is the retort temperature.
This problem occurs in chemical engineering.
Change the axes on your sketch so that the result will be a graph
for t = 735 (instead of for 212 as first used).
151. In a certain engine, the relationship between the pressure and the
volume of gas, i.e., in the space between the piston and cylinder head,
is given by the equation
p = 144 ( v 2 + ) lb. per sq. ft.,
where v is volume in cubic feet. Sketch a graph of p as a function of v
and estimate the value of the smallest positive pressure that can occur.
152. An empirical equation for train resistance R in lb. per ton of
train weight for different speeds S in miles per hour is
R = 3.5 + 0.0055& 2 16
(S + I) 2
Sketch a graph for 8 from to 60 miles per hour. What would be a
good empirical equation for speeds from 40 to 60 miles per hour?
GRAPHS OF THE CURVES : y = ax n
153. Under certain conditions the plate current in a two-element
vacuum tube is given by
/ = K - E* amp.
where E is the plate voltage, I is the plate current, and K is a positive
constant that depends on the geometry of the tube.
a. What is the general shape of this curve of / as a function of E,
irrespective of the value of /?
b. Sketch for K = 1, for K = 2, and for K = 3.
154. The length of life L of a gas-filled Mazda lamp operated on
voltage V volts is related to the life L operated at the rated voltage
V Q by the equation
L
ANALYTIC GEOMETRY 47
Sketch L/Lo as a function of V/ TV Also sketch the following equations
which are for lumen output Q, power output P, and lumens per watt K:
v ' PO ~ \v
155. A study of the formation of producer gas as made in chemical
engineering leads to the equation
(b a)v = x x b/a ,
where x is the proportion of residual water and v is the corresponding
amount of carbon dioxide (OO 2 ). The constants b and a are numerical
values that depend on the process.
If a = 3.17 and b 4.18, sketch a graph for v in terms of # as #
changes from to 1. Then estimate from your graph the value of z that
makes v a maximum,
156. The quantity of water (q cu. ft. per sec.) that flows over a partic-
ular type of spillway is given by
q = 3.Q8BHK,
where B is the width of the spillway in feet and H is the vertical distance
from the top of the water in the spillway to the water level some distance
before the water reaches the spillway passage.
Sketch a graph for q/B as a function of H.
157. A gas in an engine at a certain instant has a volume of 1 cu. ft.
and is at a pressure of 14,400 Ib. per sq. ft. The gas expands according
to the law pv = C\ until v = 2 cu. ft. and p = 7,200 Ib. per sq. ft. It
then expands according to the law pv lA 2 until v = 6 cu. ft. and
p = 1,550 Ib. per sq. ft. It then contracts according to the law pv = C 3
until v = 3 cu. ft. and p 3,100 Ib. per sq. ft. It then contracts
according to the law pv lA = (7 4 until v = 1 cu. ft. and p = 14,400 Ib.
per sq. ft.
Plot this "Carnot" gas cycle carefully on graph paper. Use v as the
independent variable.
158. The equation pv n C, where n and C are positive constants,
gives the relation of pressure to volume of a gas. Sketch p as a function
of v for n == 0, 0.5, 1, 1.25, 1.40, 1.60, and 2.
159. The following three equations give the volume, speed, and cross-
sectional area in a certain nozzle, each in terms of the pressure p:
/200V 71 - 4
V = 0.962 (~ cu. ft. per Ib.,
r f v \ 4/i - 4 T^
v = 2,500 [l - \^o6/ J ft *
per sec ''
A = sq. ft.
48
ENGINEERING PROBLEMS
Sketch these three graphs on the same axes. Use a range for p from
to 200.
160. A canal lock has vertical sides and a rectangular horizontal cross
section of area M. The water discharges through an outlet of area A.
The time (t sec.) required for the water level to fall from h^ to h\ ft.
is approximately
a. If h% hi + y ft., show that
M 2 ^ M
b. If A = 4?r sq.ft., M = 30,000 sq. ft'., and t = 20 min. = 1,200 sec.,
show that y 101 + 20.1 v/k, approximately, and sketch a graph of
y as a function of h\.
161. A loud-speaker horn is to have a fixed length and fixed radii
at the throat and mouth. Its outline is to be such that the radius y
FIG. 50.
(see Fig. 50) is a power function of the distance along the axis, measured
from some point to be determined. For convenience, take the origin
at the center of the throat, the length as one unit, and the throat radius
as one unit (of course to a different scale). Let the mouth radius be y\.
Obtain the equation for the top section of the horn by starting with
the equation
y = k(x + a)*.
ANALYTIC GEOMETRY
49
Determine the values of the constants k and a so that the curve will go
through (0,1) and (l,2/i) and show that the result is
If y\ = 10 (vertical units), compute y when x = 0.5 for n = 2, 10,
and 100.
Sketch the horn outlines on the same graph assuming that yi = 10 for
the particular values of n: 1, 2, 3, and 5.
CIRCLES
162. Figure 51 is known as "Mohr's circle diagram" for stresses and
is used in the course in strength of materials. Obtain the equation of
E
FIG. 51.
FIG. 52.
the circle in variables (<r,r) and the coordinates of the point E, all in
terms of the constants <r x , cr y , and a. The coordinates of point E are
the values of the stresses a and r on the small triangular block shown
in Fig. 52.
Remark: If one studies the stresses in a small triangular section of abeam
(as shown in Fig. 52), the stresses on a plane making an angle a as shown
/ 9 \ I _
( tan 2a = ^ ) are given by <r == -^-s - + o cos 2a,
(Tf. (Tn / & &
163. Determine the equation relating I\ and J 2 (in terms of E/R) in
the triangle shown in Fig. 53. Discuss your result.
Remark: Texts on alternating currents use circle diagrams to study the
real and imaginary parts of the vector current. The present example is
for a series circuit consisting of a resistance and inductance.
50
ENGINEERING PROBLEMS
1 9
FIG. 53.
164. a. Show that the equations
V = In [(at + a)* + ifl - In [(x - a)* + jfl,
= JL tan -. (_^\ _ t^-x f _L _ V
' 2?r \# + / 2 ?r \^ /
can be rewritten in the following forms:
(a; + a) 2 + y* = e 4 '^[(a; - a) 2 + ?/],
2 2ay
tan = -- r~T~~9 ~~^ J
g a: 2 + ?/ 2 er
and that these two equations can be written in turn in the forms [where
^ =f e 4Kv/ q an( j c = cot (2irri/q)]
x*(l - b) + 2ax(l +b) + y*(l - fe) = a 2 (?; - 1),
a 2 .
6. Identify these two curves, assuming that a, 6, and c are constants.
c. Let a = 1 unit ( = 1 in.). Sketch the first curve for 6 = 3, 1.5,
1.2, 1.1, 1, 0.9, 0.8, and 0.6. Sketch the second curve for c = 2, 1,
0.5,0, -0.5, and -1.
Remark: Your resulting graph has the following properties: Suppose that
at the point ( a,0) in your sheet of paper you allow water to flow out in all
directions (the sheet of paper to be horizontal and of unlimited length and
width). Suppose that there is a small drain pipe placed at (,0). Then
the second set of curves you drew were the "path lines" for the flow. The
first set of curves has a physical significance that will not be described here.
This same type of problem occurs in the study of the flow of heat,
electricity, and moisture, in the study of a drying process, etc.
165. Use a sheet of graph paper and draw to a large scale the circle
with center at (0.2, 0.5) and radius 1.3. Let the coordinates of any
point on this circle be designated by (x,y) and write z = x + iy, as
you may have done either in college algebra or in trigonometry in
connection with complex numbers. Read from your graph the approxi-
mate coordinates of a number of points on the circle and write each pair
ANALYTIC GEOMETRY 51
in the z form. Compute w z + (I /z) for each such value of z. Plot
the values of w u + iv on a new sheet of rectangular graph paper with
v as the vertical scale and u as the horizontal. If you are careful and
use enough points on the original circle, your resulting graph on the
u, v axes will be a possible airfoil.
PARABOLAS
166. A beam AB of length L ft. and weight W Ib. is hinged at B
and held in a horizontal position at A
(see Fig. 54). When released at A the A B
beam begins to rotate about B. At the I
instant it is vertical, the end B is re- I
leased and the entire beam falls. The
equation of the path traced by the
center of the beam after B is released,
referred to the indicated axes, is
4# 2 = QLx - 3L 2 . Identify and sketch
using y/L as a function of x/L (dimen- FIG. 54.
sionless variables).
167. Figure 55 shows a beam which is built in at both ends. The
"bending moment" (to be defined when you take strength of materials)
for this beam is given by M = wLx/2 wL' 2 /l2 wx z /2, where x ft.
is measured from the left-hand wall.
Identify and sketch a graph of Y M/wL 2 as a function of X = x/L
(dimensionless variables). Then
determine the coordinates of the
highest point and the X and Y
intercepts. Give the equivalent
values for M and x for each of
J these points.
j w Ibs./ft.
P, _ r
Remark: All these points have
physical significance. The larger of
the numerical values of the Y inter-
cept and the Y value of the vertex
determines the point in the beam
where the stress is largest. The X intercepts determine those points in the
beam where the tension or compression (pull or push) is zero.
168. A simply supported beam (Fig. 56) is 12 ft. long, is made of
yellow pine, and is rectangular in cross section, being 6 in. wide and
10 in. deep. The equation of the "curve of the beam" is given as
follows :
For x from to 3ft.:
y - 0.000,349z 3 - 0.028,27z ft.,
52 ENGINEERING PROBLEMS
Fora; from 3 to 9ft.:
y - 0.003, 142z 2 - 0.037,70z + 0.009,42,
For x from 9 to 12 ft.:
y - 0.000,349(12 - x) 3 - 0.028,27(12 - x).
a. Identify the curve of the middle portion of the beam and deter-
mine its lowest point. I
b. Plot a graph of the "curve of the beam" for the span of the beam,
i.e., for x from to 12 ft.
Curve of beam
FIG. 56.
169. A weight is thrown directly downward from the top of a high
building with an initial speed of 48 ft. per sec. Its distance below the
top of the building (neglecting air resistance) t sec. after being thrown
is s = 16J 2 + 48Z ft.
a. Identify and sketch the graph of this locus.
b. Determine the time when the weight hits the ground if the building
is 640 ft. high.
170. A cable supporting a suspension bridge hangs in the form of a
parabola. The tops of the supporting towers are 35 ft. above the floor
of the bridge and the lowest point of the cable is 5 ft. above the bridge.
The distance between the supporting towers is 60 ft. Determine the
length of a suspending cable (a vertical cable from the bridge to the
parabolic cable) 10 ft. from one of the supporting towers.
171. A bullet is fired upward at an angle with the horizontal and
with an initial velocity of V ft. per sec. (see Fig. 57).
The equation of the path of the bullet referred to axes as shown
(neglecting air resistance, etc.) is given by
x = Vt cos 0,
y = - Y + vt sin >
where g 32.2 ft. per sec. per sec., approximately.
a. Solve the first equation for t in terms of x, V, and B and substitute
for t in the second equation. Simplify and obtain
~~ \2Fv
y = ~~ sec2 e + x tan
ANALYTIC GEOMETRY
53
b. Show that the equation of the directrix of this parabola is
F 2
Remark: This result is independent of the angle 0. Hence the directrix is
the same irrespective of the angle of fire. This y value for the directrix is
the distance the bullet would rise if fired vertically.
FIG. 57.
172. An empirical equation for the resistance of a locomotive and
tender to motion on a straight level track is
L = 6.0 + 0.0035(S - 10) 2 ,
where L is resistance in pounds per ton of weight and S is speed in
miles per hour.
a. Sketch L for values of S from to 50 miles per hour.
b. What is the smallest resistance and at what speed does it occur?
173. Figure 58 shows a cable suspended from a trestle. If the curve
of the cable is a parabola, determine the lengths of AC, CD, and BD,
each in terms of L and p.
L B L C L
FIG. 58.
FIG. 59.
174. Figure 59 shows an arched truss 80 ft. long, hinged at A and on
rollers at B. The rollers roll on a plate that makes an angle of 30 with
the horizontal. The vertical stringers are 10 ft. apart.
Determine the sum of the lengths of the vertical and inclined stringers,
correct to three significant figures, assuming that both the top and
bottom of the arch are arcs of parabolas. One method of obtaining this
54 ENGINEERING PROBLEMS
result is to plot the figure carefully to a large scale and measure the
lengths of the separate stringers.
*175. A gun at A fires a projectile which pierces a captive balloon
at B and then hits the ground at C. The plane of the ground is hori-
zontal. Let D be the projection of B on this plane. The distance
AD = 2,000 ft., DC = 1,000 ft., the angle of elevation of the balloon
from A is arc tan 2. A first approximation for the curve which the
projectile follows is a parabola.
a. Determine the equation of the path of the projectile, referred to
axes through A.
b. What is the maximum height that the projectile reaches? That
is, what is the ordinate to the vertex?
c. What are the values of the slope and inclination of the chord
through A and the point on the parabola whose abscissa is x 1 ft.
and hence what is an estimate of the angle at which the projectile was
fired?
176. At what speed must an open, vertical, cylindrical vessel, 4 ft.
in diameter and 6 ft. deep and filled with water, be rotated about its
axis so that the effect of rotation will be to uncover a circular area on
the bottom of the vessel 2 ft. in diameter? The cross section of the
inner surface of the rotating water will be an arc of a parabola. The
equation of this parabola, referred to axes through its vertex (the y
axis .will be altmg the axis of the cylinder and the vertex will be below
the bottom of the vessel) is y ai 2 x z /2g, where co radians per second
is the angular speed of the vessel and g is approximately 32 ft. per sec.
per sec.
177. If y is the concentration of acetic acid in ether and x is the
concentration of acetic acid in benzene, it has been found that y 2 = kx
where k is an empirical constant. Sketch a schematic graph.
178. With reference to the 4 e c equation described in Prob. 228,
determine the coordinates of the vertex and sketch the curve.
ELLIPSES
179. In a course on strength of materials one makes use of the
" ellipse of stress" whose equation is (x 2 /2 2 ) + (2/ 2 / s i 2 ) 1> where 2
and si are constants. Sketch the graph if s 2 = 6,360 Ib. per sq. in.
and si = 2,640 Ib. per sq. in.
180. Engineering mechanical drawing courses sometimes include a
method of constructing a rough ellipse. A rhombus is constructed and
lines are drawn from the "wide" angles to the mid-points of the opposite
sides. The intersections of these lines by pairs on the line joining the
"small" rhombus angles are the centers for two circular arcs. The
vertexes at the wide angles are the centers for two other circular arcs
and the four arcs make a crude ellipse.
ANALYTIC GEOMETRY
55
a. Construct such an ellipse using vertexes at (0, 6), ( &,0), (0,6),
and (&,0) and 6 = 3, k = 6.
6. Determine the value of a (the semimajor axis) for this special
case.
c. What is the error in y in this special case when x = 4? The
percentage of error?
d. Determine a formula for a in terms of 6 and k.
181. An airplane strut is 6 ft. long and is tapered uniformly from
the middle toward both ends. Every section of the strut is an ellipse.
The axes at the middle are 2a = 1 in. and 26 = 0.75 in. and at both
ends 2a = 0.75 in. and 26 = 0.5 in.
a and 6 are each linear functions of
the distance x from the center.
a. Determine .the relations for a
and 6 in terms of x (inches).
6. Determine the cross-sectional
area as a function of x in. and sketch
its graph.
182. An arch has a cross section,
as shown in Fig. 60, with the curve
a semiellipse.
a. Determine the lengths of the
ordinates to the arch measured from the ground at every 2 ft. distance
from the point A.
6. If the arch is 10 ft. thick, determine the number of cubic yards of
concrete necessary in its construction.
HYPERBOLAS
183. Sketch a graph of X as a function of the positive values of /
if X = 27T/L l/(2irfC) for the following sets of values for L and C.
Determine algebraically and from your graph the value of / that makes
X zero.
a. L 0.00025 henry, C = 10~ 10 farad (data for a radio circuit).
6. L = 1 henry, C = 7(10~ 6 ) farad (data for a power circuit).
Remark: This equation gives the net " reactance" X in an a.c. circuit
containing inductance L and capacitance C (as well as resistance R) in series
with a sinusofdal voltage of frequency /.
184. Use is made in radio theory of the "gain" obtained by the use
of a three-element vacuum tube. The formula for this gain is
f ^ . &RL
Gain = TFT~ ; ^ *
where R P is the plate resistance and RL is the load resistance.
56 ENGINEERING PROBLEMS
a. Sketch a graph of gain/ju as a function of R P /RL and identify the
curve.
b. Sketch a graph of gain//z as a function of RL/RP and identify the
curve.
186. An indicator card, which shows how the pressure varies with
the volume in an engine cylinder, has the theoretical shape shown in
2 Feu. ft.
Fig. 61. Determine the equation of each part of the graph assuming
that the curves BC and FE are arcs of hyperbolas (p V = constant).
a. Use p in pounds per square inch and V in cubic feet.
6. Use p in pounds per square foot and V in cubic feet.
186. The following equations give the "kinematic viscosity"
(K.V.) = y in terms of 8 in "Saybolt seconds":
1 88
y = 0.002,168 - -~ for 8 from to 100,
1 35
y = 0.002,205 ^- for 8 larger than 100.
Sketch y = K.V., as a function of 8 for 8 from to 200.
187. In the study of strength of materials one has the formula
s <
(?) + G)
+
which gives the maximum tensile unit stress St when a bar is subjected
to combined tensile and twisting loading, as suggested by Fig. 62. In
this formula s t is the tensile unit stress due to the axial load (P Ib.)
and s s is the shearing unit stress due to the twisting load.
a. Assuming that s s = 400 Ib. per sq. in., identify and sketch the
graph of S t as a function of s. Use the positive value of the square
root.
ANAL YiyC GEOMETRY
57
b. Determine the relation approached between St and s t as s t increases
without limit through positive values and give the geometrical signifi-
cance of this result.
c. Determine the relation approached between St and s t as s t increases
without limit numerically but through negative values. Give the
geometrical significance.
d. In this same problem, the minimum tensile unit stress (or maxi-
mum compressive unit stress) is given by
= (j) - G
and the maximum shearing unit stress is given by
Sketch these two curves on the same graph with your first curve (for
the same value assigned to s).
\ Twisting
joad
Plbs.
FIG. 62.
188. A listening post B is 4 miles east and 3 miles south of a listening *
post at A. The explosion of a gun is heard at B 10 sec. before it is
heard at A, and the explosion sound comes from an easterly direction.
Since sound travels at about 1,086 ft. per sec., the gun is about 10,860 ft,
closer to B than to A.
a. By graphical methods construct the asymptotes for the hyperbola
defined by the given data and sketch in the curve on which the gun
must approximately lie.
6. Choose the x axis through the two listening posts and the y axis
as the perpendicular bisector of the line segment AB and determine
the equation of the hyperbola.
ROTATION OF AXES
189. The foundation bolts for a certain machine are to be placed at
the points A, B, (7, . . . , L, as indicated in Fig. 63. Determine the
58
ENGINEERING PROBLEMS
distances from the two walls, marked as the x and y axes, to each bolt
center.
y
D
r, zr
-E\
FIG. G3.
Data: A x = 3ft., A y = 5ft.,
AB = BH - 4 ft.,
AH = 7 ft., S = 30,
J5C = CD = JvL - #/? = EG = AL = 1 ft.
190. Figure 64 represents an airplane climbing upward. Let L be
the lift force, D the drag force, N
the force perpendicular to the
direction of climb, T the force in
the direction of climb, and a the
angle of climb. Show that
N = L cos a + D sin a,
FIG. 64. T = ~~"k sin a + D cos a,
and also determine a similar set of formulas for L and D in terms of
N, T, and a.
GRAPHS OF SINE WAVES
191. Given the equation y = F m cos (oo + a), in seconds.
a. Sketch the graph if Y m = 3, co = 2 radians per second, and
a = 7T/6 radians. Label the maximum value Y mj the angular f re-
gency co, the period p = 27r/co, and the phase angle a (label it as the
abscissa from cof == a; to toZ = 0). Sketch a "second graph using co
as the independent variable and label the above quantities.
b. Show that the equation may be written in the form
y = Y m sin
a +
c. If the axes are translated in your second sketch so that the equa-
tion is y' Y m cos (co')> what are the equations of translation?
192. If the total current in a conductor is the sum of the two com-
ponents: ii = /i cos cat amp., 2 = 1 2 cos (coZ 6), where /i and 7 2
are both positive constants, show that the sum i = i\ + i$ can be
ANALYTIC GEOMETRY
59
written as a cosine wave in the same form as that for t' 2; determine its
amplitude, period, and phase with respect to i\, and sketch all three
curves (schematically) on the same graph. Use co as the independent
variable.
193. In a three-phase system, three outgoing currents in three differ-
ent wires are, respectively .
ii I cos 2irft amp.,
iz = I cos ( 2irft o~ )
/ 47T\
z* 3 = I cos I 2irft 5- ) amp.,
) amp..
FIG. 65.
and all three have a common re-
turn wire which carries the current
IR i\ + ^2 + iz (f is frequency and co = 2?r/).
Evaluate i R in the form of the given currents and sketch all four waves
on the same graph. Use a common abscissa variable 2irft.
194. In a problem similar to the preceding, suppose that i R is known
to be zero and that i\ = / cos 2irft, iz I cos (2irft STT/G). Use the
equation i R = u + iz + iz = to express i 3 as a sinusoidal function
of time in the same form as the functions for i\ and iz. Sketch all three
curves on the same graph and give the amplitude, frequency, and phase
angle for the graph of i 3 . Use x ~ 2irft as the independent variable.
195. The mechanism shown in Fig. 66 is a piston at the end of a
connecting rod. The crank arm OA is 10 in. long and revolves at
co = 120 revolutions per minute = 4?r radians per second.
/ \X0=47T
1 $
J
"Jl _
1
^ i-
?~S
It $
^ r
$ i
/
1 $
u r-
r-^
FlG.
66.
a. Show that the displacement of the piston from the position it would
occupy when the crank arm is vertical is given by
x = 10 cos 4?r in. =
cos
ft.
6. By methods of calculus it can be shown that the speed and accelera-
tion of the piston at any time t sec. are given by
v = speed = (4rr)(%) sin Airt ft. per sec.,
a = acceleration = (167r 2 )(%) cos 4irt ft. per sec. per sec.
60 ENGINEERING PROBLEMS
Sketch on the same graph the graphs of x, ?', and a, each as a function
of time t sec.
c. The force that* the crank arm transmits to the piston is always
given by force = (mass) (acceleration). If the moving mechanism
weighs 100 lb., it can be shown by methods of physics that the mass is
10 % 2 and hence that the force is
Sketch a graph of F as a function of t.
196. A streetcar oscillates harmonically in a vertical direction on its
springs. The amplitude of motion is I in., the frequency is 2 cycles per
second, the loaded cab weighs 20,000 lb., and the truck and wheels
weigh 2,000 lb. The force acting on the rails at any time t sec. can be
shown by methods of mechanics and calculus to be approximately
F = 22,000 +
= 22,000 + 8,170 sin 4?rZ lb.
Sketch a graph of F as a function of the time. What is the maximum
force acting on the rails? The minimum force?
197. The terminal voltage of an a.c. generator and the current sup-
plied are, respectively,
e = 100 sin 120?rt volts,
i = 5 sin ( 1207r + o ) amp.
Sketch e and i on the same graph as functions of the common abscissa
6 = 12Chr, but with different meanings for their ordinates. Also
sketch on the same graph the product p = c i, which is the instanta-
neous electrical power delivered by the alternator. You can simplify
your work to obtain the last graph by first expressing
= c-i = 500(sin 120rrO sin l20?rt + = 125 - 250 cos
198. The impressed voltage in an electric circuit is given by
e = 100 sin (120irf)- + 20 sin sGOTrt - volts.
Sketch e as a function of 1207r radians; t is in seconds.
199. The equation of an amplitude-modulated voltage in radio
is given by
e volts = 100(1 + 0.7 cos 4,000* - 0.3 cos 8,0000 sin 4,000,000,
where t is in seconds.
ANALYTIC GEOMETRY 61
a. Sketch the boundary curves for one cycle; i.e., sketch
e = 100(1 + 0.7 cos 4,000* - 0.3 cos 8,0000-
b. From your graph estimate the peak value of the given voltage.
EXPONENTIAL AND LOGARITHMIC GRAPHS
200. The current flowing in a series circuit (with inductance L henrys,
resistance R ohms, and a voltage E volts) is given by
(TyTV
_ j (i _ e-m/L) amp .
and the power going into the magnetic field is given by
fE*\
p = -- ( e -Ri/L _ e -*nt/L) m
a. If R = 10 ohms, L = 0.0001 hein-y, and E = 15 volts, sketch
graphs of P and / as functions of the Dime t.
b. Sketch graphs of RI/E and RP/E* as functions of Rt/L. These
are dimensionless variables.
201. In the circuit of Prob. 200, the power going into the resistance
is given by
P R = PR
and the energy stored in the inductance is given by
Using the formula for / in Prob. 200 and these formulas, sketch graphs
of P R R/E* and P L R 2 /LE* as functions of Rt/L.
202. A problem in engineering required the simultaneous solution
of the two equations:
35 = E m (l - e-"),
64 - E m (l - e-*").
What method would you devise for solving these two equations simulta-
neously, if the accuracy required is two significant figures? What is
the common solution to that accuracy?
203. The following is taken from an article in Chemical Engineering
on gasoline cracking. Sketch graphs of x and y as functions of the time
tit
x = 100(1 - e~ kt ) and y
where k = 0.01076.
62
ENGINEERING PROBLEMS
204. The "magnetic flux" (in maxwells per centimeter of length)
between two parallel wires of a transmission line is given by
<p = 0.47 log e
D - r
where J is the current in amperes, D is tne distance in centimeters
between the centers of the two wires, and r is the radius of the wires
in centimeters.
Sketch a graph of tp/I as a function of r/D. What happens graphi-
cally and what happens physically when r/D = K ?
205. In a cylindrical cable in which the
inner conductor has a radius r and the outer
conductor has a radius R (see Fig. 67) the
maximum electric intensity in the insulation
(there is some loss of current through the
insulation) is given by
E m = -", -
Insulation
FIG. 67.
r log. (R/r)
and occurs at the surface of the inner conduc-
tor. (Electrical breakdown occurs when the
voltage V between the conductors is large
enough to make E m greater than a particular value which depends on
the kind of insulation.) If y = E m R/V and x = r/R, show that
1
11 :
^ X \Oge X
a. Sketch a graph of RE m /V as a function of r/R for < r/R < 1.
b. Estimate from your graph the value of the ratio r/R that gives
the smallest value for E m , assuming
that V and R are constants.
c. Sketch a graph of E m as a func-
tion of 7 if R = 2r = 0.6 cm.
206. An equation for "belt fric-
tion" is
FIG. 68.
where ju is the coefficient of friction, a
is the angle of contact, and T% and
are the "pulling" forces on the two
ends of the belt (see Fig. 68).
a. Sketch a graph of T\/T* as a function of a, if p, = 0.4. Show the
graph for a from to 4?r.
b. Compute TI if T 2 = 1,300 lb., /* = 0.3, and a = 37T/4 radians.
ANALYTIC GEOMETRY 63
207. A weight hangs at the end of a spring and vibrates so that its
deflection, y ft., from a certain position is always given by
y = e -o.o2(o.4 sin 100* + 0.3 cos 1000 ft.
t is in seconds.
a. Sketch a graph of y as a function of t from t = to t = 0.1 sec.
b. Show that the equation can be rewritten in the form
y = Ae-- 1 sin (100* + 0),
and compute the values for A and 0.
c. Determine the time t sec. after which y is always less than 0.005 ft.
(Answer this question by solving for t in 0.5e~ 2 ' = 0.005.)
208. If a gas expands in a cylinder according to Boyle's law (pV =
constant for a constant temperature), the work the gas does is given by
W (piVi) log e
where p\ is the initial pressure, V\ is the initial volume, and 2 is the
final volume.
a. Sketch a graph of W as a function of V% if p\ = 600 Ib. per sq. ft.
and Fi = 0.4 cu. ft.
6. Sketch a graph of W/piVi as a function of Vt/Vi.
c. Compute W if pi = 600 Ib. per sq. ft., V\ 0.4 cu. ft., and
7 2 = 0.85 cu. ft.
HYPERBOLIC FUNCTIONS
209. A cable of length L ft. is suspended between two points which are
in the same horizontal plane and which are a ft. apart. Given that the
length of the cable is L = 2c sinh (a/2c) and the sag is
* i T- ( a
f + c = c cosh I 2~
If L = 100 ft. and a = 80 ft., these equations become
100 = 2c sinh ( )> / + c = c cosh
Solve the first equation graphically for c by plotting graphs of y = 5a;/4
and y = sinh x, where x = 40/c. Substitute this value for c in the
second equation and determine /.
210. The following equations give the voltage E 8 and the current /
required at the sending end of a cable to yield a voltage E r and current I r
at the receiving end :
64 ENGINEERING PROBLEMS
E 8 = E r cosh (L Vfg) + I r \ sinh (L
L = I r cosh (L Vrg) + E r \ ^ -y sinh (L
where L is the length of the line in miles and r and g are constants for
the line.
If r = 20 ohms per mile, g = 0.000,02 mho per mile, E r = 100 volts,
and I r = 0.2 amp., show that these equations become
E s = 100 cosh 0.02L + 200 sinh 0.02L,
/, = 0.2 cosh 0.02L + 0.1 sinh 0.02L.
a. Compute E s and 7 S , each correct to three significant figures, when
L = 10 miles and when L = 100 miles.
b. Sketch graphs of E s and I s as functions of L for L from to 150
miles.
211. Given that sinh (a + jb) sinh a cos b + j cosh a sin 6,
cosh (a + jb) = cosh a cos b + j sinh a sin 6,
compute the values of
(a) sinh (0.785 +#.87),
(6) cosh (0.785 +#.87).
This type of problem occurs in senior electrical engineering courses in
both power and communications.
212. A weight rests upon a rough horizontal table at a point, on
an assumed set of axes, whose coordinates are (Q,h). A string of length
h is attached to the weight and the other end is held at the origin.
The free end of the string is then pulled along the positive x axis and
the weight follows along a curved path whose equation is
x h secli" 1 I T J
\h/
Show that this equation can be written in dimensionless-variable form,
X = x/h and Y = y/h, as
X = sech- 1 Y - VT^Y\
Sketch this last equation by the addition of abscissas. The curve is
called a "tractrix."
POLAR COORDINATES
213. Secure either a picture or an actual chart from a recording
pyrometer and study it. The radius represents temperature, and the
angle is marked in units of time.
ANALYTIC GEOMETRY 65
214. A block weighing 10 Ib. rests on a horizontal surface for which
the coefficient of friction is 0.3. A force P, inclined at an angle B with
the horizontal (Fig. 69), acts on this block and is just enough to cause
motion of the block to impend. The
value of P is given by the equation
Q
p = cos B - 0.3 sin 0.
a. Sketch a graph in rectangular
coordinates of 1/P as a function of B.
b. Sketch a graph in polar coordinates of 1/P (radius vector) as a
function of 8.
c. Is there any value of 8 (acute) for which P is infinite, and hence
1/P zero? If so, locate on each sketch.
215. A cam is to be built with a cross section defined by
p = 4 + 2 cos 8 in.
Sketch the cross section of the cam and also the "layout" curve, i.e.,
the curve whose rectangular coordinate equation is
y = 4 + 2 sin x.
216. Antenna radiation patterns, such as can be found in modern
radio textbooks, may be plotted from equations such as the following:
cos
( 5 cos )
V2_Z,
sin
fr COB g)
sm
sin (2?r cos 0)
sin0
a. Plot each of these graphs on polar coordinate graph paper.
6. Plot each of these on rectangular coordinate graph paper using
as abscissa and F(6) as ordinate. Plot for < < 2ir.
217. Assignment for a mechanical engineering student: Secure from
your engineering library a copy of a textbook on machine design. Look
over the discussion on cams and layouts and take the textbook to class
to show the diagrams to the other members of the class.
218. a. Sketch the polar coordinate graph for p e~ QMQ for the
range from = to = 25.
b. Let OP be a vector with at the pole and P some point on the
curve drawn in part (a). Show that the horizontal component of OP is
x = e-- 046 cos 0.
66 ENGINEERING PROBLEMS
c. Now suppose that OP rotates about with an angular speed of 2ir
radians per second so that 6 = 2wt radians. The preceding equation
becomes
x = e-o- 281 * cos 2wt.
Sketch the rectangular coordinate graph of x as a function of t as t
increases from to 8 sec.
Remark: This concept of a rotating vector is used in texts on vibrations to
discuss damped free vibrations. If the vibration were undamped, the
point P would move along a circle instead of along the spiral drawn for
part (a). This problem may also be interpreted in terms of a transient
alternating current.
PARAMETRIC EQUATIONS
219. A particle moves in a counterclockwise direction around a circle
of radius 10 in., starting at the right end of the horizontal diameter.
At the end of t sec. the particle has moved through s = 4 2 in. Deter-
mine the parametric equations for x and y in terms of t (the abscissa
and the ordinate measured from the center of the circle).
220. A railway easement curve (used to join a straight track to a
uniform or circular track) has the following parametric equations when
designed for a speed of 33 miles per hour. Plot, taking u at intervals
of 0.1 from to 1.
(?/ 5 \
u ~~ To/ ft "'
' / 7 \
y = 200 (u* - ~) ft.
221. A point on the rim of a wheel of radius 2 ft. moves in such a
way that its coordinates (referred to axes through the center of the
wheel) are always given by x = 2 cos 3t ft., y = 2 sin 3t ft. The
velocity at any time t sec. in the x direction is given by v x = 6 sin 3 ft.
per sec. and in the y direction by v v = 6 cos 3t ft. per sec.
Plot the curve in polar coordinates (r,0) which is given in para-
metric form by r 2 = w* 2 + % 2 = (-6 sin 3J) 2 + (6 cos 3*) 2 and
tan = Vy/v x . Label the points on your graph which correspond to
t = 7T/6, 7T/3, 7T/2, and 2?r/3 sec. (t = corresponds to = ir/2, r = 6).
Remark: In mechanics the polar coordinate curve plotted in this problem
is called the "hodograph" for the given motion. It will be defined in that
course.
222. A point moves around a curve whose parametric equations are
y = 16 t 4 ft., x t* ft., where t is in seconds. v x = 2t ft. per sec. and
v y = 4/ 3 ft. per sec. (See Prob. 221 for the meaning of v x and v y .)
ANALYTIC GEOMETRY
67
a. Sketch the graph of the given motion curve. Locate those points
which correspond to t = 0, t = 0.5, and t = 1 sec.
b. Sketch the curve in polar coordinates whose polar parametric
equations are r 2 = v x * + v y 2 , tan 6 = v y /v x . Locate the points on this
second curve that correspond to t = 0, 0.5, and 1 sec.
Note: When t = 0.1, r = 0.2 and 6 = -19'.
c. At the point that corresponds to t = 1 sec. in the second graph 7
draw the tangent line by aid of a straightedge and compute its slope.
223. Figure 70 shows a crank arm OB of radius r, which rotates
counterclockwise with a constant angular speed of o) radians per second.
FIG. 70.
A connecting rod BA has a length of L ft. and the end A moves up and
down the x axis. Assume that r/L = H.
a. Determine the parametric coordinates for the point B as a function
of the time t sec., measured from the time B was at the right-hand end
of the horizontal diameter of the circle.
6. Show that the abscissa for the point A is always given by
x = r cos <j)t + (L 2 r 2 sin 2 o>0^ ft.
c. Let z cos cot. Show that the result in part (6) can then be
written as
x = rz +
- r
2 + rV = 0.2Lz + L Vo.96 + 0.04z 2 .
Sketch x/L as a function of z. (Observe that z can vary only between
1 arid + 1 and also that the sign in front of the square root is to be
positive.)
d. What are the parametric equations for the curve which the mid-
point of the connecting rod traverses? Eliminate the parameter and
sketch this curve by the method of addition of abscissas.
224. Observing a point on a vibrating pedestal of a rotating machine
through a microscope reveals that such a point usually describes an
68
ENGINEERING PROBLEMS
elliptic curve as illustrated in Fig. 71. This ellipse may be considered
as clused by simultaneous harmonic motions in the horizontal and verti-
cal directions (Lissajous figure). The vertical and horizontal motions
have a certain phase displacement a with respect to time.
a. Determine A, B, and a so that the parametric equations for the
ellipse are
x = A cos ut, y B cos (co a).
b. Determine a condition on the given dimensions (a,&,c, and d)
so that this type of motion will be possible.
225. Graph the following Lissajous figures (see Prob. 224):
(o)
0)
(c)
x = sin 2?r, y = 2 cos TT;
x a sin STT^, y b sin 2irtj
x = 6 sin 2irt, y 8 sin ( 2?r ^ ).
226. A cylindrical water tank is to be kept full of water. At a
distance of 9 ft. below the top of the tank is a small circular opening.
The water is allowed to start flowing out of this hole. It is known, from
fluid mechanics, that the path of the water is a parabola (to a good
approximation) with its vertex at the hole.
If the origin is taken at the hole, x measured in feet outward from the
hole, and y measured in feet downward from the hole, the parametric
equations for the water path are
y
TT, X
Vt,
where g is approximately 32.2 ft. per sec. per sec. and v is the speed of
the water as it leaves the hole.
a. Eliminate t and sketch the curve.
b. If x 8 ft., y 2 ft., is a point on the water path, determine the
initial speed (v in feet per second).
ANALYTIC GEOMETRY
69
227. The crank OP in Fig. 72 is 10 in. long and rotates around in
the plane XOY with a constant angular speed of co radians per second.
AP = PB = 10 in., and A and B are
free to slide along the x and y axes,
respectively. The point M is mid-
way between A and P. Determine
the parametric equations for the
curves traced by the points P and
Ml Sketch the curves and identify
them.
228. The plate current i b in milli-
amperes in a three-element vacuum
tube is given in terms of the grid
voltage e c in volts (for an assumed
resistance load in the plate circuit) by
FIG. 72.
A sinusoidal voltage, e c = 20 + 20 sin where 6 = l,00(hrf, is
impressed in the grid circuit.
a. Show that the equation for i b in terms of 6 is
i b = 3 + 4 sin - cos 20,
and sketch i b as a function of on rectangular-coordinate graph paper.
6. Usually the i b -e< relationship is available in graphical and not
I Y0
FIG. 73.
algebraic form. Hence a graphical method should be used to obtain
the required graph. The graphical method is described below and the
student is asked to perform the indicated graphical steps and to obtain
approximately the same graph as he obtained in (a).
First plot i b in terms of e e . Then plot the graph of e c in terms of 6
and divide the interval from = to = 2n into any convenient num-
ber of equal parts, say 12. Construct axes for the graph of i b in terms
of 8, and lay off these 12 division points on the 6 scale. Find the i b
70
ENGINEERING PROBLEMS
point that corresponds to each e c point graphically, as suggested in
Fig. 73. Then sketch the required graph.
Note: The ib-e c graph is zero for e c to the left of the e c intercept (vertex in
this case) . The instructor can vary this problem by changing the impressed
voltage. If e c = 30 -f 20 sin 0, the result will be a " blocked" wave.
If ec = 40 -{- 10 sin 0, another " blocked" wave will result.
229. A conchiodograph consists of a rod AB, one end of which moves
in the slot DE (the x axis) and the other end passes through a pivoted
guide at N. The distance between the slot DE and the pivot at N is a.
Mi and M 2 are points on the rod such that A M\ = a and AM 2 = a/2.
Determine the parametric equations for the curves which M i and M%
describe and eliminate the parameter in each case (see Fig. 74).
FIG. 74.
*230. The magnetic field quantities B and H are related (in iron-core
transformers or inductances) by a curye of the form shown in Fig. 75.
This figure is called a hysteresis loop. The branch or part abc gives
the values when B increases; branch cda when B decreases,
a. If B, caused by an alternating voltage (in a transformer, for
example), is given (in gauss) as a function of time by
B = 10,000 cos 120rrZ,
3lot the curve for H as a function of time. (The magnetizing current
as proportional to H.) This plot is to be obtained in the following way:
Let B = 1207rt. When = 0, B = 10,000 and the point on the hystere-
sis loop is at c. Plot // = 1.92 when 6 = 0. Now as 6 increases from
to TT, B decreases from 10,000 to 10,000 and the point is moving
along the branch cda. Compute a number of values for B for assigned
values of 0, read the corresponding values of H from the loop graph 9
and plot these in terms of the assigned values for 6. The values for
H as B changes from TT to 2?r are to be read from the loop branch abc
(since B is now increasing).
ANALYTIC GEOMETRY
71
b. Sometimes it is convenient to approximate the graph of H as a
function of time by an equation of the form
H = H m cos (ut + <p).
Since B in terms of t is given to be of the form B = B m cos co, eliminate
FIG. 75.
the parameter t and show that the hysteresis loop for this hypothesis
is an inclined ellipse. Notice that B mj II m , w, and <p are constants,
EMPIRICAL CURVES
231. The data given below were taken on a compressive test of a
concrete cylinder with diameter 6 in. and height 12 in. s is unit stress
in pounds per square inch (total load divided by the area) and is the
unit strain in inches per inch (total amount compressed divided by
original height). Let K.= 10,000e.
K
1.10
3.58
5.00
6.25
8.10
10.60
15.00
s
101
430
1,250
1,700
2,030
2,460
2,840
3,090
72
ENGINEERING PROBLEMS
Use the method of averages to determine a, b, and c if
s = a + bK + cK 2
and then transform the equation to one in terms of s and e. Estimate
the highest point on the curve.
232. A portion of the stress-strain curve showing the results of a
tension test of a mild steel bar is given by the following data, s is
unit stress in pounds per square inch and c is the unit strain in inches
per inch. Let S = 0.001s.
FIG. 76.
S
7.5
14.1
19.8
25.0
31.1
35.0
37.2
37.5
e
0.002
0.004
. 006
0.008
0.010
0.012
0.014
0.016
a. Determine, by the method of averages, the value for b if 8 6c.
Then transform the equation to s = l,0006e. The value of 1,0006
has very important physical significance. It is the slope of the straight-
line part of the graph and is the modulus of elasticity of the material
(denoted by E in engineering notation).
b. Determine, by the method of averages, the values of a, b, and c if
S = a + be + ce 2 and then transform the equation by aid of S = 0.001s.
Estimate the largest value for s.
233. The following data were obtained from a filter-press experiment
in a chemical engineering laboratory. V is volume in liters that has
been filtered at the end of sec.
V
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
6.3
14.0
24.2
37.0
51.7
69.0
88.8
110.0
134.0
a. Determine empirically the values of a, 6, and c so that 6 will be
represented as a quadratic function of V; i.e., 6 aV 2 + bV + c.
ANALYTIC GEOMETRY 73
6. Transform your resulting equation to the form
(V + V C Y = k(6 + O c ).
V c is a "mythical" volume that is due to the filter cake and the,
effect of the filter cloth.
234. The following data give ib in terms of e e for a vacuum tube as
described in Prob. 228.
e c
-20
-16
-12
-8
-4
ib
0.1
0.7
1.4
2.2
3.1
4.2
a. Form a difference table and show that t& can be expressed to a good
approximation as a quadratic function of e c .
b. Determine a, h, and c if i b = ae c ' 2 + bc c + c.
235. The following data give the discharge Q in cubic feet per second
over a rectangular weir for a given head H in feet.
//
0.166
0.509
0.989
1.152
1.792
3.970
Q
0.93
5.58
13.85
17.52
34.05
107.60
The formula for Q is known to be Q = CLH n , where C and n are con-
stants and L is the length of the weir in feet and is 4.26 ft. for these
data. Determine C and n by the method of selected points. C is
called the "mean value" of the coefficient of discharge.
236. The electron emission current from a hot filament is found
experimentally to vary with temperature as given in the following
table. T is absolute temperature in degrees Kelvin, i is the emission
T
i
1400
1600
1800
2000
2200
2400
2600
2800
6.61(10- 9 )
9.51(10- 7 )
4.55(10~ 5 )
0.001,003
0.01328
0.1160
0.7160
3.53
current density in amperes per square centimeter. The following
equations were found theoretically:
= a
i = AT 2 e-/ T ,
the first being the older.
a. Determine the constants for each equation to fit the given data.
6. Compute the residuals and compare the fitting by the two curves.
74
ENGINEERING PROBLEMS
237. Froelich's equation, B = aH/(b + H), is sometimes used as
an analytical representation of the magnetization curve for ferro-
magnetic materials. Determine a and b so that this equation will
fit the data in the following table. Plot a graph using the given data;
'using the derived equation; compare. In most practical cases the
agreement is not close over the whole range of the curve.
//
1.0
2.0
3.0
4.0
5.0
6.0
8.0
10.0
15.0
20.0
30.0
R
n
5 00
9 00
10 30
11 32
11 80
12 15
12 69
13 08
13.72
14 18
14.75
238. Assignment for any engineering student: Ask your engineering
adviser to suggest a textbook in your chosen field of engineering in which
use is made of curve fitting or of graphs on special coordinate paper.
Remark: Use is made in engineering of log log and semilogarithm paper not
only for curve-fitting purposes but also to show the graph of a function
where one or both variables have large ranges. For example, about 75 per
cent of chemical engineering data are plotted on log log paper.
239. For very smooth pipes the following formula relates the Reynolds
number R to the frictional factor / for the pipe. (These terms will be
defined when you take fluid mechanics.)
lo glo R = ^~ - \ logic/ + 0.40.
Texts on fluid mechanics show the graph of this equation on log log
paper for a range 0.010 to 0.050 for / and 5,000 to 5,000,000 for 12.
a. Plot the -graph on log log paper for this range, or plot logio R as a
function of logic / on ordinary paper.
6. Show that the following method will yield the graph for the given
equation. Let x = log/ and y log R. Then the given equation
may be written in the form
y = Q) (10-*/ 2 ) - ~ + 0.40.
Use a sheet of log log paper but plot y as a function of x (by the method
of addition of ordinates) using uniform scales on both x and y axes-
This will entail some care that use is not made of the logarithm rulings.
240. Plot on ordinary graph paper and on log log paper the two
following equations. Show a range of to 1,000,000 volts for V on
the first graph and from 1 to 1,000,000 for V on the second graph.
v = 597,000
v = 300,000,000 \/l
v
_ - _
(1+0.000,001,95F) 2
ANALYTIC GEOMETRY 75
where V is the voltage in volts in an " electron gun" and v is the speed
of the particle in meters per second.
Note: The former equation is arrived at from Newtonian theory and pro-
vides an excellent approximation for voltages below 10,000 volts. The
latter equation comes from modern physics.
ELEMENTARY FORMULAS IN SOLID ANALYTIC GEOMETRY
241. A force of 80 Ib. acts along a line directed from (10,0,8) toward
(5,3,0).
a. Sketch a figure and show the two points and the given force.
6. Determine the direction cosines and the direction angles for the
line of action of this force and label these direction angles on your figure.
c. Determine the components of this force in the x, y, and z directions
by multiplying the force 80 Ib. by each of the direction cosines in turn.
242. A force of 18 Ib. acts from the origin toward (2,1,2). A force
of 12 Ib. acts from the origin toward (4,4,2).
a. Sketch a figure, show these two forces as vectors properly direc-
tioned and scaled, and construct their resultant by constructing the
parallelogram that has these two given vectors as sides. Draw the
diagonal of this parallelogram, which starts at the origin. This diagonal
represents the resultant of the two given forces.
b. Determine the x, y } and z components of each force.
c. Determine the sum of the x components of both forces, the sum
of the y components, and the sum of the z components. Then determine
the vector (in magnitude and direction, i.e., direction numbers or cosines)
that has these sums as its x, y, and z components.
d. Determine the acute angle between the given forces, correct to
the nearest tenth of a degree.
243. (Similar to Prob. 242.) The following four forces are each
directed towards the origin from the point given with each force:
Fi = 100 Ib., (2,2,1); F 2 = 150 Ib., (3,2, -2);
F 3 = 50 Ib., (-4, -3, -3); F 4 = 200 Ib., (0,0,4).
a. Sketch a figure and show all four forces.
6. Determine the sums of the #, y, and z components for all four forces.
Then determine the resultant (that force which has these sums for its
x, y, and z components). Give the magnitude correct to the nearest
three significant figures and the direction angles each correct to the
nearest tenth of a degree.
244. A horizontal bar ABC supports a load of 1,000 Ib. as indicated
in Fig. 77. The bar is supported by a ball-and-socket joint at B and
by two cables CD and AE. If axes are chosen as shown, the coordinates
of D are (14, -6, 12) and of E are (-9, -10, -10). The unit is 1 ft.
76
ENGINEERING PROBLEMS
The problem in mechanics would be to determine the tensions in the
two cables and the reaction at B. You are to study and complete the
following solution and notice the use of methods of analytic geometry.
Solution: Let the force or reaction at B have components B x , B y , B z
(each assumed to be positive). Assume the tension in CD to be T Ib.
a*nd that in AE to be P Ib. The x, y, and z components of these tensions
D(14,-6,12)
X-9,-10,-10) I
FIG. 77.
are determined by multiplying the tension by each of the direction
cosines of the line segment. Thus the components are found to be
2T
T'
37; 67 T
7' 7 ;
2P
T'
2P
3 '
The sum of the forces in the x direction must be. zero and also the
sum of the forces in the y and z directions. Hence
= 0:
0:
. = 0:
2T
3T 2P
6T 7 2P
-y - -j- + B, - 1,000 = 0.
ANALYTIC GEOMETRY
77
The sum of the "moments (turning effects) with respect to each
axis" must be zero. (The moment of a force is the product of that
force and the perpendicular distance from the axis to the line of action
of that force.) This yields
= 0,
5,000-^-^-0,
1,M Z = 0:
8P 30T
= 0.
Solve these two sets of equations simultaneously.
245. The following data give the dimensions in inches of half an
airplane landing gear. Compute the lengths and direction cosines for
each member and sketch the half gear.
Coordinates
Point
X
y
z
A
10
B
-30
-20
30
C
- 6
10
30
D
- 6
-40
30
E
- 5
The members are AO, BE, CO, DO, and EO.
246. A street corner is at the base of a hill from both the north and
the east direction. The grade of the street north is 20 per cent and
that for the east direction is 10 per cent. A water main having the
same grades as the streets is to turn the corner. Determine the angle
required for the elbow at the turn.
247. Two tunnels start from a common point A in a vertical shaft.
Tunnel AB bears N.60W., falls 10 per cent and is 300 ft. long. Tunnel
AC bears S.30W., falls 12 per cent and 1s 200 ft. long. The ends of the
tunnels are to be connected by a ventilating shaft.
a. Sketch a figure with origin at A and determine the x, y, and z
coordinates for B and (7.
b. Determine the following quantities for the ventilating shaft :
(1) The true length, correct to three significant figures.
(2) Its bearing, correct to the nearest minute, with reference to B.
78 ENGINEERING PROBLEMS
(3) Its percentage of grade, correct to the nearest tenth of 1 per cent.
(4) The angle it makes with the horizontal, correct to the nearest
tenth of a degree.
248. A vertical mast AB i guyed by three guy wires to anchors at
C, D, and E. The distance is 150 ft. up to the point B where the three
wires are fastened.
Point C is 60 ft. east, 40 ft. north, and 10 ft. above A,
Point D is 80 ft. west, 20 ft. north, and 10 ft. below A,
Point E is 10 ft. east, 70 ft. south, and 15 ft. above A.
Determine (assuming the guy wires to be straight) :
a. The length of each guy wire, correct to three significant figures.
b. The angle each guy wire makes with the horizontal.
c. The angle each guy wire makes with the east direction and with the
north direction.
249. Points A, J5, and C are points of outcrop of a vein of ore. A is
450 ft. north and 300 ft. west of B and has an elevation of 2,500 ft.;
C is 100 ft. north and 250 ft. east of B and has an elevation of 2,200 ft.;
and the elevation of B is 2,700 ft. Point 7), not on the vein, is 400 ft.
north and 150 ft. east of B and has an elevation of 2,600 ft.
It is desired to drive the shortest possible tunnel from D to the vein.
Determine the following:
a. The coordinates of A, B, C, D referred to x, y, z axes chosen in a
convenient manner. (One such choice would be to take z upward, x
positive in the east direction, and y positive in the north direction.)
b. The equation of the plane ABC.
c. The perpendicular distance from 7) to that plane and the direction
cosines and angles for that perpendicular.
250. Two mining shafts AB and CD are determined by A (0,0,0),
5(800,600, -800), (7(100,900, -600), (700, 100, -100).
a. Sketch a figure and show the two shafts.
b. Show that a set of parametric equations for the shaft AB are
x = 800J, y = 60CX, z = -800Z; for CD the equations are x = 100 + 600s,
y = 900 - 800s, z = -600 + 500s.
c. Determine the length of the tunnel joining the point on the shaft
AB that corresponds to t = t to the point on CD that corresponds to
s = s. Also determine expressions for the direction numbers for this
line segment.
d. If the tunnel in (c) is level, show that St = 6 5s and hence that
the length of the tunnel can be reduced to
L = 25 V2,225s 2 - 2,372s + 724.
e. Identify and sketch a graph of L as a function of s. What value
of s gives the smallest positive value for L? What are the coordinates
of the ends of the tunnel for this value Of s?
ANALYTIC GEOMETRY 79
SURFACES IN SOLID ANALYTIC GEOMETRY
251. Plot a careful graph for the gas law: pv = kT, where p is pressure,
v is volume, T is absolute temperature, and k is a constant. Use 1 in.
equal to one unit on the v axis (positive to the right), K in. equal to one
unit on the v axis (positive perpendicular to your sheet of graph paper),
and 1 in. equal to l/k units on the T axis (positive upward). Give a
careful discussion about the traces in the coordinate planes and in planes
parallel to the coordinate planes and answer the following questions:
a. What traces on the graph correspond to Boyle's law?
b. Charles's law has two different parts. What are these and to what
traces do they correspond?
c. Show that the trace of this surface in the plane p = v is a parabola.
d. If axes in the p-v plane are rotated through an angle of 45 about the
T axis so that the new variables x, y are given by
x y x + y
p _ ^ v _
y V2 V2
show that the equation becomes # 2 y 2 = 2kT or, if z = 2kT,
x 2 7/ 2 = z. Identify this quadric surface.
262. Sketch the surface E = Ir (Ohm's law in electrical engineering) .
Will your same graph do for the equation P = El (another equation
from electrical engineering) ?
253. Will your graph for Prob. 252 do for the graph of the important
beam equation 5 = M(c/I) or for the important equation for torsion
s = T(c/J), if c is a constant? Both these equations will be derived
when you study strength of materials.
Remark: The three preceding problems indicate how widely diverse
engineering problems or theory may have a common mathematical discussion.
254. When water flows turbulently through a rough pipe, the speed
at any distance from the center of the pipe (of radius R) is given closely
by the following rule : Let the speed at the center (the maximum value)
be V m . Then the speed at a distance r from tKe center of the pipe is
the ordinate to a surface of revolution: The bottom part of the surface
is a cylinder of radius R and height V m /2. The top part is half an
ellipsoid of revolution whose semiaxes are /, V m /2, and R.
a. Show that the speed at a distance r from the center is given by
V-^(l +
b. Sketch the surface.
c. Sketch a graph of V/V m as a function of r/R and explain the rela-
tion of this plane graph to the surface.
80 ENGINEERING PROBLEMS
255. Refer to prob. 446 and sketch the surface described in the first
paragraph of that problem. Assume 26 = 1.50 in. and 2a = 0.75 in.
256. Airy's stress function for the torsion (twisting) in a bar with an
elliptical cross section (semimajor and semiminor axes a and 6, respec-
tively) is
where M is a positive constant.
a. Determine the value of <p for points on the boundary of the ellipse.
b. Sketch a graph of the surface. Show only that portion for which
<p is positive.
257. A grain elevator has a rectangular hopper 12 by 20 in. and a
circular feed pipe with diameter 16 in. The axis of the pipe and the
hopper coincide, but the end of the pipe is 30 in. above the hopper.
Sketch the setup and show a connecting pipe that is circular on one end
and rectangular on the other.
Suppose the circular end is in the plane z = 30, the rectangular end
in the xy plane, and the common axis is the z axis. Suppose the rec-
tangular end in the xy plane has its vertices at (10,6,0), (10, 6,0),
( 10, 6,0), and ( 10,6,0). Show that a portion of the connecting
pipe could have for its equations in parametric form the following:
x = 10 + t(8 cos r 10),
y = 10 tan 6 + t(8 sin 0-10 tan 0),
z = 30,
where t can vary from to 1 and from tan" 1 (%) to + tan"" 11 (%).
Discuss the section when t = 0, when t 0.5, and when t 1. Would
this be a reasonable way to design the connecting pipe? Should the
connecting pipe be a "ruled surface"?
258. An equation used in strength of materials is
X 2 , 7 2 , Z*
where the quantities in the three denominators are constants. Would
the name ellipsoid of stress be a reasonable name, assuming that the
equation has something to do with stress? Sketch the surface.
259. Let be the North Pole of the earth (assumed a sphere with
radius R). Let A be any point on the surface of the earth and let
the line OA intersect the plane of the equator at point P. Assume
the origin at the center of the earth and the z axis through the North
Pole. Show that the coordinates of the point P are XP = XAR/(R ZA),
yp = yAR/(R ZA), and Z P = 0, where (XA^A^A) are the coordinates
of point A.
ANALYTIC GEOMETRY 81
Note: The method of "projection" indicated in this problem is the basis
for a Mcrcator map of the earth.
260. In Prob. 259, suppose that the point is at the center of the
earth and, instead of using the equatorial plane, that we use & plane
tangent to the earth at the North Pole (it could be tangent at any point
on the earth for the purposes of the projection). If the line OA inter-
sects this tangent plane at the point P, show that the coordinates 01
the point P are
X A R VAR &
X P = -> y p = > Z P = R.
ZA ZA
Note: This problem forms the basis for a gnomonic type of map of the
earth.
*261. For a map of the earth, as indicated in Prob. 260, show that
every great circle on the earth projects into a straight line on the tangent
plane or gnomonic map.
Note: The converse of the statement of this problem is also true. This
property makes the use of gnomonic maps of great importance in charting
courses.
FIG. 78.
262. Figure 79 shows a "contour map" of a surface, such as is shown
in Fig. 78. The surface represents grid voltage e c and plate voltage e b
plotted in the horizontal plane and plate current ib along the vertical
axis.
82
a.
Of e&,
ENGINEERING PROBLEMS
Plot a second contour map showing the contours for various values
b.
of ib.
10 20 30 40 50 60 70 80 90 100 110
e^ in Volts
FIG. 79.
Plot a third contour map showing the contours for various values
PART IV
DIFFERENTIAL CALCULUS
USES AND INTERPRETATIONS OF DERIVATIVES
263. If specific heat is defined to be dQ/dt, where Q is the quantity of
heat necessary to raise the temperature of 1 gram of a substance from
to tC. and if for ethyl alcohol
Q = 0.5068* + 0.001,43 2 + 0.000,001,8 3
(valid for the range from to 60C.), determine the specific heats of
ethyl alcohol at t = 10C., 20, 30, 40, and 50.
FIG. 80.
v
P"
M + AV
L
c
< Ax->
V+AV
FIG. 81.
*264. A section of a beam is shown in Fig. 81. The beam (Fig. 80)
is loaded with a uniform load of w Ib. per ft. and with some concentrated
loads whose magnitudes and positions are not shown. (The quantities
M and V will be defined in strength of materials.) If one equates to zero
the "first moment" of all the forces with respect to the point 0, one
obtains
M +
- (w
Simplify, divide by A#, and obtain
AM
Ax
V -
+ (M + AM) = 0.
w Ax
83
84 ENGINEERING PROBLEMS
Let A# approach zero and obtain
r AM dM
hm - = -- = V.
z ax
Remark: In the language of strength of materials, this equation states
that, "The rate of change of bending moment M with respect to the distance
(measured from some point on the beam) is equal to the vertical shear F."
Or, "The derivative of bending moment with respect to x is equal to the
shear."
Notice that this derivation, while containing the language and notation
of the course in strength of materials, is the important delta process of
differential calculus.
265. If V = dM/dx and if
2Px
M = -= for < x < 10 (P is a positive constant),
M = WP -~forlQ<x < 14,
a. Sketch a graph of M/P as a function of a;. for x from to 14.
b. Determine V for < x < 10 and for 10 < x < 14. Then sketch
the graph for V/P as a function of x for x from to 14.
c. Does the graph of V as a function of x have a discontinuity in the
range < x < 14?
-*~x
y
t 10'
p
\
I 10 >,
FIG. 82.
Remark: The beam and loading for this problem are shown in Fig. 82.
The mathematical analysis of such beams gives a shear graph (V as a
function of x) with discontinuities at each point where there was a concen-
trated load. The mathematical analysis of such problems assumes that
the concentrated load is applied at a point (which is physically impossible).
However, the results from this analysis are accurate enough for most
purposes.
266. In a resistance of r ohms the current i amp. is given in terms of
the voltage (e volts) by the equation (Ohm's law) i = e/r. In a con-
denser (C farads) the relation (on discharge) is
DIFFERENTIAL CALCULUS
85
If e EQ T i/Cr , determine equations for i in terms of t for the case
of a resistance and for the case of a condenser (C, r, and E are constants) .
(e is the letter used in electrical engineering for 2.71828 . . . .)
Sketch graphs of e as a function of t/Cr, i as a function of t/Cr for
both the resistance and condenser cases.
267. "Power" is defined as the rate of doing work with respect to
time. If W denotes work, t time, and P power, give the mathematically
equivalent definition for power.
Curve of beam
FIG. 83.
268. The beam in Fig. 83 supports a uniform load of w in pounds
per foot of beam. If axes are chosen as indicated, the equation of the
"curve of the beam" is
_ T
Ely =
wLx s
__
"24*
wL*x
~2T'
where E t /, w, and L are positive constants.
In strength of materials it will be proved that the "bending moment"
is given by M = EI(d*y/dx*), the "shear" by V = dM/dx, and the
load by EI(d*y/dx 4 ). Determine M, V, and EI(d*y/dx 4 ) for this beam
and sketch these three curves (these variables as functions of x) on
the same graph.
iPlbs.
- L / 2 -
Curve of beam
FIG. 84.
269. The beam in Fig. 84 has a concentrated load at mid-span. The
'curve of the beam," is given by
Ely =
Ely =
12
P(L
PL*x
16
for < x < ->
PL*(L - x) . I. ^ T
~ for 75- < x < L.
10 &
Determine M = El^y/dx*) and V = dM/dx = EI(d*y/dx*) for
the entire range from x to x L and sketch M and V as functions
12
86 ENGINEERING PROBLEMS
of x on the same graph. For what value of x is the graph of V as a
function of x discontinuous?
270. The "kinetic energy " acquired by a body falling from an infinite
distance to a distance r from the center of the earth is given by W = k/r,
where ft is a constant. Determine the force F = D r W.
271. "Power" is denned as the rate of doing work, i.e.,
If the work being done by a force is W = 3t 2 + 4t + 6 (t in seconds
and W in foot-pounds), find the power at t = 2 sec.
TANGENT AND NORMAL
272. A parabolic arch is 10 ft. high and 20 ft. wide, as shown in
Fig. 85. A brace AB is inserted as
shown in the figure. Find its length.
273. Euler's column formula from
strength of materials is
P_ _ 7T 2 ff
A - (L/r) 2 "'
where P is the total load, A is the
cross-sectional area of the column,
IG " * E is a property of the material from
which the column is made (the modulus of elasticity), L is the length of
the column, and r depends on the shape of the cross section.
a. Sketch a graph of P/A as a function of L/r. E is a positive
quantity.
6. Determine the equation of the tangent to this curve at the point
where L/r - (3ir*E/p)^ p/A - p/3.
Remark: p is the load required to crush the column. Your resulting
straight-line equation is known as "the straight-line column formula."
274. Write Euler's column formula (see Prob. 273) in the form
y = a/x 2 , where y = P/A and x = L/r, a being a constant. In this
same notation, another column formula is of the form y = b mx*,
where b and m are constants. Determine the relation between a, 6,
and m so that the graphs of these two formulas are tangent, and find
the coordinates of the point of tangency.
Note: In the practical design of columns, the second formula is used for
values of x from to the point of tangency; the first formula is used for all
larger values of a?.
275. A catenary is the curve that a cable assumes when hanging
between two supports. If the two supports are of equal height and
DIFFERENTIAL CALCULUS 87
are attt = a, and x = +a, the equation for the curve is
y = c + k cosh ~ = c + gj (e*/ a + e~*/ a ).
Determine an expression for the angle that the cable makes with the
vertical support at x = +a.
276. Using the data of Prob. 228,
a. Determine the value of di b /dO = 2 radians.
6. Determine the equation of the tangent line to the i b e c graph
at e c = -20.
277. Using the data of Prob. 146, show that the two equations have
the same first derivative value at x = 8 ft. and also the same ordinate.
Then write the equation of the common tangent line.
278. In constructing a certain type of cam for accelerating a lift,
it is necessary to find two parabolas that have a common tangent at
points on two given abscissas. Find a and b so that the tangent to
x 2 = ay at x = 2 shall coincide with the tangent to
(x - 10) 2 = b(y - 8.5) at x = 9.
279. Devise a graphical solution for the common tangent line in
Prob. 278 based on the following theorem for parabolas. Also prove
the theorem.
THEOREM: The tangent at the vertex of a parabola bisects the seg-
ment of any other tangent which is included by the principal axes and
the point of tangency.
^ Curve of beam
FIG. 86.
280. A cantilever beam of length L ft. bears a uniform load of w Ib.
per ft. for the length L/2 ft. next to the wall as shown in Fig. 86. The
equation for this part of the " curve of the beam" is
EIij = 1 1
24 12 16 24 48 '
i.e., this equation is valid for x between L/2 and L. If the weight of
the beam itself is neglected, the part of the beam to the left of this load
will be straight and will be along the tangent to the preceding curve at
the point whose abscissa is x = L/2.
88 ENGINEERING PROBLEMS
Determine the maximum deflection; i.e., find the largest numerical
value of y in the entire range (which is clearly the ordinate at = 0).
281. A rock is tied to the end of a string of length 3 ft. and is whirled
in a clockwise direction, and the free end of the string is held steady at
a certain point. If the origin is taken at the point where the free end
is held and if the string breaks when the string makes an angle of
+ 135 with the horizontal (right-hand direction is positive), determine
the equation of the path that the rock will take for a short time (neglect-
ing the effect of gravity).
282. In mechanics, the ellipse whose equation is
is called the ellipse of inertia of a plane lamina referred to axes through
the origin. R x and R v are known as the principal radii of gyration with
respect to the x and y axes. Find the equations of the tangents to this
ellipse which are parallel to the line y = mx. Express the square of
the distance between one of these tangents and the line y mx in terms
of m, R x , and R lf . (This perpendicular distance is the radius of gyration
for the axis or line whose equation is y ~ mx.}
283. The point (#0,2/0) is in the plane of the ellipse described in Prob.
282. Two tangents are drawn to the ellipse and their points of tan-
gency are connected by a straight line. Show that the equation of this
line is
Note: The line xx$/Ry* + yy /R x * = 1 has an interesting association
in the study of a vertical column hearing an axial load which is eccentrically
placed, i.e., is placed at (#0,2/0). It is known as the line of "zero stress" or
the line of "stress reversal." The areas of the column sections on opposite
sides of this line are, respectively, in tension and compression.
284. If the current flowing in an electric circuit is given by
where / and a are positive constants, determine the length of the sub-
tangent to this curve at the time t 0.
Remark: This result is of importance in electrical engineering. It is
called the "time constant" T of the electric circuit and has the following
properties:
1. It is the length of the subtangent at t = 0.
2. It is the time required for the ordinate (current) to change from an
arbitrary value A to the value A/e (a decrease of about 60 per cent). In an
interval equal to 3T the ordinate decreases from A to A/e 3 , i.e., to about
DIFFERENTIAL CALCULUS
89
5 per cent of the value at A. Five time constants of time would r6duce the
current to about 0.67 per cent of the value at the beginning of that time
interval. Since this last result is often negligible as compared to the starting
value by ordinary standards of engineering accuracy, the following state-
ment is apparent: "The duration of the current, if of exponential form, is
five time constants/ 7
3. The time constant is the time it would take i to reduce to zero if i
decreased at a constant rate equal to the rate at which i is decreasing when
t = 0.
4. The area of the rectangle whose base is the time constant and whose
altitude is along the i axis from i to i I is equal to the area under the
curve i = Ie~ at in the first quadrant.
Prove that properties 2 and 3 are true. Property 4 can be established
after you have studied integral calculus.
*285. A rhumb line on a polar gnomonic chart (see Prob. 260) has
the equation
p = ke~ e <i>,
where c is a constant specifying the course traveled, k is a constant
related to the scale of the map, and
that this rhumb line makes equal
angles with the radial lines on the
map given by < = constant.
286. When benzene vapor is dis-
solved from flue gas 'by oil, it is
found that
is a meridian of longitude. Show
H-P
FIG. 87.
_ Hx
y ~ P - (H - p)x
where x is the concentration of the
benzene in the liquid and y is the
concentration of the benzene in
the flue gas. P is the total force
and H is Henry's constant in
Henry's gas law.
a. Sketch a graph for y in terms of x, assuming that
(1) H is larger than P,
(2) # is smaller than P.
b. Use your curve for II > P and determine the slope (dy/dx) at
x = 0, and then the equation of the tangent line at this point. What is
the ordinate to this line at x = 2 ?
c. Determine the equation of the tangent to the curve f or H > P at
(#2,2/2) as determined in (b). What is the x intercept, # 3 , for this second
tangent line?
90
ENGINEERING PROBLEMS
Remark: The tangent line as described in (6) is called the " operating
line" in chemical engineering terminology.
*287. In designing rolled products it is necessary that the rolled
contours be very accurately constructed. For the U section shown in
Fig. 88 determine the x and y coordinates of the center (each to the
nearest 0.001 in.) of the 1-in. end miller.
Arc AB is circular with radius 10 in. Arc BC is made by the miller
with diameter 1 in., and straight line CD makes an angle of 10 with
the horizontal.
-4.5
FIG. 88.
CURVE SKETCHING. MAXIMUM, MINIMUM, AND FLEX POINTS
288. A simply supported beam of length 12 ft. weighs 100 Ib. per ft.
and is loaded with concentrated loads of 2,000 Ib. at 3 ft. from the left
support, 4,000 Ib. at 6 ft., and 6,000 Ib. at 7 ft., as shown in Fig. 89.
2000 4000 6000
6600 Ibs.
Curve of beam
FIG. 89.
6600 Ibs.
The " bending moment 7 ' M (to be defined in strength of materials)
is given by the following equations :
For < x < 3:
M = 6,600z - 50s 2 ,
DIFFERENTIAL CALCULUS
91
for 3 < x < 6:
for 6 < x < 7:
for 7 < x < 12:
M = 4,600z + 6,000 - 50z 2 ,
If - 600z + 30,000 - 50z 2 ,
M = -5,400z + 72,000 - 50z 2 .
a. Sketch a graph of dM/dx as a function of # for < x < 12. (This
is called the " shear " diagram.)
b. Sketch on the same graph a graph of M as a function of x.
c. If, for the equation of the "curve of the beam/ 7 M = EI(d*y/dx*),
determine the abscissas of the points of inflection. E and 7 are positive
constants.
d. For what values of x is M a maximum or minimum? The require-
ment in this engineering problem is to determine the largest values of M,
not merely those for which dM/dx is zero but also those at the end of an
interval.
Remark: The answers to (c) and (d) are important in design. The stress
at any point in a beam, whether tension or compression, is given by s = Me/ 1,
where c is the distance of the point in question from the "neutral" or cen-
troidal axis and, / depends on the shape of the cross section of the beam.
It should be clear that s will be zero at every point along a vertical section
through a flex point of the "curve of the beam" and that s will be largest or
smallest depending on the behavior of M, and hence of d z y/dx 2 . E and /
are constants.
289. The beam in Fig. 90 supports a uniform load of w Ib. per ft.
The equation of the "curve of the
beam" is
Ely
wLx 3
~T2~
~24
wx*
24
Curve of beam
m-
I
where E and / are positive con-
stants that depend upon the ma-
terial from which the beam is made
and upon its cross section. . FlQ - 90 -
a. What is the largest numerical value of M = EI(d 2 y/dx 2 ) and where
does it occur?
6. For what values of x does M = (EI)(d 2 y/dx 2 ) equal zero.? These
are the sections in the beam where the tension or compression is zero.
c. Sketch graphs of Ely/w as a function of x, (EI/w)(dy/dx) as a
function of x, M/w as a function of x, and (1 /w) (dM/dx) as a function
of x. Use a common abscissa but different vertical scales.
92
ENGINEERING PROBLEMS
290. The electric field intensity on the axis of a uniformly charged
ring is found to be E ~ Qx' ( x 2 + a 2 )'>~, where Q is the total charge
on the ring. Also, a and Q are
constants (see Fig. 91).
a. Sketch E as a function of x.
b. At what value of x is E a
maximum?
c. What is the value of x which
makes d 2 E/dx 12 - zero?
FIG. 91.
Remark: There is need for an under-
standing of how to sketch the first
derivative curve, in general form, directly from the graph of the original
equation. Also there is need in engineering for the ability to compute
dy/dx by graphical means. These abilities can be developed by problems
such as the following two problems.
291. The current i amp. for a condenser of C farads is given in terms
of the impressed voltage e volts by the equation
i = C
y /de\
(dt)
If e is an alternating voltage having the wave form shown in Fig. 92,
sketch the wave form for i. C is a positive constant.
volts
\\
t seconds
FIG. 92.
292. In the following table
t
10
20
30
40
50
60
70
s
156
608
1,308
2,180
3,132
4,076
4,942
s is the distance in feet traversed by a body in t sec. (the body moves
along a straight line). Plot the graph carefully, draw the tangent lines
(to the best of your ability) at each of the given time values, compute
the slopes of these tangent lines (ds/dt) , and finally plot ds/dt as a f unc-
DIFFERENTIAL CALCULUS 93
lion of t. Use the same abscissas that you used for your original graph
but choose the vertical scale so that the resulting graph will be of a
reasonable size.
293. An airfoil is to be designed so that it is symmetrical with respect
to the x axis, with its nose at the origin, and with its tail toward the
right. The equation
y
= do \/x
is to fit the top half of the foil curve and to satisfy the following
conditions :
1. The maximum ordinate y = 0.1 is to occur at x 0.3 (two condi-
tions implied).
2. The ordinate at the trailing edge x = 1 is to be y = 0.002.
3. The trailing edge angle is to be such that at x = 1, dy/dx = -0.234.
4. The ordinate at x = 0.1 is to be y = 0.078.
Determine the required values for the a's. Then plot the curve.
294. An airfoil is to be plotted with a " camber" or skewed foil.
The mean line of the foil is given by
y = p (2px - x 2 } for x ^ p,
V = (i p y (1 - 2p + 2px - x 2 ) for x ^ p.
a. Show that the curve is smooth at x = p; i.e., that the two equations
have a common ordinate and a common tangent line at x = p.
b. Plot this curve from x = to x = 1 on the assumption that
p = 0.3 and m = 0.06.
c. Now plot the airfoil itself according to the following directions:
Locate any point on the mean line curve, for example, at x 0.1
and y = 0.033. Draw a perpendicular to the mean line curve at this
point. Now compute y for x = 0.1 from the result for Prob. 293
and measure this distance in both directions along the perpendicular
line. Repeat at every tenth for x from to 1 and obtain the required
airfoil graph.
MAXIMUM AND MINIMUM PROBLEMS
295. The total annual cost of a pipe line may be expressed as
C = Ad 2 + B/d*, where d is the diameter of the pipe, and A and B
are positive and are substantially constants for a given range of diameter
values. Determine the diameter for minimum cost.
296. (Taken from a physics text.) Given that
D = siny 1 (u sin r) + sin" 1 [u sin (A r)] A >
where u and A are constants. Show that D is a minimum when 2r = A.
94 ENGINEERING PROBLEMS
297. It is shown in hydraulics that the quantity q of water (q cu. ft.
per sec.) that flows over a certain type of spillway is given by
q = BD 2g(H - D),
where B is the width of the spillway, D is the depth of the Water flowing
over the spillway, g is the gravity constant, and H is the head of water.
Assuming that B and H are constants, determine the value of D that
makes q a maximum.
298. Solve Prob. 92d by aid of calculus.
299. The equation e = tan X ( - TT~~ V i /) gives the efficiency
e for a worm drive which has a lead angle X, a pressure angle <, and
friction /.
a. Show that the given equation can be rewritten in the form
2f
r -^ = cos <t> sin 2X + / + / cos 2X.
L 6
b. Show that the value of X that makes the efficiency e a maximum is
given by tan 2X = cos <//.
Note: The result in part (a) will facilitate the solution to part (6).
300. Let c cost of one Mazda lamp plus installation charge in
cents, b cost of power for the lamp in cents per kilowatt-hour,
V = actual operating voltage, F = rated voltage for lamp, jR = watts
input at voltage V , F = luminous output in lumens at voltage V .
The cost per lumen for 1,000 hr. (assuming 1,000 hr. of life on the rated
voltage of the lamp) is
bp<>\ / y
( c\( F
= w \r
where the B's are constants that are determined experimentally.
a. Determine the value of x = V/Vo that makes y a minimum.
b. For Mazda-Clamps from 60 to 150 watts: B 2 = 3.613; 3 = 1.523;
B$ = 13.50. If b = 6 cents per kilowatt-hour, P Q 100 watts, and
c = 20 cents, determine ow and hence determine the best value for
the rated voltage Vo if the operating voltage V is 120 volts.
301. An electric . battery whose internal electromotive force is E g
volts and whose internal resistance is r ohms has the terminal voltage
E = E ff Ir volts, the output power P == E g l 7 s r, and the efficiency
77 = P/(E g I) as functions of the current I amp. Sketch the graphs of
E, Pj and tj each in terms of I in the range from open circuit (/ = 0)
to short circuit (E 0). Determine by inspection the largest and
smallest values of E, P, and 77 that occur at the end points of this range
and whether any occur at intermediate points. Then differentiate to
DIFFERENTIAL CALCULUS
95
determine the latter. Tabulate all such values together with the values
of / at which they occur. (Notice that the "maximums" for E and I
actually occur, while the " maximum" for t\ does not, since it arises as
an indeterminate form.)
302. The power output of an electric generator is E I, where E
is the constant terminal voltage and / the current. The power loss
in the generator is the sum of a constant component P and a variable
component (due to heating loss) PR, R being the internal resistance of
the generator. The efficiency of the generator may be written
77 =
El
output __
input ~ output + losses ~ 'El + P +
output
Determine the maximum value for the efficiency as the current / varies
and express the result in terms of E, R, and P . Also give the relation
between the fixed and variable losses (P and PR) when the efficiency
is a maximum.
303. The relation between magnetic force F and air-gap separation
s in an electromagnet is shown in Fig 94. The magnetic force cor-
responding to an initial separation s is counterbalanced by a constant
applied force F . If the armature A moves toward the core, work of an
amount F^SQ is done on the load. (The excess of F over F , when
s < s , is not utilized.) It is desired to determine So to obtain the
greatest useful work.
I
J, t
1
t A
1
1
| Load
FIG. 93.
Fia. 94.
Solution: We have that F s (the shaded area in Fig. 94) is to be a
maximum (F is shown as a function of s in the graph). Hence (after
differentiating), = s(dF/ds) + F, or
s =
F
dF/ds
Hence, show that the maximum value for F is located at a point whose
abscissa is equal in length to the subtangent.
304. In a cylindrical cable (Fig. 95) in which the inner conductor
has a radius r and the outer conductor has a radius R, the maximum
96 ENGINEERING PROBLEMS
electric intensity in the insulation is
m ~ r log e (R/r)
and occurs at the surface of the inner conductor. Determine the value
of r/R that makes (E m R/V) a maximum or minimum.
Remark: Electric breakdown occurs when the voltage V between con-
ductors is large enough to make E m larger than a particular value which
depends on the kind of insulation.
'Insulation
FIG. 95. FIG. 96.
305. A solenoid has a fixed internal diameter of D in. and has a
laminated core of the form shown in Fig. 96. Determine the dimensions
s and t of the core so that the cross-sectional area of the core will be a
maximum.
306. A source of light is to be placed directly over the center of a
circular table of diameter 20 ft. The intensity of illumination at any
point on the circumference of the table varies directly as the cosine
of the angle between the vertical and the light ray, and inversely as
the square of the distance of the point from the light. How high should
the light be placed above the table to obtain the maximum intensity
at the edge of the table?
307. One of the factors considered in choosing the size of wire for a
transmission circuit is cost. The larger the cross-sectional area, the
greater will be the first cost and hence also the annual charges for inter-
est, taxes, and depreciation. At the same time the larger the cross-
sectional area, the lower will be the cost of lost power since the heating
losses will be lower. For bare wire the investment is directly propor-
tional to the area, and the lost power is inversely proportional to the
area; hence the total part of the line cost depending on wire size can
be written as
C =
+ ,
where a is the area and ki and kz are positive constants.
DIFFERENTIAL CALCULUS
97
Show that the area of the wire which makes C a minimum is that for
which the two terms are equal, i.e., that for which kia = k 2 /a. Illus-
trate by sketching the two components and their sum, all on the same
axes.
Remark: The basic law as stated in this problem is known as Kelvin's law.
308. Determine the value of x = p*/p\ that will make the expres-
sion y = (p2pi) 2/fc (p2/pi) ( * +1)/ *, k > 1, a maximum and thus deter-
mine p 2 in terms of p\ when y is a maximum.
Remark: This problem occurs in thermodynamics in the study of flow in a
nozzle.
What is the value of the coefficient of pi in your result if k = 1.4?
309. A block that weighs 100 Ib.
rests on a horizontal surface for which
the coefficient of friction is ju. A
force of P Ib. acts on the block as
shown in Fig. 97, the action line of the
force making an acute angle 9 with FlG> 97
Mie horizontal. The force just neces-
sary to start this block in motion can be shown, by methods of physics
and mechanics, to be
p = _ _ _100/i .
cos +
sin 9
If jj, = 0.2, determine the largest and smallest values for P. Can you
give a physical interpretation for each of your results?
3000 Ibs.
Surve of beam
FIG. 98.
310. A beam is loaded with a concentrated load as shown in Fig. 98,
The equation of the " curve of the beam" is given as follows:
For x from to 7 ft. :
Ely = -25z 4 + 650z 3 - 38,650z,
For from 7 to 10ft.:
Ely = -25o; 4 + 150z 3 + 10,500z 2 - 112,150s + 171,500,
where E and / are positive constants.
98
ENGINEERING PROBLEMS
a. Determine the value of x that makes y a minimum, i.e., makes
the deflection a maximum.
b. Sketch on the same graph:
(1) y as a function of x from to 10 ft.
(2) dy/dx as a function of x.
(3) M as a function of x if M = EI(d' i y/dx y ).
(4) dM/dx as a function of x.
311. A beam 2L ft. long weighs w Ib. per ft. and is supported at both
ends and at the middle as shown in Fig. 99. The ordinate to the
FIG. 99.
"curve of the beam" at x ft. to the right of the center support is given
by
where E, I, and w are positive constants. The " curve of the beam"
to the left of the support is the reflection of the given curve in the y axis
through the middle support.
a. Determine the minimum value of y (maximum deflection of the
beam),
b. Sketch the entire curve of the beam.
312. A weight of 100 Ib. is to be raised by means of a lever which is
uniform in cross section and which weighs 2 Ib. per ft. of length. The
force is to be applied at one end, the fulcrum is to be at the other end,
and the weight of 100 Ib. is to be at a distance of 1 ft. from the fulcrum.
How long should the lever be to make the applied force smallest?
313. Let d denQte the diameter of the wire from which a coil spring
is made and let D denote the mean diameter of the coil. When the
spring is compressed by an axial load P, the material in the spring
experiences its greatest stress S at those points on the wire surface
nearest the center line of the coil. S is given by the formula:
S =
SPD
D
4 W """
For any given value of D, what value of d makes S the smallest?
DIFFERENTIAL CALCULUS 99
*314. The " moments of inertia" of an area A with respect to axes
u and v inclined at an angle 9 with the x and y axes are, respectively:
J u = I x cos 2 + I y sin 2 - P xy sin 20,
I v = I x sin 2 6 + I v cos 2 + P xy sin 26,
and the "product of inertia of area" with respect to the u and v axes is
given by
Puv = M(I - I y } sin 20 + P xv cos 20.
If /*, I v , and P* tf are constants,
a. Determine the angle that makes one of I u , I v a maximum and
the other a minimum. Show test and compute the values of I u and I v .
What is the value for P uv for this critical angle 01
b. Determine the angle 6 that makes P vv a maximum or minimum and
show that the value of tan 26 for such angles is the negative reciprocal
of the values found in (a). What relation follows between the angles
found in (a) and the angles that make P uv a maximum or minimum?
315. The density s of water is given with sufficient accuracy by the
equation s = s (l + at + bt 2 + ct s ), where So is the density at 0C*
and t is temperature in degrees centigrade. If a 6.95(10~ 5 ),
b = 9.85(1Q~ 6 ), and c = 15(10~ 7 ), estimate the temperature at which
water has a maximum density.
316. The study of the formation of producer gas in chemical engi-
neering leads to the equation
(6 a)v = x x b / a ,
where b and a are constants that depend on the process, x is the pro-
portion of residual water remaining undecomposed and v is the corre-
sponding value for the amount of C0 2 .
a. Determine a relation for the maximum value for v (amount of
CO 2 ), assuming that b is larger than a.
b. If a 3.17 and b 4.18, determine the maximum value for v
correct to slide-rule accuracy.
317. In a study of solid carbon reactivity it is found theoretically
that the amount A of C0 2 present is given in terms of the time t (in
seconds) by the equation
where k\ and k* are constants. Determine the maximum value for A
and the time at which it occurs.
318. The total cost of manufacturing a certain article is fixed by:
1. The fixed organization cost which is $90 per day.
2. The unit production cost of each article which is $0.09.
100
ENGINEERING PROBLEMS
3. The cost of repairs, maintenance, etc., which is # 2 /10,000 per day
(in dollars) as estimated by past records, x is the number of articles
produced per day.
a. Show that the total cost for each article in dollars is
b. Determine the number of articles to be produced each day to
make the unit cost least.
VELOCITY AND ACCELERATION. RECTILINEAR MOTION
319. Figure 100 shows a crank arm OA, which revolves at the constant
rate of co radians per second and has a length of r ft. The connecting
FIG. 100.
rod AB has a length of L ft. I? is a piston which moves along the
horizontal axis.
a. Show that x = r(l - cos Q + L - (L 2 - r 2 sin 2 wt)M.
b. Expand the binomial to two terms by aid of the binomial theorem
and obtain the approximate expression for x :
x = r(l cos 0)0 + jr- (1 cos 2coO.
c. Determine dx/dt and d 2 x/dt 2 , using both the original and the
approximate expressions for x.
d. Tabulate the values for dx/dt and d 2 x/dt 2 from both results in
(c) when ut = 0, T/4, ?r/2, TT, and 3?r/2. Assume that r = I ft.,
L 5 ft., and co = J^ radian per second.
e. When is the acceleration of the piston a maximum and hence what
is the maximum inertia force which the piston can transmit? (The
maximum inertia force is given by F = ma, where m is the mass of the
piston and a is the maximum acceleration.) Use the approximate expres-
sion for d 2 x/dt 2 to answer this question.
DIFFERENTIAL CALCULUS
101
320. The mechanism shown in Fig. 101 is an air compressor. The
displacement of the piston from its position when the crank arm AB is
vertical is given by x = r cos co, where r is the length of the crank arm
and the crank arm is rotating at co radians per second.
a. If r = 0.8 ft. and o> = 4?r radians per second, sketch the space-time,
velocity-time, and acceleration-time graphs and label the amplitude and
period of each.
b. If the weight of the piston and connecting rod is 100 lb., what is
the force transmitted to the piston when ut = 7r/6? (Use force =
mass X acceleration; the mass = 10 % 2 , approximately.)
c. What is the maximum inertia force which the piston can transmit
to the crank arm and connecting rod?
FIG. 101.
321. A 1-lb. weight is suspended from a spring fixed at the upper end
and the spring stretches 0.375 in. The weight is pulled 0.1 ft. farther
downward and then released with an initial velocity downward of
1.59 ft. per sec. Resistance to motion is always Jieo of the speed in
feet per second. The displacement of the weight downward from its
position of equilibrium is then given approximately by
y =
cos 32Z + 0.05 sin 32J).
a. Find the time t sec. at which the speed is largest and give the
corresponding displacement.
b. Sketch a graph of the displacement as a function of time.
c. Let t = ti be a value of t that makes y a relative maximum. Let
t = 2 be the value of t that makes y a relative minimum and such that
there are no other maximum or minimum values between t\ and t z .
Also suppose that t 2 is larger than ti. Determine an expression for
2/2/2/1, where 2/2 corresponds to t z and 7/1 to h.
322. A streetcar oscillates harmonically in a vertical direction on its
springs. The amplitude of motion is 1 in., its frequency is 2 cycles
per second, the loaded cab weighs 20,000 lb., and the truck and wheels
weigh 2,000 lb. Determine the force acting on the rails. Sketch.
102 ENGINEERING PROBLEMS
Solution: Let y denote the displacement of the cab from its equi-
librium position. Then
y = H2 sin 4irf ft.,
sm 47rZ ft- P er sec - P er sec
The force acting on the rails at any time t is the sum of the total weight
of cab, truck, and wheels and the force required to produce the acceler-
ation (which can be obtained by aid of Newton's second law of motion).
Thus,
= 22,000 - 8,180 sin 4art Ib.
VELOCITY AND ACCELERATION. CURVILINEAR MOTION
323. Determine the velocity and acceleration, in magnitude and
direction, of the middle point of the connecting rod AB in Prob. 319
at the time t % sec. Use the data in Prob. 319d.
324. A point P moves along the parabola y 2 = 36z according to the
parametric equations x 4t 2 , y = 12t^ where x and y arc in feet and
t is in seconds. Let the projections of P on the x and y axes be A and B,
respectively.
a. Determine the velocities of A and B when t 2 sec.
6. If, at the instant t = 2 sec., A is given (in addition to the motion
due to P) a velocity equal in magnitude but opposite in direction to
that of B, what will be the resultant velocity of A ? In mechanics this
resultant velocity is called the " velocity of A relative to that of B. })
325. A weight moves around the curve
x = 5 cos lOrrZ cos
y = 5 sin 10rr sin S
where x and y are in feet and t is in seconds.
a. Sketch the curve and determine the period of motion.
b. Determine the speed at any time t.
c. Determine the magnitude of the acceleration at any time t.
d. When is the magnitude of the acceleration a maximum or minimum
and at what positions on the curve do these values occur?
*326. The gear C has a diameter 20F of 16 in. and is fixed. OE.
rotates about at the rate of 12 r.p.m. Gear G is in mesh with gear C
and has a diameter of 4 in. The connecting rod PB, 50 in. long, is
pivoted to gear G and P and causes piston B to move along the x axis-
a. Determine the parametric coordinates for P at any time t, assuming
that P is at H at time t = 0.
DIFFERENTIAL CALCULUS
103
b. Determine OB as a function of the time t.
c. What is the velocity in magnitude of point P at time t = 0?
d. What is the acceleration in magnitude and direction of point P
at time t = 0?
e. What are the values of the velocity and acceleration of B at time
* = 0?
FIG. 102.
327. Same as Prob. 325 but for the curve
x = 3 cos 10x + cos 307r, y = 3 sin IQirt sin 3Qwt.
*328. A wheel 4 ft. in diameter rolls without slipping on a horizontal
straight track. The velocity of the center is constant and is 6 ft. per
sec. directed toward the right. Write
the parametric equations of motion, in
terms of the time t sec., for a particular
point on the rim of the wheel, measuring
the horizontal distance or abscissa from
a point at which this rim point would be
on the ground. Then find the velocity
and acceleration of this rim point at an/
time after it passes this reference point
on the ground. Also determine the
maximum numerical values of the veloc-
ity and acceleration.
*329. A ladder is 8 ft. long and rests
against a wall as shown in Fig. 103.
It starts slipping in such a way that
dO/dt 2 radians per second and
d*0/dt* = 3 radians per second per Fia. 103.
second.
Determine the velocity and acceleration in magnitude and direction
of the center of the ladder at the instant the ladder starts slipping (the
104 ENGINEERING PROBLEMS
ladder makes an angle of 60 at the instant it starts slipping). Work
this problem by two methods :
a. By aid of the parametric equations for the path of the mid-point.
b. By the use of at and a n to obtain the acceleration, where a t is
the tangential component of the acceleration and a n is the normal-
component.
*330. A link mechanism is shown in Fig. 104. The member AD
is fixed in a horizontal position. AD = 1 ft., AB = DC = 2 ft.,
BC = 3.828 ft. The members are
free to turn at the pins A, B, C,
and D. If member DC is rotating
about D with a constant angular
speed of 4 radians per second, find
the velocities and accelerations of
points C and B and the angular
, speed and acceleration of member
AB, all at the instant that the
member BC is parallel to the fixed
member AD, i.e., when ir/4- and <p = 37T/4.
*331. Show that the most general equation of rectilinear motion
which has constant acceleration is
s = a + U + ct\
Identify the graph of s as a function of the time t and sketch the graph,
if a = 2, b = I, and c = 1.
Let (tijSi) and (fa,Sz) be two points on this space-time curve. Deter-
mine the average rate of change of s with respect to t as t increases from
ti to t 2 and show, on your graph, the geometrical equivalent of this
result.
Determine the average of the instantaneous rates of change of s
with respect to t at ti and fa. t Show that your result is the same as
previously obtained.
What geometric property of the motion curve makes this true?
Show, by means of an example, that the equivalence is not true in
general if s is a polynomial in t of degree higher than two.
332. The Le Due equation for the velocity v ft. per sec. of a projectile
at a distance of x ft. in the bore of the gun is
v " b~'+~x = dt'
where a and b are empirical constants. Determine the distance x ft.
where the total force imparting velocity to the projectile is a maximum.
(Use force = mass X acceleration.)
DIFFERENTIAL CALCULUS 105
*333. A circular disk in a horizontal position slides freely down a
vertical wire. At the same time it rotates uniformly about its center
at co radians per second. Determine the magnitudes of the velocity
and acceleration of a point of the disk r units from the center at the time
t sec. Assume that the disk starts from rest.
334. A particle moves with simple harmonic motion in a groove in a
horizontal bar and the bar falls freely. Find the magnitudes of the
velocity and acceleration of the particle at the end of t sec. if the particle
starts from the center of its path at the same time that the bar starts
to fall. Assume that the particle makes a complete vibration in the
groove in b sec. and that the amplitude of vibration is r.
335. At the instant a dive bomber releases a bomb the airplane makes
an angle of 60 with the horizontal, the bomber is traveling at 500 miles
per hour, and the airplane is approximately 2,000 ft. above the ground.
Determine a first approximation for the parametric equations of motion
of the bomb (neglect air resistance, wind, spiral motion of the projective,
etc.). Also determine the point of impact on the earth, with respect to
the position where the bomb was released.
DIFFERENTIALS
336. Bernoulli's equation from fluid mechanics may be written
where p is pressure, p is the constant density of the fluid, v is velocity,
and // is a constant. Obtain an approximate formula for the change in
pressure Ap due to a small change in velocity Ay.
337. The equation of the LL-3 scale on a 10-in. log log slide rule is
y in. = 10 logio (log e x),
where the numbers that appear on the scale are values of x, and y is
measured from the left-hand side of the rule. Use differentials to
determine the approximate distance in inches (Ay) between the numbers
marked x = 3 and x = 3.2.
Note: The true length is 25 cm.
338. van der Waals' equation for real gases is
+ J (*-&)= nRT,
where a, 6, n, R are constants, p is pressure, v is volume, T temperature.
a. Determine an equation for dv if T is a constant.
b. Determine an equation for dv if p is constant.
339. A sphere is to be fired (baked) out of a certain kind of clay
that has a linear shrinkage of 4 per cent. Determine the approximate
106 ENGINEERING PROBLEMS
radius of the sphere before firing (baking) if the final radius is to be
6 in. Determine the approximate change in volume.
340. The heat Q required to raise the temperature T of a certain liquid
1K. (one degree on the Kelvin or absolute centigrade scale) is given
by Q = a + bT + cT 2 . Determine an expression for the amount of heat
necessary to heat the material from T = 300.0K. to T = 300. 1K.
341. Determine the approximate change in tan if 6 increases from
4510 / to 4511'. Then check with your tables.
342. If y = sin x and x is to increase by 1 min., determine the value
of x that makes the largest change in y. Then refer to your tables and
check.
343. The force acting on a projectile from a gun after firing is
F 4/(t + 0.05) 4 Ib. and is valid so long as the projectile is in the bore
of the gun. The projectile weighs 5 Ib. Use Newton's second law of
motion (F ma where m = % 2 m this case and a is the acceleration)
to find an approximate value for the speed 0.01 sec. after firing.
CURVATURE
344. The following statement is from " Resistance of Materials,"
by Seely: "The error made in approximating
_
d 2 y/dx 2
by R = l/(d 2 y/dx 2 ) is never more than about 4 parts in 1,000 since the
tangent of the angle to the elastic curve (the curve which a loaded
beam assumes) is probably never larger than one part in twenty."
Assuming that the tangent to the elastic curve never has a slope
(i.e., dy/dx) numerically larger than 0.05, what is the largest percentage
error possible in the value obtained for R by the preceding approximate
formula?
r
i
^~- ....
Curve of beam
FIG. 105.
345. Figure 105 shows a simply supported beam loaded with a uni-
form load of w Ib. per ft. The equation of the "elastic curve" (the
curve of the beam) is
wLx* wx* wL 3 x
-W~~2i~ ~24~'
DIFFERENTIAL CALCULUS
107
where L is the length of the beam, / is a property of the cross section,
and E is a property of the material from which the beam is made.
E, I, and w are positive constants.
Determine the radius of curvature at x = L/2. Assume that
L = 12 ft., E = 288,000,000 Ib. per sq. ft., w = 400 Ib. per ft., and
I = 0.008 ft. 4 .
346. Determine the curvature and radius of curvature at u = 0,
at u = 0.5, and at u ~ 1 to the railway easement curve which is given
by the following parametric equations :
' ( u - S) ft "
y = 200 (u* - ~) ft.
x = 600 i
347. Use the answer for Prob. 293 and determine the radius of
curvature at x = 0, the leading edge of the airfoil.
INDETERMINATE FORMS
348. Figure 106 shows a flat plate in the form of an isosceles trapezoid.
When this plate is subjected to an axial tensile load (pull) of P Ib., the
elongation of the plate is given by
PL
A =
, c
tE(c - a) l ge a
where E is a constant and depends on the material of the plate.
IP
T
FIG. 106.
The formula for the elongation of a flat plate with a rectangular cross
section is A = PL/AE, where A = a t = area with width a and
thickness t.
Evaluate lim A in the first formula and obtain the second formula.
Remark: It is common engineering practice to check new formulas with
formulas for simpler situations that are already known. This process of
108 ENGINEERING PROBLEMS
checking often involves, as in the present problem, the evaluation of indeter-
minate forms.
349. A weight W Ib. hangs by a spring whose spring constant is
k Ib. per ft. A harmonic force F = P sin ut Ib. is applied to the weight.
Let p 2 = kg/W, where g = 32.2 Ib. per sec. per sec. and q = Pg/W.
At the time t sec. the displacement of the weight from its equilibrium
position is given by
y __ (3^ w _ sm pl f
p 2 co 2
This equation is valid provided that p 7* co.
Determine the limiting value of the displacement as p approaches
the value co (resonance case).
350. The current flowing in a series circuit, consisting of a resistance
of R ohms arid an inductance of L henrys connected to a battery of
E volts, is given by
Determine the limiting value for i as R approaches zero and thus
obtain the equation for the current, i amp., that would flow in a purely
inductive circuit.
351. Determine the limiting values for each of the antenna radiation
patterns in Prob. 216 as 6 approaches zero.
352. A "fluid foil" for a section of an airplane wing is given in para-
metric form as follows:
Xp = Xs = (gin 0) |J (A - B) + B] + (tan 0) (l - ^ 0) (1 - A),
y s = (cos 0) [~ (A-B) + B] -A (l - \ 0),
y f = (cos 0) [~ (A - B) - B] -(A- 2B) (l - \ 0),
where A and B are constants. x s and y s are parametric coordinates
for the "suction" face of the wing section, and x p and y p for the "pres-
sure" face.
Show that the two sets of parametric equations define the same
result when = and when = 90 = w/2 and determine the results
in terms of A and B.
353. In Prob. 398 the bar of equal strength is approximated by
a bar having a large number of constant-section segments each of
equal thickness. It is shown in elasticity that the area at the nth sec-
tion is
DIFFERENTIAL CALCULUS 109
A n = 7 T \~~>
"(l-^Y
\ ma/
where m is the total number of segments and L is their combined length.
Let x Ln/m and find the limiting value for A n as n increases without
limit. Note that m also increases without limit and that x/L is a
constant for the limiting operation process.
354. The deflection of a clamped circular plate of radius a supporting
a load P at its center is
**l+\v -<*)]>
where r is the radial or polar coordinate distance and N is the so-called
"rigidity constant' 7 for the plate.
a. Evaluate the deflection w, the slope div/dr, and the "bending
j.i> TIT AT [ d*W , 1 dw~\ , , v ... , ~
moment M ~ N \ -r-r- + r- at the origin, i.e.. at r = 0.
L dr* r dr J
6. Sketch the graph of the deflection as a function of r and find an
inflection point.
c. Evaluate the bending moment M at r = a. This last result leads
to the interesting fact that the bending moment at the edge of the
plate is independent of the size of the plate.
Note: The deflection equation has another interpretation: w is also the
deflection at the center of the plate caused by a load P at a distance r from
the center.
355. The equation W - 2 y - - - gives the work done by a gas
JL ~~~ il
as it expands from pressure p\ and volume V\ to pressure p 2 and volume
V 2 . The constant n is used in the equation
and the given equation for work is valid if n ^ 1.
a. Show that the equation for work may be written in the form
W =
6. Use this last equation to determine lira W and thus obtain the
formula that should be used when p\V\ =
356. Use the data of the preceding problem and the equation
Ti _
110 ENGINEERING PROBLEMS
to determine
n >1 1 ^
and simplify your result by aid of the equation pV = wRT.
357. In Prob. 161 about the loud-speaker horn, start with the
equation
y = [x(yi l ' - 1) + 1]",
notice that y\ > 1, and determine the limit of y as n increases without
limit through positive values and through negative values. Then
sketch the horn outline for your resulting curve or curves.
RELATED RATES
358. If the quantity of wood in a tree is approximately proportional
to the cube of the diameter at its base and if the diameter increases
approximately 0.8 in. per year, what is the approximate rate of change
i:i the volume? Give your result in terms of the constant of variation
and the radius of the tree.
359. A given quantity of gas is expanding according to the adiabatic
law: pV lA = k = constant. If the volume is 10 cu. in. when the
pressure is 20 Ib. per sq. in. and if the pressure is increased at the con-
stant rate of 0.5 Ib. per sq. in. per sec., what is the rate of change of
the volume when the volume is 5 cu. in.?
360. The relation between altitude above sea level (h ft.) and the
pressure (p Ib. per sq. ft.) at a certain place on the
earth and at a certain time of year is given by
p = 2,140e-- 000 ' 035 ' 1 .
If an airplane is climbing at this particular spot on
the earth and at the stated time at the rate of 200
miles per hour, what is the rate of change of the
pressure due to change in altitude when the airplane
is 3 miles up?
361. A boat is pulled in by means of a rope wound
around a windlass on the dock which is 20 ft. above
the deck of the boat. If the windlass is pulling the boat in at 10 ft.
per sec., determine, when there is 100 ft. of rope out:
a. The speed of the boat.
6. The acceleration of the boat.
PARTIAL DIFFERENTIATION
362. Airy's stress function for the torsion (twisting) of a bar with
an elliptical cross section with semimajor and serniminor axes a and 6,
respectively, is
DIFFERENTIAL CALCULUS 111
where M is the " torque" or the amount of twist that is applied at the
end of the bar.
a. Determine the value of p for points (x,y) on the boundary of the
ellipse.
b. Sketch a graph of the surface <p as a function of x and y. Show
only that portion of the surface for points inside or on the boundary
of the given ellipse.
c. Evaluate r xz = > r yg =
5y dx
and give their geometrical significance. These two quantities are
measures of the " shearing stress" in the bar at any point of a cross
section.
363. Given the function F = (y mx)(y + nx)(x a), which is
Airy's stress function for a beam with a triangular cross section.
a. Sketch a graph of the surface, F as a function of x and y. Assume
that m, n, and a are all positive constants. Show only the portion of
the surface whose projection on the xy plane is the triangle.
d 2 F d 2 F d 2 F
b. Evaluate cr x -^-^ <r y = =-,-> r xy = ~-~~,
dy 1 ox 2 oxdy
which are measures of the tensile stress in the x and y directions and
the shearing stress.
c. What are expressions for the curvature at a point on the surface
in the x direction and in the y direction? What is the approximate
relation between these results and those from (b) ?
364. "Flow around a cylinder" may be characterized by the equation
<p = ax/ (x 2 + 2/2).
a. Determine the velocity components v x d<p/dx and v y 5<p/dy
and give the magnitude of the velocity that has these for components.
b. Determine the velocity in magnitude and direction at several
points on the circle x 2 + y' 2 = 4. Assume that a = 1.
Remark: This type of analysis sometimes appears in undergraduate texts
on fluid mechanics.
365. The "stream function" and "velocity function" for a uniform
translating flow past a circular cylinder are, respectively,
- -as (l
112 ENGINEERING PROBLEMS
Show that -~ = ~ (= v x ), ~^= -~(=^), and give the expres-
ox oy by ox
sions thus twice obtained for v x and v y , the velocity components.
366. Given the function
\l/ = x 2 + xy t/ 2 + x y.
Show that -r-~ + = 0. (Taken from a text in aeronautics.)
5x 2 dy 2
*367. In thermodynamics, c p is the specific heat at constant pressure
and c v is the specific heat at constant volume. For oxygen,
c p - 0.1904 + 0.000,056,57^ - 0.000,000,009,81 T\
c v = 0.1284 + 0.000,056,57 7 - 0.000,000,009,81 T\
where T is absolute temperature on the Fahrenheit scale and c p and c v
are in B.t.u. per pound per degree Fahrenheit.
If c p = ( TT?- ) and c v ( Tp ) ' sketch the figures and compute the
approximate change in Q.
a. If T increases from 400 to 402 at constant pressure.
b. If T increases from 400 to 402 at constant volume.
368. A rectangular box has a metal plate as its top. This plate has
a hole in it of an elliptical shape with major and minor axes 2a and 26,
respectively. A soap film is stretched across the hole and the pressure
in the box is increased so that the soap film bulges upward. The ordi-
nates to the soap film are given by
provided that the pressure is not too large, k is a positive constant
which depends on the values of a and b and the pressure.
a. Sketch a graph of this surface (soap film).
b. Determine the slope to this surface in
(1) The <p x plane at x = a.
(2) The <p y plane at y = b.
369. The equation for a perfect gas is pV = kT, where p is pressure,
V is volume, T is temperature on an absolute basis, and k is a positive
constant.
a. Determine dT/dp, ST/dV, and 5P/5V.
b. Give an expression for the total differential dT, first in general
terms by aid of partial derivatives and then in terms of p, V, k, dV,
and dp.
*370. The total heat Q in thermodynamics is a function of the tem-
perature T, the pressure P, and the volume V. How many different
DIFFERENTIAL CALCULUS 113
first partial derivatives are there of Q with respect to any one of these
three variables? In answering this question, be careful to notice that
P V = RT where R is a positive constant. Then verify the following
different equations for d!Q, in which each equation involves two of these
first partial derivatives :
-($),
where the letter subscript denotes the quantity that is held constant.
Thus, (5Q/5T)v indicates the first partial derivative of Q with respect
to T with constant volume and the pressure was first eliminated by
PV = RT.
*371. Use the data in the preceding problem and express each of
(8Q/SV) T , (5Q/5P)r, (5Q/5P)y, and (dQ/dV) P in terms of P, 7, R,
c p = (8Q/dT)p, and c v (5Q/5T) V . Notice, for example, that
(5Q/dP) v = (dQ/dT)v(dT/5P) v . Also, since
dV
it follows that (6Q/67) r = (dQ/SPM-ftT/F 2 ) + (dQ/dT) P (P/R).
372. Use the " contour graph" in Prob. 262 and determine approxi-
mate values for each of the following:
(a) b : ~ the plane e c = 1 and at e ft = 60 volts,
(6) the plane e b = 50 and at e c = 0.
PART V
INTEGRAL CALCULUS
CONSTANT OF INTEGRATION
373. The ram of a pile driver hits a pile and travels with it. The
pile is driven 4 in. and moves that distance in 0.05 sec. Assuming
the deceleration of the driver to be constant, determine the speed of
the ram at the instant of impact.
374. The motorman of a streetcar increases the power of the motors
by gradually cutting out resistance. The tractive effort on the rails
increases steadily at the rate of 24 Ib. per sec., the car weighs 16,100
lb., and the frictional resistance to motion is always 400 Ib. Determine
the distance (s ft.) that the car moves as a function of the time (t sec.)
since the motorman started to increase the power, if the speed of the
car at that instant was 10 ft. per sec.
Solution: By aid of Newton's second law of motion:
Total horizontal force = (mass) (acceleration in horizontal direction).
Hence
375. A ship, while being launched, slipped down the skids with a
constant acceleration. If the ship slid the first foot in 10 sec. and if
the skids were 400 ft. long, determine the time it took for the launching.
376. An automobile is moving along a straight road at 60 miles per
hour when the driver decides to stop the car. If the brakes can slow the
car down with a constant deceleration and if it takes 75 ft. to stop the car,
what was the deceleration and how long did it take to stop the car?
377. The braking resistance of a streetcar is 200 lb. for each 1,000 lb.
of weight. If a streetcar weighs 15,000 lb. and is traveling at 24 miles
per hour, determine the time required to stop the car and the distance
it will travel (in a horizontal direction). Use Newton's second law of
motion to start the solution of the problem.
378. A 4-in. marine gun fires its shell weighing 38 lb. with a muzzle
velocity V Q = 2,300 ft. per sec. Actual trajectories of the shell are
shown in Fig. 108.
1. For a gun elevation of 45.
2. For a gun elevation of 75.
114
INTEGRAL CALCULUS
115
{ y miles
4 6
FIG. 108.
10,
miles
a. Determine the equations of the two theoretical trajectories neglect-
ing air resistance, etc.
6. Determine the theoretical altitude and range in both cases.
c. What is the approximate error made in altitude and in range for
both cases?
379. During an interval of 4 sec. 10
after the current is shut off, the
angular velocity co of a certain elec-
tric motor is given with sufficient
accuracy for engineering purposes
by the equation
co = 200 - 20 + 0.5 2
radians per second, where t is the
*time in seconds since the current
is turned off.
a. What is the angular accelera-
tion at t = 3 sec.?
b. How many revolutions does the motor make during the 4-sec.
interval?
c. Does the angular acceleration increase or decrease during this
interval? Why?
380. A circuit (Fig. 109) consists of an inductance L = 0.2 henry con-
nected to a generator with an electromotive force e = 100 sin ut volts
(^ == 60 cycles per second = 120?r radians
per second). The switch is to be closed at
the time t = 0; hence when t = 0, the
current i = amp.
Given that L(di/dt) = e = 100 sin co,
determine a formula for i in terms of the
FlG 10Q> time t sec. Sketch the graph of i as a
function of t for t from to J^ sec.
Also determine the current flowing in the circuit when t = >f so sec.
381. When a condenser (C farads) is charged from a battery (E volts)
through an inductance (L henrys) and resistance (R ohms), the current
is given by
^ =
E
VI
~
which, by the introduction of new symbols, may be written
i =
Li,)*'
sin gt.
116 ENGINEERING PROBLEMS
Integrate the expression for condenser voltage
* dt
and obtain e c (EgQ/g)e~ at cos (gt r?) + k. You are to obtain
explicit expressions for g Q and rj.
Also determine k if e c = when = 0.
382. A ship whose weight is 100 tons is traveling at the speed of
40 ft. per sec. when the power is cut off. The acceleration at any time
t sec. later is given by
1,600
~+ 50)2
P cr sec - P er
and this is valid for about 5 min. Determine the velocity and distance
traveled (assuming that the ship moves along a straight path) as func-
tions of the time t sec. What are their values at the time t = 4 min.?
383. A bomber is traveling horizontally at a height of h ft. with a
speed of 300 miles per hour = 440 ft. per sec. Neglecting air resistance,
etc., determine how far ahead of a target the bomb should be released
if (a) h = 10,000 ft.; (b) h = 20,000 ft.; (c) h = 30,000 ft. Also
compute for each case the angle that the bomb makes with the vertical
on impact with a horizontal target. What is the theoretical speed of
the bomb for each of these three cases just before impact?
The following table gives the observed striking speeds and angles of
impact :
Altitude,
feet
Striking speed,
miles per hour
Angle of impact
with vertical
2,000
540
50
4,000
620
40
6,000
690
34
8,000
750
30
10,000
790
26
12,000
830
24
15,000
870
21
20,000
940
17
384. Given that power = rate of change of work = dw/dt, that
w = mv z /2, and that v = dx/dt = velocity. Derive the equation of
motion of a body which is accelerated with a constant or uniform
expenditure of power. Assume that v VQ and x = when t = 0.
m is the constant mass of the body and x is the rectilinear displacement.
INTEGRAL CALCULUS 117
Solution: Since dw/dt = P = power = constant, w = Pt + c\.
When t 0, v = VQ, and hence c\ = mv Q 2 /2.
nM , mvo 2 mv z
Hence rl -\ " = w = -fr ,
_ dx __ /2P<
/ m \ /2PZ
Then x = ^p J ^ + i
Since x when t = 0, determine 2 = (m/3P)(^o 8 ) and hence
Now sketch a graph of x as a function of t. You will find it convenient
for this purpose to sketch 3Px/mvo s as a function of 2Pt/mvo z .
385. Given the equation (mv)(dv/dx) = mkx, where m is a constant
mass and ft is a constant of variation. Integrate this equation and
transpose all variable terms to the left-hand side. If kinetic energy
(energy due to motion) is mv^/2 and if potential energy (energy due to
position) is mkx' 2 /2, what does your resulting equation state?
386. A beam of length L ft. is simply supported and is loaded with a
concentrated load of P Ib. at a point located 2L/3 ft. from the left
FIG. 110.
support. If axes are chosen as indicated in Fig. 110, the following
equations hold:
For < x < 2L/3:
^ _ (JL\ (*}
dx* \EIJ V3/'
For 2L/3 < x < L:
^ED
where E and / are positive constants.
Determine the equation for the "curve of the beam," i.e., for y as a
function of x. Notice that at x = 0, y = 0; at x = L, y = 0; and at
x = 2L/3 the curve is continuous and smooth, i.e., the ordinates
obtained from the two equations are the same and the slopes are likewise
equal.
118
ENGINEERING PROBLEMS
387. A simply supported beam is loaded with a load w in Ib. per ft.
that varies directly as the distance from the left support as indicated
in Fig. 111. Given, from strength of materials, that
----*.
where E = 200,000,000 Ib. per sq. ft., I '= 0.00800 ft. 4 , and y is the
ordinate to the curve of the beam at abscissa x ft. from the left support
-10'-
5000
Ibs.
10,000
3
Ibs.
FIG. 111.
Given that (d*y/dx*)(EI) = 5,000/3 when x = 0, tfy/dx* = when
x = 0, and y = when x = arid when x = L. Determine the
equation for y as a function of x, i.e., "the curve of the beam." Also
determine the maximum deflection of the beam from a horizontal
position.
388. Two weights of 7 Ib. and 9 Ib. hang on the ends of a string that
passes over a smooth pulley. The smaller weight moves upward for
5 sec. at which instant the string breaks. How far will it continue to
move upward after the string breaks? Use g = 32 ft. per sec. per sec.
Notice that the velocity and acceleration of the smaller weight are
always numerically equal but opposite in direction to those for the larger
weight. Notice also that the total force acting on the smaller weight
at any instant is equal to 7 Ib. downward combined with the constant
tension T Ib. in the string.
H
->-#
Fio. 112.
389. A wooden beam is 12 ft. long, 4 in. wide, and 8 in. deep, and is
loaded with a uniform load of 400 Ib. per ft. as shown in Fig. 112.
INTEGRAL CALCULUS
119
From strength of materials one can obtain the following equation :
d 2 v
-Ji = 0.00204z - 0.000, 170z 2 ,
where x and y are in feet.
Determine the equation for y in terms of x and also determine the
minimum value of y, i.e., the largest deflection of the beam.
2000 Ibs. /ft.
10,000 Ibs.
5' j 5'
5'
51 1
17,500 Ibs.
FIG. 113.
r
2500 Ibs.
390. Figure 113 shows a beam supported at points 5 ft. from each end.
The beam carries a uniformly distributed load of 2,000 Ib. per ft. for
the first 5 ft. and a concentrated load of 10,000 Ib. at the mid-span.
Figure 114 shows a graph of the " shear " V as a function of x. The
first integral of " shear" is " bending moment" M and, in this problem,
the bending moment is zero when x = 0.
7,500
1
\
-10,000
1 1
-2500
FIG, 114.
Sketch a graph of the bending moment M as a function of x without
obtaining any equations. Make use of the following facts:
1. The derivative of the curve you are to sketch is the given " shear"
curve.
2. The first integral of a horizontal line is an inclined straight line.
The first integral of an inclined straight line is an arc of a parabola
with axis vertical, and the position of the vertex of the parabola is
determined by the abscissa where the straight line crosses the x axis.
3. The first integral represents the area between the " shear" curve
and the x axis, from x to x x.
391. Figure 115 shows the speed of a train t min. after it left the start-
ing point.
a. Sketch the graph of distance as a function of time (t min.) without
writing any equations. Make use of facts such as those stated in
Prob, 390,
120
ENGINEERING PROBLEMS
b. Obtain the equations for the distance from the starting point as a
function of the time t min.
392. The velocity of a trimolecular reaction (discussed in physical
chemistry) is given by
j = k(a - x)(b - x}(c - x),
where a, b, c, and k are constants. If x = x when t = 0, determine an
equation for t as a function of x.
,,V miles/min.
0.8
10
100 105 120
FIG. 115.
393. The speed of "inversion" of cane sugar is given by
dr
jj- = 0.00154(10.023 - x),
where t is time in seconds and x is the amount in grams of the new type
of sugar present at time t.
If x = when t = 0, obtain an equation for x as a function of t and
for t as a function of x. Sketch the graph for t as a function of x for t
from to 30 min. = 1,800 sec.
394. The decomposition of such materials as radioactive substances
obeys the following law: The rate of decomposition at any moment is
proportional to the quantity present at that moment. If one starts
with an amount q and if at time t T the amount is #/2, obtain an
expression for this special time, known as the period of half life.
395. Use the law stated in Prob. 394. If one-fourth of the material
has decomposed in 2 hr., what is the value of the constant of propor-
tionality and what is the period of half life?
396. A study of a corrosion-time relationship for iron requires the
integration of dy/dt = p/y where y is the thickness of the layer of
corrosion at time t and p is a constant. Integrate if the material is
clean at time t = 0.
INTEGRAL CALCULUS
121
397. Kick's law for the work done in reducing the size of a given
amount of material (crushing rock, for example) assumes that the
energy required for subdivision of a definite amount of material to be
the same for the same fractional reduction in average size of the indi-
vidual particles. Thus, if 10 hp.-hr. ( = E) is required to crush a
given amount of a certain material from 1 to ^ in., average diameter,
the energy required to reduce it from Yi to ) in. or from Y to % in.
would be the same. Then E = b log (L]/L 2 ) where LI is the initial and
L 2 is the final average linear dimension arid 6 is a positive constant to
be determined experimentally. (It also depends on the base used for
the logarithm.)
ItdE/dL = -C//A
a. Derive Kick's law. Use n = 1.
b. Derive Rittinger's law if n = 2: E = C(Li - L 2 )/LiL 2 .
c. A general law. Assume that n ^ 1 .
d. Determine the limiting value of the
equation for E in (c) as n approaches the
value one (an indeterminate form) and
again derive Kick's law.
398. A bar of variable cross section is
suspended from the upper end and sup-
ports a load P at its lower end (see Fig.
116). The total force on any section is
the load P and the weight of the bar
below this section. The most favorable
condition exists when each section is equally stressed. This result
follows from integrating
dA _ Ay
dx a
where A is the area of the section of the bar at a distance x from the
lower end, 7 is the weight per unit volume of the bar, and a is a constant
allowable stress.
a. Find A if the area at x is A = P/a.
b. Find the area at the fixed end x = L = 12 ft., if P = 36,000 Ib.,
cr = 16,000 Ib. per sq. in., and 7 = 450 Ib. per cu. ft.
399. A sled is being pulled along level ice by a rope, which is inclined
at an angle of 11 with the horizontal. The force pulling the sled
varies thus with the time : F = 24 *- 0.9 2 (F in pounds and t in seconds) .
The change in "momentum" of the sled is defined to be the product
of the force component in the direction of motion multiplied by the
time interval during which this constant force acts. Find the change
in momentum of the sled during the interval from t = 1 sec. to i = 5 sec.
122 ENGINEERING PROBLEMS
EVALUATION OF DEFINITE INTEGRALS
400. Evaluate the following definite integrals that were found in
textbooks in chemical engineering and in technical journals for chemical
engineering:
(a) I y n dw if y n = e ELw (y n o Y n -i) + F n -i and E } L, y nQ) and
F n _i are constants.
(b) A = 232 + r (0.000,374* + 0.251) dt.
/176
(c) V TT I x 2 dy if x dy = 2k dx. where k is a constant and x = a
/o
when y 0.
(d) C w . dw JT-.
(e) If h(x) = 2.995e- 14 - G27 * + 2.1&?- 82 - 22 * + l.OOGcr 212 *, obtain an
equation for 6(x) if
1 - 0(x) = 4
(/) Evaluate I TPO ? \27/'* \' ^< ; * s a constant.
401. An electric circuit contains in series an inductance of L 0.5
henry and a resistance of r = 20 ohms. At the time t the current
i = 0.6 amp. Given, from electrical engineering, that L(di/dt) + ri =
(for this problem), show that the current i at the time t = 0.01 sec.
is given by the definite integral
r { di /o.ot
I -r- = 40 I dt, and evaluate i.
Jo.6 ^ Jo '
Also solve this problem by the constant of integration method.
402. In a circuit containing resistance and inductance in series, the
power supplied to the magnetic field of the inductance is given by
p = (*
Show that the total energy stored in the magnetic field, a quantity
defined by W = f^Pdt, has the value W = L/ 2 /2 where / = E/R.
/o
403. If a battery of E volts and zero internal resistance is connected
to a long uncharged submarine cable of capacitance C farads and
resistance R ohms (each per mile), the battery current t sec. later is
given by
INTEGRAL CALCULUS
123
a. Sketch a graph of i as a function of t. To show the general shape
of this curve, sketch i/E as a function of irRt/C.
b. Are the current i and the power P = Ei undefined (momentarily
infinite) when the battery is first connected?
i*T (*T
c. Determine the charge Q = I i dt and the energy W = I P dt
taken from the battery up to the time T sec. Are these ever undefined
(infinite)? Sketch each as a function of T.
Remark: A mathematically "infinite" current is not physically possible,
since no circuit can actually have zero resistance. However, the resistance
can be so small that the momentary
current is eno mous compared with
the normal current in the circuit and
may be called "physically" infinite.
404. The potential energy (P.E.)
stored in a beam is given by
Fia. 117.
where E, /, and L are constants.
Determine the potential energy stored in a cantilever beam, which has
for its equation
y = 2/0 (l - cos g).
405. The force ejecting a projectile from a gun changes with the
time after firing according to the equation
F =
4.35
(0.05 + t)'
Ib.
(t in seconds).
The total "momentum" given the projectile (momentum is defined as
mass times velocity) during the 0.04 sec. required for the projectile to
pass through the bore of the gun is obtained by evaluating the definite
integral
.35 dt
rtz /o.04 4 ?
L**-i m
(0.05 + O 4
Evaluate this definite integral.
40'6. On a Mercator map of the earth, the distance on the map of a
parallel of latitude from the equator is given by
S
R sec 4> d<l>,
where S is in units corresponding to the unit on the map which repre-
sents 1 min. of longitude on the equator. Evaluate this integral for
ENGINEERING PROBLEMS
latitude L - 60 and R = 10,800/7r units. What is the corresponding
distance on the earth itself?
407. The potential energy stored in a rod because of a torque (twist)
which has been applied to one end (the other end of the rod being fixed)
and which twisted the bar through an angle of 30 is given by
,. Evaluate.
408. Evaluate correctly to the nearest third decimal the following
definite integral which appeared in a mechanics text:
-r-,7 da.
409. The two following empirical formulas were found for the "specific
heat" at constant pressure for hydrogen:
Cp = 3.45 - 0.000,055,17* + 0.000,000,073,67 7 ' 2 ,
c p = 2.86 + 0.000,028,7 T + =.;
Compute I ' c p dT for the two approximate formulas and thus
J 1,000
determine the total heat required to raise the temperature of 1 Ib. of
hydrogen from 540 to 1040F. (T absolute temperature = degrees
Fahrenheit plus 460.)
410. Evaluate the following definite integrals that appeared in a
text on aeronautics:
()
where R, G, and r are constants.
(ft) T-
411. The general theory of arch dams (for example, dams built
between the walls of a canyon) requires the solution of the equation
j 7 =
for r n in terms of t and c. Show that r n = ; ; 715 TC where
in K in \ti t)
R t/2 + c + r n . When numerical values of R and t are given, this
equation can be solved by approximate methods for r n .
INTEGRAL CALCULUS
125
DEFINITE INTEGRALS. SIMPSON'S RULE AND TRAPEZOID RULE
412. In calculating the capacity of absorption towers in chemical
engineering it is necessary to evaluate certain definite integrals by
approximate methods. Evaluate the following definite integrals by
use of Simpson's rule and the trapezoid rule using the given data
(Xi is an empirical function of x, yi is an empirical function of y) :
r 12 dx r
J2 Xi - x Jo.c
>.026
f 0.010 \
dy
/0.02
JO.OH
026(1 +y)(l
0.010 17 ft
<%.
1
ti
1
X
x\
Xi X
y
/
y - yi
2
5.10
0.322
0.010
. 0008
108
3
5.55
392
0.012
0.0040
125
4
6.13
0.469
0.014
0.0082
172
5
6.70
0.587
0.016
0.0122
263
6
7.40
0.715
0.018
0.0156
416
7
8.15
0.869
0.020
0.0183
588
8
9.05
0.952
0.022
0.0203
588
9
10.10
0.909
0.024
0.0220
500
10
11.40
0.715
0.026
0.0233
370
11
13.25
0.444
12
16.00
0.250
413. The length of an "indicator card" is 3.6 in. The widths of the
diagram at intervals 0.3 in. apart are 0; 0.40; 0.52; 0.63; 0.72; 0.93;
0.99; 1.00; 1.00; 1.00; 1.00; 0.97; 0. Determine the area of the "indi-
cator card" by Simpson's rule and the trapezoid rule and divide by
the length of the card to obtain the "mean effective pressure." Work
this problem using 6 subdivisions and 12 subdivisions and compare
your results.
414. a. The table of data gives the "specific heat" s of water at
/12
temperature 0C. Evaluate the definite integral I s dQ and thus
/o
determine the total heat required to raise the temperature of 1 gram
of water from to 12C.
9
2
4
6
8
10
12
s
1.00664
1.00543
1 . 00435
1.00331
1.00233
1.00149
1.00078
6. Now evaluate the same definite integral assuming (the common
assumption) that s = 1 calorie per degree centigrade for the given range.
126
ENGINEERING PROBLEMS
c. How do your two results compare with the result obtained by
assuming that s = a + bO, where the values of a and 6 are to be deter-
mined by the method of averages?
Remark: In engineering it frequently happens that one has the data and
not the function required for the integration and hence that the use of
Simpson's rule or the trapezoid rule is indicated (or the use of a planimeter).
415. Evaluate by an approximate method (you may evaluate the
denominator by an exact method) :
r /
_ _
Of(6)de
_ _____ _ ______
f*/2 . '
I sin cos 6 dO
Jo
if
e
10
20
30
40
50
60
70
80
90
/(*)
1.00
0.97
0.91
0.80
0.65
0.47
0.30
0.13
0.01
0.00
This problem occurs in a text on illumination.
416. Estimate the number of cubic yards of crushed rock necessary
to make a roadbed of the dimensions shown in Fig. 118. The road
-20-
FIG. 118.
is to be 1 mile long. Assume that any other material added merely
fills up the voids. Also assume that the crown of the pavement is an
arc of a parabola.
417. The data for a "magnetization curve" (for an iron core induct-
ance) are as follows in c.g.s. units:
H
8
10
15
20
30
40
60
80
B
13.0
14.0
15.4
16.3
17.2
17.8
18.5
18.8
where H is the number of gilberts per centimeter (a measure of the field
intensity) and B is the number of kilolines per square centimeter (a
measure of the flux intensity).
INTEGRAL CALCULUS
127
a. Plot a graph of B as a function of H.
/*18.8
6. Evaluate w = (M?r) I H dB,
/o
(1) By use of the trapezoidal rule for each interval.
(2) By aid of Simpson's rule and about 10 subdivisions. Take the
necessary data from your graph.
c. Plot 1/B as a function of l/H. The points will approximate a
straight line. Hence 1/B = a + b/H, approximately. Determine a
and b by the method of averages and then substitute for H in terms of
B in the definite integral and evaluate.
Remark: The value of this definite integral gives the energy stored in the
magnetic field. The results by either method (a) or (6) are probably just
as accurate as those by method (c) and are much more rapidly calculated.
418. In traveling by automobile a passenger checked the speedometer
mileage by reading the rate of speed in miles per hour every 10 min.
The readings were
Time
12:00
12:10
12:20
12:30
12:40
12:50
1:00
Speed
40
30
25
15
22
24
19
Estimate the distance traveled by the car in the 1 hr. Assume that
the given ordinates belong to a continuous curve.
AREAS. AVERAGE ORDINATE
419. The voltage and current in an electric circuit are, respectively,
given by
e = 160 sin ut volts, i = 2 sin ( co ^ ) amp.
a. Determine the " average power," defined as
O Jo
o eidt >
where T is the period of both the voltage and current (T is to be deter-
mined).
b. Sketch the voltage and current waves from t to t 2-rr/o).
Sketch on the same graph the curve for "instantaneous power," defined
by p = e i. Then show the graphical meaning of the preceding
definite integral.
420. The voltage in an electric circuit is given by
e = E sin co volts,
128 ENGINEERING PROBLEMS
where E and o> are constants. You may assume, if you wish, that
co = 60 cycles per second = 120rr radians per second.
a. Determine the average voltage for the interval of time from
t = to t = TT/CO. Also determine the average voltage from t = to
t = 27T/CO.
b. Find the root-mean-square value of the voltage; i.e., find the square
root of the average value of the ordinate to the curve y E 2 sin 2 cot
from t = to t = 27T/CO.
c. Use your result from (b) to determine to three significant figures the
value of E so that the root-mean-square value will be 120 volts.
d. Sketch graphs of e = E sin ut and y = E 2 sin 2 co, each for a
complete period, and indicate the average ordinate for each curve for
the complete period.
421. An alternating voltage is given by
e = 100 sin lOOrrt + 50 sin 300?r^ + 10 sin 500 irt volts.
a. Sketch a graph of the voltage wave for one complete period.
b. Determine the average value of the voltage from
(1) t = to t = 0.01 sec.
(2) t = to t = 0.02 sec.
c. Determine the root-mean-square value of the voltage over a com-
plete period.
*422. An alternating current is given by (k is an odd number)
i = /i sin oil + Iz sin 3co + +/* sin kut amp.
Determine formulas for
a. The average value of the current from t to t = TT/CO sec.
b. The root-mean-square value of the current from i = to i = 2?r/co
sec.
*423. The voltage in the circuit discussed in Prob. 422 is given by
e = E\ sin co + Ez sin 3co ++-/* sin kwt volts.
Determine the " average power" defined by .
where T = 2?r/co.
Remark: Texts on alternating currents derive comparable formulas but
use equations for the voltage and current that involve both sine and cosine
terms.
424. The instantaneous rate of heat production of a current i amp.
flowing in a constant resistance r ohms is i*r watts. Determine the
average rate of heat production over a cycle (one complete period) for
the following periodic curves. Thus, determine the average ordinate to
the curve y = t 2 r for a complete period.
INTEGRAL CALCULUS
129
a. i I sin 27r/, 7 is a constant.
b. i 7i sin 2irft + 7 3 sin GTT/^, I\ and 7 3 are constants.
c. i = 7i sin cot + 7s sin (3cotf 0), 7], 7s, and 6 are constants.
2T ZT
FIG. 119.
Portions of rectified sine waves (output of a controlled rectifier)
T
JL
2
7T
FIG. 120.
27T
Remark: This heat production is the basis of the definition of the effective
(root-mean-square) value of a periodically varying current. That is,
( Effect ive) 2 W is the average rate of heat production over one cycle (one
complete period).
425. A ball is thrown directly upward with an initial speed of 80 ft.
per sec. If the y axis is positive upward measured from the ground
and if the ball is thrown at time t = 0, show that y = 80 gt 2 /2 ft.
(g = 32 ft. per sec. per sec., approximately).
a. Find the average value of the
speed with respect to the time from
t sec. to the time at which the
ball is at its highest point.
6. Find the average distance
with respect to the time for the
same time interval as before.
426. Figure 121 shows a canti-
lever beam of length L ft. loaded p IG 12 i.
with a concentrated load of P Ib.
at the free end. The equation of the "curve of the beam" with respect
to the indicated axes is
Ely = (f ) (3Lz* -
where E and 7 are constants.
130
ENGINEERING PROBLEMS
Figure 122 shows a similar cantilever beam of the same length and
cross section but loaded with a uniform load of w Ib. per ft. The
equation of the curve of the beam in this case is
FIG. 122.
Ely = ~
where E and 7 are the same constants as before.
Suppose that the total load is the
same in both cases so that P = wLlb.
Show that the average ordinate to
the first curve for the span L ft. is
equal to the largest ordinate (the
ordinate at x = L) to the second
curve. Hence show that for any
cantilever beam the mean deflection
produced by a vertical load applied
at the free end is equal to the deflec-
tion at the free end caused by the same load distributed uniformly over
the length of the beam.
427. The velocity of flow of water out of a small circular hole on the
side and at the bottom of a barrel is given by
v c \/~2gh,
where c is an empirical constant and h is the distance from the center
of the hole to the top of the water.
Determine the "average head" h &v with respect to time and the
11 average velocity" v &v with respect to time as the head decreases from
hi to A 2 . Assume that the cross-sectional area of the hole is A sq. ft.
(A is small) and that the water is in a circular barrel of radius r ft.
Remark: The two terms h &v and v &
fluid mechanics.
occur in chemical engineering and in
428. Use the data of the preceding problem to determine a formula
for the time required to empty the circular tank if the water was
originally at a height of h ft.
429. Taken from thermodynamics: If c = a + $T + yT 2 = specific
heat (a, jft, and y are constants), determine c me an as temperature T
increases from T\ to Tz.
430. In the equation for the flux density at any point in a torus:
B x = 2NIu/x (u, N, and 7 are constants), why is it not accurate to find
an average flux density by finding the flux density at x = r\ and at
x = r 2 and taking the mean?
INTEGRAL CALCULUS 131
What type of relation would have to hold between B x and x for this
procedure to be correct? Obtain a formula for the average flux density
between x = r\ and x = r 2 .
431. If the particles in a ceramic clay are assumed spherical with
diameters ranging from d to D, determine the mean volume and the
diameter d &v that corresponds to this mean volume.
Use the results of this problem to determine rfa v for a ceramic clay
in which the diameters vary from 0.015 to 0.058 mm.
Remark: Notice that d &v is the value of the diameter that corresponds to
the mean volume. The mean diameter would be a quite different value.
432. If the pressure and volume of a gas are related by the equation
pv n = k, where k and n are constants, determine expressions for the
average value of pressure with respect to volume if n I and if n ^ 1.
(n is a positive quantity.) Assume that the volume increases from vi
to v 2 .
What is the mean value of the pressure if pv 1 - 57 550 and v increases
from v = 4 to v = 22 cu. in. (p is in pounds per square inch)?
LENGTH OF ARC. AREAS OF SURFACES OF REVOLUTION
433. In engineering it is common practice to approximate the length
/*#2 / ~ -
of arc integral: s = I V 1 + (dy/dx)* dx by the integral
Jx\
= f
Jx\
[I + ( l A}(dy/dxY] dx
for curves whose slopes are small for the interval in question.
a. Expand the integrand in the first integral by the binomial theorem
to three terms and write the resulting approximation for s. What
must be true of (dy/dx) z for xi ^ x ^ x% to ensure that this series will
converge ?
b. Determine the percentage of error made by the use of the second
integral for the length of the curve y = # 2 /100 for x from to 2 and for
x from to 50.
434. A piece of steel of length 10?r in. is 5 in. wide and 0.1 in. thick. It
is initially fastened in a horizontal position by clamping its ends and
there is no initial stress in the steel. It is then deformed into a half
sine wave with amplitude 1 in. (the ends remaining fixed). (A change
in temperature could be the cause of the deformation.)
a. Determine the increase in the length of the steel strip by aid of the
approximate length of arc formula
132 ENGINEERING PROBLEMS
b. Determine the force that would be required to increase the length
in a horizontal direction by the amount determined in (a). Use Hooke's
law: s/e = E = 30,000,000 Ib. per sq. in., where s is pull per square inch
of cross-sectional area, E is Hooke's law constant or the modulus of
elasticity, and e is the stretch per inch of original length of the steel strip.
435. A steel spring is 1 in. wide, KG i n - thick, and lOx in. long. It is
fastened in a horizontal position and is then deformed into (a) a half
sine wave with amplitude 1 in., (b) an arc of a parabola with ordinato
at its vertex of 1 in., (c) an isosceles triangle with altitude 1 in. Deter-
mine the approximate increase in length for each case.
Remark: These are all approximations
to the correct shape that the steel spring
would assume. The shape that the
-p 12 Q spring would actually take would de-
pend on the manner in which the load-
ing was applied. Figure 123 shows the more probable shape of the deformed
spring.
436. A straight steel rail 100 ft. long increases in length by 0.2 ft.
because of an increase in temperature. If the ends of the rail are fixed
so that this increase in length must be taken care of by the bending of
the rail, determine the maximum deflection of the rail from its straight
position if the curved part takes the form of (a) an arc of a circle, (6)
an arc of a parabola, (c) a half sine wave, (d) an isosceles triangle, (e)
the curve y = ax*(x 100) 2 from x = to x = 100.
Remark: The results of this problem give a good reason for leaving spaces
between rails or sections of payment.
437. A corrugated piece of steel is 4 ft. wide and 10 ft. long. A
section through the corrugation has the form of a sine wave with ampli-
tude 0.2 in. and period 2 in. The steel is 0.05 in. thick. Determine
the weight of the piece of steel if this steel weighs 477 Ib. per cu. ft.
What would be the weight of a flat piece of steel with the same length,
width, and thickness?
438. An arched truss AB is 60 ft. long, weighs 20,000 Ib., and is
hinged at A. At B it rests on rollers as indicated in Fig. 124. The two
curved parts ACB and ADB are parabolic with vertex at C and D,
respectively. Determine the lengths of the curved parts of the truss,
each correct to the nearest three significant figures. Why would it be
wrong to use the approximate length of arc formula given in Prob. 434a?
*439. Determine the elongation due to its own weight of a tapered
vertical wire of length L ft., which is suspended from its top. At a
distance y ft. from the top the change in length per unit length is
k(L y) where & is a constant for the wire. It is assumed that the
INTEGRAL CALCULUS
133
change in length, although important in itself, can be neglected in
comparison with L.
D
FIG. 124.
440. If a flexible cable is suspended from two points as shown in
Fig. 125 and carries a load that is distributed uniformly in a horizontal
direction, the curve assumed by the
cable is a parabola. If the span is
a and the sag /, show that the length
of the cable is given by
Also determine the length of a para-
bolic cable with span of 100 ft. and sag of 4 ft. Integrate by exact
methods.
441. Work the numerical part of the preceding problem by expanding
the integrand by the binomial theorem to three terms and integrating.
What percentage of error is made if three terms are used; if just the first
two terms are used?
442. The shape of an automobile headlight reflector is determined by a
parabola revolved about its axis. If the reflector is 6 in. wide at the
front end and is 6 in. deep, determine the area of the reflecting surface
correct to the nearest third significant figure.
VOLUMES
443. A hemispherical bowl of diameter 6 ft. and full of water is rotated
about its vertical axis at 30 r.p.m. The surface of the water assumes
the shape of a paraboloid of revolution, and the equation of a cross
section through the center and referred to axes through the vertex
134
ENGINEERING PROBLEMS
of the parabola is y = W V/20, where co is the angular speed in radians
per second and g is approximately 32 ft. per sec. per sec. Determine
the amount of water that overflows.
444. A right circular tank of diameter 4 ft. and height 5 ft. revolves
about its geometrical axis at 10 r.p.m. If the tank was originally full
of water, how much will spill out by the time the speed reaches the
stated amount? Use the equation from Prob. 443.
446. A concrete culvert is shown
in Fig. 126. If the curve ABC is
an arc of a parabola, find the num-
ber of cubic yards of concrete
necessary for the culvert. Ignore
shrinkage.
*446. In chemical engineering use
is made of "English saddles" in
absorption towers. Several of these
saddles were available, and it was
desired to determine their total sur-
face areas. Each of these saddles
seemed to have a common thickness (approximately) so that the total
surface area could be approximated by doubling the "top" surface
area.
In order to determine the surface area it was found that, at least to
a good first approximation, these saddles were of the form
2az x 2 7/ 2 .
The first step in the solution of the problem was to determine the
surface area on the surface 2az z 2 y 2 , which is bounded by the
four planes x 6, x 6, y = 6, and y 6.
a. Show that the total (doubled) surface area is given by
-12 >\2'
FIG. 126.
b. Sketch the volume represented by the preceding double integral,
and then show that, in cylindrical coordinates, this volume is
sec 9
[(1 +
sec*
1] M,
where c = b/a.
c. Assume that c = 2 and compute the approximate value of the
preceding definite integral by the use of Simpson's rule. Use
INTEGRAL CALCULUS
135
A = 7.5 = 7T/24 radians. Then tabulate the values of S for 26 = 0.25,
0.50, 1.00, and 1.50 in.
FIRST AND SECOND MOMENTS
447. A triangular corner whose area is 25 sq. in. is cut from a square
10 in. on a side. What are the dimensions of the triangle if the centroid
of the remaining area is 4 in. from one
side of the square?
448. a. Determine, in terms of R and
r, the coordinates of the centroid of
the area in the first quadrant as shown
in Fig. 127.
6. Let r approach zero and obtain the
coordinates of the centroid of a quarter
circular area.
c. Let r approach R. What is the FIG. 127.
significance of your result?
449. Determine the coordinates of the centroid of the two areas in
Figs. 128 and 129.
FIG. 128.
FIG. 129.
*450. A simply supported beam has a length of L ft. and supports
a load of w Ib. per ft. as shown in the upper part of Fig. 130. The
" equation of the curve of the beam" is
Ely -
wx*
"24
wL z x
where E and 7 are constants. From this equation one obtains
wLx wx 2
Let the left-hand member of this last equation be M and obtain
M _ ( J-\ ( w ^ x wx *\
EI ~ \El) \ 2 ~ T/
The graph of M/EI as a function of x is shown in the lower part of
Fig. 130.
136
ENGINEERING PROBLEMS
a. Show that the difference in slopes at any two points on the curve
of the beam is equal numerically to the area under the curve in the
M/EI diagram between the two points.
b. Show that the distance of any point A on the curve of the beam,
measured normal or perpendicular to the original position of the beam,
from a tangent drawn to the curve of the beam at any other point B,
FIG. 130.
is equal numerically to the moment of the area under the M/EI curve
between the two points and with respect to an ordinate through A.
Remark: The two statements in this problem form the basis in strength of
materials for the moment-area or slope-deflection method of solving beams.
So long as the loading on a beam is a combination of uniform and concen-
trated loading, the M/EI diagram will be made up of inclined straight lines
and arcs of parabolas. The student can memorize the areas and centroids
of such areas and so can use the two principles of this problem to determine
the maximum deflection.
The basic method for solution of beam problems is the double-integration
method.
c. If the point A is at x = and the point B is at x = L/2 (the middle
of the beam), determine the slope of the tangent to the curve of the
beam at the origin and the deflection of the beam at the middle of the
beam by aid of the statements in (a) and (b) and verify from the given
equation for the curve of the beam.
INTEGRAL CALCULUS
137
y
h
FIG. 131.
461. a. What are the coordinates of the centroid of the area shown
in Fig. 131? What are the radii of gyration of this area with respect to
the x and y axes?
b. What would be a reasonable
definition for the " third mo-
ment " of this same area with
respect to the x axis? What
would be the analogue of "radius
of gyration" for third moment?
Answer the same questions for
the "nth moment/' In parti-
cular show that the quantity
which, when raised to the nth
power and multiplied by the area
of the rectangle to give the "nth moment" with respect to the x axis,
is h/\/n + 1.
452. The "product of inertia" of an elementary area dA with respect
to the x and y axes is defined by
dP xy dA xy dx dy.
Determine the "product of inertia" of the area in Prob. 451 with respect
to the x and y axes.
*453. A particle moves along a straight line so that its acceleration
is given by d*x/dt* = /(O, where f(t) is a continuous function of t.
When t = 0, x = XQ and dx/dt = VQ. Show that the following equation
gives the value of x at any time T:
X = XQ -\- VQ ' 1 -
Also show that the following equation gives the velocity at time T:
v = t>o+ J7 /(*)*
Now show that the integral in the second equation is the area between
the curve y = /()> the t axis, from t = to t = T. Then show that
the integral in the first equation gives the first moment of this area with
respect to the line t T.
464. Use the two formulas in Prob. 453 to determine at time t T
the displacement of a particle if the acceleration is given as follows:
(a) -jjp = +4 ft. per sec. per sec.;
(6) -jTj- = 4 ft. per sec. per sec.;
when t = 0, x = 0, and v 0.
when t = 0, x = 0, and v = 0.
138 ENGINEERING PROBLEMS
(c) -p- = 1 + T; ft. per sec. per sec.; when t = 0, x = 0, and v = 0.
(d) -J~ 20t P ft. per sec. per sec.; when t = 0, x = 0, and v = 0.
456. The kinetic energy of a rigid body rotating about an axis
is given by K.E. = /a> 2 /2, where co is the angular velocity in radi-
ans per second and J is the moment of inertia of the mass of the
body with respect to the axis of rotation.
a. A homogeneous solid disk of radius R ft. and thickness L ft. weighs
k Ib. per cu. ft. and rotates about its geometrical axis with a constant
angular speed of co radians per second. Determine the kinetic energy.
6. A flanged wheel rotates with a constant angular speed of co radians
per second about its geometrical axis. The weights of the spokes and
hub may be neglected and the total weight of the flange is the same as
the weight of the solid disk wheel in (a). The width of the flange (in a
direction perpendicular to the geometrical axis) is quite small. Deter-
mine the radius of the flange so that the kinetic energy will be the same
as was obtained in (a) and give the significance of your result.
456. Given that the kinetic energy of a rigid body rotating about an
axis is K.E. = Ja>'*/2, where co is the angular speed in radians per
second and / is the moment of inertia of the mass of the body with
respect to the axis of rotation.
An* elliptical disk with thickness h = 0.2 ft. and with an outline as given
by the equation 9# 2 + 25i/ 2 = 2.25 (x and y in feet) is homogeneous
and weighs 100 Ib. per cu. ft.
a. Show that the mass of the disk is (iro6A)(100)/32.2 = 0.293 slug.
6. Determine the kinetic energy of the
disk if it rotates about its geometrical
axis with a constant angular speed of 20
radians per second.
c. What would be the result in (a) if
the disk were circular with radius 0.4ft.?
d. What would be the result in (b) if
the elliptical disk revolved about an axis
through one focus?
457. If a right circular disk revolves
about its geometrical axis, then T = Jo.,
where J is the moment of inertia of the
mass of the disk with respect to the axis
of rotation, a is the angular acceleration,
and T is the moment of the external force.
Figure 132 shows a homogeneous disk with radius 0.6 ft., thickness
0.2 ft., and weight 500 Ib. per cu. ft. A weight of 50 Ib. is hanging
FIG. 132.
INTEGRAL CALCULUS
139
from a cord that passes around the disk as shown. At a particular
instant the weight is allowed to fall. Determine the angular acceler-
ation of the disk at the instant the weight starts falling. Neglect any
axle effect and friction.
Solution: Substitute in T = Jo. and obtain
\ in f*n\ f ^
(0.6) (50) = (a)
^(32:2)
'
I
k
JL
T
FIG. 133.
stress with respect to the x axis will be
y
458. Figure 134 shows a vertical section of a beam together with a
side view (Fig. 133) of the beam from
which this section was taken. Sup-
pose that the unit stress (push) on
the top fibers of the beam is 8 and
on the fibers in the element dA is
s y . The graph shows the manner
in which the unit stress s v varies
and from this graph it is seen that
s v = sy/c.
The total stress on the element
dA is s v dA and the moment of this
\
I
k
FIG. 134.
dM =
(y*dA).
Hence
The integral is the second moment of the area of the cross section
with respect to the x axis and hence M = sl x /c.
Remark: This is a rough outline of a derivation to be found in texts on
strength of materials. It illustrates how integrals for second moments or
moments of inertia arise. Because of the frequency with which integrals
for first and second moments arise in engineering derivations, it is convenient
to use special names for them and to learn quick methods for their evaluation.
140
ENGINEERING PROBLEMS
*469. The following theorems are established in engineering courses
and suggest how first and second moments will arise. Prove these
theorems :
a. The total fluid force (sometimes miscalled the total fluid pressure)
on one side of a vertical submerged area is equal to the weight of a cubic
unit of the fluid multiplied by the area multiplied by the depth from
the fluid level to the centroid of the area.
b. The total work done in pumping fluid from a horizontal pipe into
a tank of any shape is equal to the weight of a cubic unit of the fluid
multiplied by the volume of pumped fluid multiplied by the distance
from the horizontal pipe to the centroid of the fluid in its final position.
460. Restate the two theorems of Pappus in moment language
instead of in centroid language. These theorems are probably stated
in your calculus text.
461. Figure 135 shows an airfoil approximation by aid of elliptical
and parabolic arcs. If the parabolic arcs have their vertices on the
FIG. 135.
y axis, determine the moment of inertia of the enclosed area with respect
to the x axis.
TOTAL FORCE
462. Determine the total gas force on the inside wall of a spherical
shell if the pressure is 20 Ib. per sq. in. and
the inner radius of the shell is 2 ft.
463. A sphere of diameter 5 ft. is im-
mersed in water so that the water level is
2 ft. above the top of the sphere. Deter-
mine the total external force on the sphere.
Start with the force on a zone of the sphere.
464. The vertical cross section of one
side of a vessel filled with water is shown
in Fig. 136. The length perpendicular to
the plane of the figure is 4 ft.
a. Determine the total horizontal force
Show that this horizontal component is
FIG. 136.
on the side of the vessel shown.
INTEGRAL CALCULUS
141
the same as the total horizontal force on the side of a rectangular area
of height 5 ft., width 4 ft., and top side at the water level.
b. Determine the total vertical component of the force.
c. Determine the total force in magnitude and direction.
WORK
465. A weight of 100 Ib. is attached to a rope which weighs 2 Ib. per ft.
If 60 ft. of rope hang from the top of a building (with the weight at the
end of the rope) and if the weight is raised a distance of 40 ft., determine
the work done.
466. A derrick lifts a shovel of sand through a vertical distance of
30 ft. The sand in the shovel weighs originally 400 Ib. and leaks out
at a rate directly proportional to the square root of the distance tra-
versed. If 320 Ib. of sand reaches the top, find the work done.
467. A bead of weight w Ib. slides without friction on the arc of a
circle, which lies in a vertical plane.
The radius of the arc is R ft. Start-
ing with the expression for the work
done by gravity as the bead moves
from A to B (see Fig. 137),
ds
XB
w cos 6 ds,
L
B
where ds is an element of length of
the circular wire, and is the angle
which the tangent to this arc makes
with the vertical, show that W is
equal to the product of the weight w
and the vertical distance between the FIG. 137.
points A and B.
What would be your result if the points A and B were joined by a
straight line? By an arc of a parabola with vertex at A?
468. During a certain process in an engine, the pressure p Ib. per sq. ft.
changes with the volume (V cu. ft.) according to the law,
p =
yy 2
Determine the work done by evaluating W = I p dV if the volume
J Vi
changes from V\ = 1 cu. ft. to 7 2 = 3 cu. ft.
469. If the pressure (p Ib. per sq. ft.) changes with the volume (V
cu. ft.) during a certain process in an engine according to the law
pV n = C, where C and n are constants, show that the work:
W
l-n
142 ENGINEERING PROBLEMS
is given by the two following equations :
(a) n 5* 1:
W = 22
(b) n = 1:
Tf = pil
Remark: The proofs of these two results are to be found in texts on thermo-
dynamics and the resulting equations are fundamental in that study.
14,000
3 4
V Cu. ft.
FIG. 138.
Data:
A., V = 1 cu. ft., p = 14,400 Ib. per cu. ft., Ci = 14,400.
B. V - 2 cu. ft., p = 7,200 Ib. per cu. ft., Cz = 19,000.
C. V == 6 cu. ft., p = 1,545 Ib. per cu. ft., C 3 = 9,270.
D. V = 3 cu. ft., p = 3,090 Ib. per cu. ft., CU = 14,400.
INTEGRAL CALCULUS 143
470. Figure 138 shows the relation between pressure and volume in a
cylinder, as the piston completes a full cycle. Determine the net
work done by finding the shaded area.
OTHER APPLICATIONS OF INTEGRAL CALCULUS
Remarks: The engineering student may be quite surprised at the few
illustrative problems from engineering showing the uses of areas, volumes,
surfaces of revolution, polar coordinate areas, etc. Aside from mathematical
development, there are two important engineering reasons for studying
these topics in calculus:
1. The use of these topics appears in centroids, moments of inertia, fluid
force, work, etc.
2. The student is being taught the art of setting up definite integrals.
The basic principle involved in each case is obtained from plane or solid
geometry (area of rectangle or triangle, volume of disk or washer or thin
shell, lateral surface area of a frustrum of a right circular cone, etc.). In
junior or senior engineering courses the student will need to set up definite
integrals for specific problems in which the fundamental principle will be
more likely to be taken from physics or chemistry. These uses will be
illustrated in the problems of this section.
It is of extreme importance that the engineering student learn to set up
definite integrals directly from the figure by aid of some fundamental
principle and not by memorizing definite-integral formulas.
471. Every text on fluid mechanics derives formulas for the amount of
water that flows over various typos of spillways and through various
types of orifices. The basic principle used for that purpose is that
dq = c \/2gz dA,
where q is the quantity of water in cubic feet per second, c is an empirical
constant, and z is measured in feet from the water level to the elementary
section dA . Verify the equations for q in the following orifice problems :
a. Rectangular orifice (Fig. 139):
= (I)
p JJ '
Water Level
t
S////////////////////////////////S
H
dA
1
FIG. 139. FIG. 140.
144 ENGINEERING PROBLEMS
b. Triangular (isosceles) orifice (Fig. 140) :
c. Rectangular orifice (Fig. 141):
q = (^) V2 (hj* -
= (cbd) V^ (1 - JJ
)
where h =
hi
: and d = hz hi.
Water Level
Water Level
i_o
FIG. 142.
FIG. 141.
d. Circular orifice (Fig. 142):
q = (cA) V%h (l -
where A = area of circle.
472. In mechanics one learns that the differential expression for the
"frictional moment" dMf is dM f = ppdN, where dN is the normal
pressure on the element dA (shown as a ring with radius p) and JJL is
the constant coefficient of friction.
A collar bearing is shown in Figs. 143 and 144 with radii r\ and r?.
The load W Ib. is to be distributed over the area between the two circles
of radii r 2 and ri and hence the pressure per unit area is
W
7r(r 2 2 -
7rr 2 - n
Then the normal pressure on the ring element is dN = p dA. The
INTEGRAL CALCULUS
Motional moment will be given by
dM f =
145
Integrate this from p = r\ to p = r 2 . Also determine the limiting
values obtained by letting TI approach zero and by letting 7*1 approach
the value TZ.
FIG. 143.
FIG. 144.
473. In thermodynamics the differential change in " entropy" is
given by dS = (cM/T) dT, where M is the weight of the substance,
c is its specific heat, and T is absolute temperature.
Determine the change in "entropy" f or M = 1 (mol) of CO 2 in being
heated at constant pressure from 40
to 340F. (T l = 40 + 460 = 500
and T 2 = 800) if the specific heat
for CO 2 at constant pressure is given
by the empirical formula:
c = 7.15 + 0.0039T -
0.000,000,60T' 2 .
Also sketch or plot a graph show-
ing an area that is equivalent in
magnitude to the required result.
474. The temperature on the out-
side of a pipe (inner radius r and
outer radius R) is To and on the inside
is Ti (see Fig. 145). The application of Fourier's heat law leads to
FIG. 145.
where fc is a constant (the thermal conductivity), T is the temperature
146
ENGINEERING PROBLEMS
on the elementary area dA, and Q is the amount of heat transferred per
unit of time and is a constant.
C R dp C T
Write this equation in the form Q I = 2?r& J T . dT and derive
a formula for Q in terms of fc, R, r, Ti, and To.
475. When drying conditions
in a slab are such that the water
diffuses to the surface before
evaporating and when certain
T other conditions are assumed to
c be fulfilled, the moisture distribu-
tion curve in a cross section will
be a parabola as shown in Fig.
146.
_>. x a. If c is the concentration at a
distance x from one face, c m the
FlG 146 concentration at the middle, and
c s is the concentration at either
face, and assuming that the thickness of the cross section is 2L, show that
2L
c
- C s
(x -
<&. Evaluate dc/dx at x and c = c s .
c. Since the "average concentration"
show that c av = c s + (%)(c m c,),
(1) By integration.
(2) Directly from the figure by
aid of Simpson's rule.
476. The current i L amp. for an
inductance coil of "self indue-
e H
FIG. 147.
e volts
r\
\J
-*t
One period -
FIG. 148.
tance" L henrys and negligible resistance is given in terms of the
impressed voltage e volts by the equation e = L(di L /dt). If the wave
form for the voltage e is that shown in Fig. 148, sketch the wave form
for u.
INTEGRAL CALCULUS
147
Start the graph for IL at any convenient height (assume L = 1 for
the purpose of the sketch) and, after the graph has been completed,
draw a horizontal axis midway through it, so that the resulting graph
will represent a purely alternating current, i.e., a current whose average
value over a complete period is zero.
477. The magnetic field strength H on the axis of a circular con-
ductor carrying a current / (Fig. 149) is
zj - ( 2 + X *Y/*
where a is the radius of the ring that
carries the current and x is the distance of
the point on the axis from the plane of the
ring.
If the mechanical work done in mov-
ing a unit pole a distance dx along the axis
of this conductor is dW = II dx, determine
the work done in moving a unit pole from
#]= oo to x = + .
478. Water is flowing along a horizontal circular pipe of radius a,
and the speed of the water at a distance r from the center line of the
pipe is v /(r), the units being feet and seconds. Set up an integral
for the number of cubic feet of water discharged per second from the
pipe.
Note: The number of cubic feet discharged per second by a stream flowing
at any given speed is equal to the product of this speed and the cross-sectional
area of the stream flowing at this speed.
479. a. Use the formula given in Prob. 254 and your result in Prob.
478 to determine the quantity of water discharged per second from
that pipe.
b. What would the result in (a) become if the water had uniform
speed?
480. From a theorem in elasticity it is known that the deflection
expression in Prob. 354 gives not only the deflection at a distance r
from the central point load on a circular clamped plate but also the
deflection at the center for a load P at a distance r from the center of
the plate.
Suppose that, instead of a concentrated load P at a distance r from
the center, the load is uniformly distributed over a ring area element
of width dr, inside radius r, and center at the origin. Then the deflec-
tion at the center is given by
148
ENGINEERING PROBLEMS
2wbr dr
where 6 is the load per unit area on the ring element.
Determine the deflection at the center due to a uniform load of b
per unit of area on a circle with center at the center of the plate and
radius c. What is the deflection if c a!
481. In Prob. 480 suppose that the load P is uniformly distributed
over a small circle of radius c whose center is at a distance c from the
center of the plate. Evaluate the following integral, which gives the
deflection at the center of the plate ;
w
P
4w 2 c*N Jo
2 /*2c cos e
Jo
*482. Figure 150 shows a top and side view of a 1-ft, section of a
gravity dam. The dam will be supposed to be homogeneous arid to
Headwater
Top view
FIG. 150.
weigh w Ib. per cu. ft. Show that
Wi = total weight of dam
/*2/2 /*i/
= 11
JO Jj/a
n
tan <p y tan <
, ,
w dz ay.
The moment of this weight with respect to a line through the point A
(perpendicular to the plane of this section) is
where Fa and FI are the limits previously used.
The water force acting horizontally in a downstream direction on the
upstream (right-hand) face is
INTEGRAL CALCULUS 149
where w is the weight of 1 cu. ft. of water. The moment of this hori-
zontal force with respect to a line through A perpendicular to the side
section is
w(y - 2/1X2/2 ~ y) dy.
The water force acting vertically on the upstream face is
W% = I yz w(y 2/1) tan 6 dy
/o
and the moment of this force with respect to the same perpendicular line
through A is
A/Y = | w(y 2/1) (tan Q)(y* tan <p + t + y tan 0) dy.
/o
The student should study the figure and the relation to these definite
integrals.
Show that these integrals evaluate to the following results:
et,
INFINITE SERIES
483. Problem 440 concerns a parabolic cable whose length was found
to be
a. Expand the integrand by the binomial theorem and integrate.
b. Observe that the resulting series for L is an alternating series.
What is an error term if the first two terms are used to compute L?
c. Show that the series converges if |//a| < >^.
Remark: Although this question of convergence can be answered by testing
for the interval of convergence of the series in (a), a simpler method is to
notice that the series for the integrand will converge if 64/ 2 # 2 /a 4 < 1. The
largest value that x z can have is a 2 /4 and hence the integrand series will
converge for all values of x in the interval from to a/2 if
16.P ^,
150 ENGINEERING PROBLEMS
The theorem about the interval of convergence on integration of a power
series is needed to complete the solution.
484. Expand the integrand of the length of arc formula:
by the binomial theorem. Show that the resulting series is an alter-
nating series. Assuming that \dy/dx\ < 1 for the required values of x,
determine an error formula for the error made by using only the first
two terms of the series. Does the requirement that \dy/dx\ < 1 for
XL ^ x Xz ensure that the series will converge?
485. The two following definite integrals arise in fluid mechanics
derivations. Expand the integrands into power series and show that
the given series are correct.
(a) Q = 2 V%g(E - *) Vr* - z 2 dz
(b) Q = B r /2 . V2g(E - z) dz
Jzo/Z
- B, VfcB (l - ggi ---- >
486. In a series circuit containing resistance R ohms, inductance
L henrys, and a d.c. voltage E volts, the current i amp. is given in
terms of the time t sec. by
Show that
. /Et\ /_ Rt
1 = (T) I 1 ~ 2i
J_
FIG. 151.
r
o
For what values of Rt/L does this series converge? What is an error
formula if the first two terms are used? If the first three terms are
used?
Let x = Rt/L and y = Ri/E. Plot a graph of y as a function of x:
(a) using the exact equation and (6) using the first three terms of the
series.
487. The time t sec. required for a certain pendulum of length L ft.
to swing from its lowest point through an angle is given by
dz
s in z
INTEGRAL CALCULUS 151
where g = 32.2 ft. per sec. per sec, A is the greatest angle reached in
the swing, and sin <f> = Sm If L = 2 ft., A = 60, find t for
sin
e = 30.
488. The chance, in throwing 100 coins, of throwing exactly 50 heads
100!
and 50 tails is given by V 50 ?\ 2(9100^' ^e v&lne of this quantity is very
closely given by
2 fo.i
~ V27T JO
Evaluate P, correct to three significant figures.
Remark: Books on mathematics of statistics will explain how this definite
integral was obtained.
489. Show that the area bounded by the curve y = x n and the x axis,
from x = Xi to x = Xz (both x\ and x% are positive), is
/j. n+1 X\ n+l
If n is close to 1 in value, the computation of the area by the first
equation is likely to be inaccurate if ordinary tables are used.
Obtain a series for calculating the value of A when n is close to 1.
Solution: A = (*!-) 1 ""*" *
since (x 2 /Xi) n+1 = e<" +1 > ln <* 2 /* 1 > and e* = l+z + z 2 /2l + .
Use this series result and also the original formula to calculate the
value of A if Xi = 2, # 2 = 5, and n = 0.99.
490. The rate p r at which a black body radiates heat is found by
integrating
r Ci dx
Pr = Jo x*(e**<"r* - 1)'
where Cz and Ci are positive constants and T is absolute temperature
and is constant for this problem. Make the substitution
z ~ ~ Tx
and obtain
(CiT 4 \ /*- zV dz
17) Jo n^'
152
ENGINEERING PROBLEMS
Expand e'/(l e') = e* + e 2 * + e 3 * + , substitute in the inte-
grand, and obtain
p r =
Evaluate the integral by integrating term by term and obtain
Remark: There are a number of steps in this mathematical procedure that
would require justification.
491. In the derivations for curved beams in strength of materials
it is necessary to evaluate the two integrals :
- y
M - 2 r /2 '
J-A/2
-y
Evaluate both integrals by expanding the integrand into series. Also
evaluate the first integral by an exact method.
492. If KPy = x/(l - a;) (Dalton's law), show that
KPy = a? + z 2 + z 3 + - - - .
If x is small, show that KPy = x, approximately (Raoult's law in gas
theory) .
493. In railway surveying it is customary to use a formula for the
difference in length between the arc ABC
and the chord ADC (see Fig. 152). Show
that this difference is given exactly by
q = rO 2r sin -
and that this can be reduced to r0 3 /24,
which is too large by a number less than
r0 5 /l,920.
494. A belt connects, without crossing
_ itself, two pulleys of diameters D and d.
FlG. 152. mu r j. u I n
The distance between centers is C .
a. Show that the length of the belt (assumed taut) is
= (
2C cos
where 6 is the angle between the straight section of the belt and the
line joining the centers of the two pulleys.
INTEGRAL CALCULUS 153
b. Expand the preceding formula for L and neglect terms in 6 of
degree higher than two. What is a formula for the maximum possible
error.
495. Gay-Lussac's law in the theory of gases states that the increase
in the volume of a gas at any temperature jPC. foi a rise of 1 tempera-
ture is a constant fraction of its volume at 0C.; i.e.,
V = 7 (1 + <*T).
Dalton's law states that the increase in the volume of a gas at any
temperature for a rise of 1C. is a constant fraction of its volume at that
temperature; i.e., V = V^e aT .
Show that if second and higher powers of T are negligible, for ordinary
gas calculations, then these two gas laws are equivalent.
496. The electrical engineering derivation of the theory of the elec-
trical breakdown of gases requires the evaluation of the following definite
integral :
where a and b are both positive constants. Obtain a formula for E by
expanding the integrand into a series and integrating term by term.
497. Two ships have masts whose tops are each 100 ft. above the
water. How far is one masthead visible from the other?
498. If a straight tunnel were to be bored from Chicago to Detroit
(about 300 miles apart) how much distance would be saved? What
would be the greatest depth?
499. The process of running a level introduces an error due to the
curvature of the earth. What is the correction per mile?
500. A certain road has a 5 per cent grade. What is the actual length
of the road along the incline correct to the nearest 0.01 ft. in 100 ft.
of horizontal distance? Devise and use an approximate formula to
answer this question and prove your accuracy.
501. Assume that the earth is a perfect sphere with radius of 4,000
miles. A band is passed around the earth at the equator so that the
band is everywhere 1 yd. away from the earth.
a. What is the difference in lengths of the band and the circumference
at the equator?
b. How tall a pole would be required to pull the band taut?
c. What angle (in b) at the center of the earth does the straight part
of the band subtend?
HYPERBOLIC FUNCTIONS
502. (Taken from the magazine Industrial and Engineering Chemis-
try.) Given that dy/dx = 2k(x* - 4& 2 )~^, show that
154 ENGINEERING PROBLEMS
x = 2k cosh ((7+ ~),
where C is the constant of integration.
603. A charged condenser (C farads) is connected at time t to an
inductance L henrys in series with a large resistance R ohms. The
current that flows is given by
where Q is the initial charge on the condenser, a = R/2L, and
k = V(ft/2L) 2 - (1/LC) where A; is a real quantity, (e = 2.718,
approximately.)
a. Determine the voltage across the inductance CL if e = L(di/dt).
b. Determine the total charge passing through the inductance
Q.L if
c. Find the voltage across the condenser at any time t if
d. Sketch graphs of CL, ec, and the voltage across the resistance
(en = Ri), each as a function of time and show that the last is equal
to the sum of the first two.
e. What does the expression for i become if the quantity under the
radical for k is negative, i.e., if k jp (where j = \/ 1)?
504. In studying the rate of formation of carbon monoxide in gas
producers, Clement and Haskins (1909) obtained the equation
= _
dt b
with the initial condition that the amount of CO at time t = is
x = 0. Show that x = b tanh at.
505. The vector voltage E 8 at the sending end of a transmission line
is related to the voltage E r and the current I r at the receiving end by
the equation
E, = E r cosh + I r Z<> sinh Q,
where B depends on the length of the line and Z is the " characteristic
line impedance."
Evaluate E, if E r = 127,000 + JO volts, I r = 200 + jO amp.,
Z = 398 - J35, and = 0.017 + j'0.200.
INTEGRAL CALCULUS
155
Remark: All these quantities are given in the form a + jb where j \/ 1-
The values of sinh and cosh 6 must be determined in this same form. For
this purpose the student may find it necessary as a supplementary problem
to verify the following identities:
sinh (a; + jy) = sinh x cos y + j cosh x sin t/,
cosh (x + jy) = cosh x cos y + j sinh x sin y.
506. The " propagation constant" F of an electric wave filter is given
by cosh r = (Zi + 2Z 2 )/(2Z 2 ).
Zi and Z 2 are the series and shunt impedances of one section of the
filter. The quantity F is complex and may be written F = +a + jtp.
a. If #1 = J80 and Z 2 = j'20, evaluate F (show that <p must be zero).
6. If Zi yiO and Zi = ^'20, show that a must be zero and evaluate F.
c. If Zi = J50 and Z 2 == jlfy show that <p must be mr and
evaluate F.
607. The two following equations give the voltage E s and the current
I 8 at the sending end of a particular cable to yield a voltage of 100 volts
and current of 2 amp. at a distance of L miles from the sending end.
E 8 = 100 cosh 0.02L + 2,000 sinh 0.02L volts,
7 3 = 2 cosh 0.02L + 0.1 sinh 0.02L amp.
a. Sketch graphs of E s and L as functions of L for L from to 200
miles.
6. Expand by Taylor's series in powers of L 200. Stop with the
term (L - 200) 2 .
c. Write E. = l,050e- 02i +/(!/), L = 1.05e- 02 ^ + g(L) and obtain
series expansions for/(L) and 0(L) through the term in (L 200) 2 .
FIG. 153.
508. A catenary cable, y = c cosh (x/c), has a length of 100 ft. and a
span of 80 ft. (see Fig. 153). Determine the sag/.
r 40 /
Solution: First show that 100 = 2 J \
= 2c smh I
sinh 2 - dx
c
156 ENGINEERING PROBLEMS
The curve goes through, x = 40, y = / + c and hence
/ + c c cosh ( 1
Solve the first equation for c, substitute this value in the second
equation, and compute /.
609. A catenary cable, such as shown in Fig. 153, has for its equation:
y = 4,500 cosh (z/4,500) 4,420, x and y in feet, y measured from
the ground.
a. What angle does the cable make with either tower if the supporting
towers are 800 ft. apart?
6. Determine the length of the cable between the towers, 800 ft. apart.
c. Determine the height of the supporting towers.
d. Determine the sag /.
e. Obtain a parabolic approximation for the cable equation that will
be approximately correct for the span (x = 400 to x = + 400 ft.).
/. Now try to answer question (6) by aid of your approximate formula
from (e). You may use either exact integration or Simpson's rule.
Remarks: When a cable or chain hangs from two supports of the same
height, the curve assumed by the cable is an arc of a catenary. For certain
types of computations it is convenient to approximate the catenary by a
parabolic curve. Problem 509 illustrates one type of computation in which
the hyperbolic form is much simpler.
510. A tangent line is drawn through the origin with positive slope
and tangent to the curve y cosh x. Determine the inclination of
this tangent line:
a. By a graphical method.
b. By some numerical approximation method.
511. One peculiarity of the catenary cable, y = c cosh (x/c), is that
the total stress (pull) at any point (x,y) is equal to T = wy, where
w is the weight of the cable per foot. Sketch a graph showing the total
stress as a function of x if the cable weighs 4 Ib. per ft. The equation
of the cable is
y = 400 cosh
(y is not measured from the ground), and the supports are 600 ft. apart.
Remark: A different manner of stating the equation T = wy would be to
say that if at any point (x,y) on the cable, the wire (assumed perfectly
flexible) were passed over a frictionless pin and the length of wire allowed to
hang downward were just sufficient to reach the x axis, then the cable
would not shift its position when released.
INDEX
(Numbers refer to problems)
To avoid the duplication of long lists of numbers, cross references have
been cited for many equivalent names and overlapping topics.
Acceleration (see Motion)
Acoustics, 161, 188, 357
Aerodynamics (see Aeronautics;
Fluid flow)
Aeronautics, airfoils, 165, 293, 294,
347, 352, 461
force analysis, 37, 84, 190
general, 43, 84, 360, 366, 410
motion, 89-92, 298
radius of action, 46-48, 92
Amplifiers (see Vacuum tubes)
Arc length, 433-441, 494
(See also Cables)
Area, 442, 446
Astronomy, 25
B
Ballistics, 171, 175, 188, 332, 335,
343, 378, 383, 405
Beams, 6, 32, 55, 56, 69, 129, 143,
146, 167, 168, 264, 265, 268, 269,
277, 280, 288, 289, 310, 311, 344,
345, 38G, 387, 389, 390, 404, 426,
450, 458, 491
Belts and ropes, 27, 206, 494
Bernoulli's equation, 10, 336
Bridges, 120, 173, 174, 182, 272, 438,
445
Cables, catenary, 209, 275, 508-511
parabolic, 12, 170, 173, 440, 441,
483<
Cams, 78, 215, 278
Carnot cycles, 157, 470
Center of gravity and moment of
inertia, 121, 314, 447-461
Ceramics, 42, 339, 431
Chemical engineering (see Chemical
reactions; Fluid flow; Thermo-
dynamics; and similar specific
topics)
miscellaneous, 233, 397, 400, 412,
446, 502
Chemical reactions, 14, 15, 33, 141,
148, 155, 177, 203, 286, 316,
317, 392-396, 475, 504
Civil engineering (see Bridges; High-
ways; Hydraulics; Railroads;
and similar specific topics)
Columns, 5, 31, 273, 274
Cost analysis, 295, 300, 307, 318
Cranks, 99, 103, 195, 223, 227, 319,
320
Curvature, 344-347
D
Dimensional analysis, 36, 37, 72, 73
Dynamics, energy, 270, 385, 407,
455, 456
friction and resistance, 27, 107,
109, 126, 152, 172, 206, 214,
309, 472
momentum, 399, 405
power, 267, 271, 384
stability, 7, 75
158
ENGINEERING PROBLEMS
Dynamics, work, 107, 459, 465-467
(See also Thermodynamics,
Work)
(See also Aeronautics; Ballis-
tics; Kinematics; Motion;
Vibrations)
E
Efficiency, 107, 299, 302
Elasticity (see Stress-strain analysis)
Electric circuits, a.c. circuits, gen-
eral, 11, 34, 77, 95, 97, 163, 183,
419, 423, 424
impedance, 17-19, 88, 506
rotating vectors, 102, 112-119
transmission lines, 210, 211, 307,
505, 507
wave form, 96, 98, 100-102, 192-
194, 197-199, 216, 291, 351,
420-422, 476
d.c. circuits, 131, 252, 301, 302
general theory, 266, 291, 380, 381,
401, 476
(See also Vacuum tubes)
Kirehhoff's laws, 34, 54, 65-68
transients, 26, 49, 200, 201, 284,
350, 403, 486, 503
Electrical engineering (see Electric
circuits; Electromagnetism;
Illumination ; Vacuum tubes ;
similar specific topics)
miscellaneous, 30, 76
Electromagnetism, electrostatics, 85,
205, 290, 304
general, 130, 164, 236, 240, 496
magnetism, 110, 204, 230, 237,
303, 305, 402, 417, 430, 477
Ellipses, 179-182, 224, 258, 282, 283,
362, 461
Energy, 270, 385, 402, 404, 407, 455,
456
Engineering method (see note after
123)
Entropy, 473
Fluid flow, nozzles, 1, 8, 39, 159, 308
orifices, 13, 38, 41, 160, 427, 428,
471
pipes, 28, 57, 73, 239, 254, 478, 479
weirs, 36, 40, 58, 59, 156, 235, 297,
471
Fluid force, 459, 462-464
Fluid mechanics (see Fluid flow;
Hydraulics)
Force analysis, 70, 80-84, 86, 87, 93,
107, 109, 190, 214, 241-244,
309, 312
Friction, 27, 107, 109, 206, 214, 309,
472
G
Gas flow (see Fluid flow)
Gas laws (see Thermodynamics)
Geometry, plane, 106, 279
solid, 7, 9, 53, 60, 61, 358, 497-501
(See also Maps; specific topics)
H
Harmonic motion (see Electric cir-
cuits; Vibrations)
Heat (see Thermodynamics)
Highways, 52, 416
Hydraulics, fluid force, 459, 462-464
general, 28, 41, 139, 246, 411, 482
(See also Fluid flow)
Hydrodynamics (see Fluid flow)
Hyperbolas, 183-188, 250
Illumination, 154, 300, 306, 415
(See also Optics)
Impedance, 17-19, 88, 506
Indicator cards, 157, 185, 413, 470
K
Kinematics, 78, 99, 195, 223, 227,
229, 319, 320, 323, 326, 329, 330
Kirchhoff's laws (see Electric cir-
cuits)
Fluid flow, general, 40, 44, 149, 164,
165, 186, 336, 352, 364, 365, 485
Light, 111, 306
(See also Illumination)
INDEX
159
Lissajous figures, 224, 225
Locus, equation of (see note after
123)
M
Machine design, 105, 189, 287, 299,
313
Magnetism (see Electromagnetism)
Maps, 259-261, 285, 406
Mechanical engineering (see Dynam-
ics; Fluid flow; Force analysis;
Stress-strain analysis; Thermo-
dynamics ; similar specific
topics)
miscellaneous, 132
Mechanics (see Beams; Columns;
Dynamics; Force analysis*
Plates; Stress-strain analysis)
Meteorology, 360
Mining, 247, 249, 250
Modulus of elasticity (see Stress-
strain analysis)
Moment of inertia (see Center of
gravity)
Momentum, 399, 405
Motion, 35, 45-48, 50, 72, 78, 89-92,
109, 124, 126, 166, 169, 171, 175,
176, 212, 219, 221, 222, 226, 281,
292, 298, 324-328, 331-335, 343,
361, 373-379, 382-384, 388, 391,
418, 425, 453, 454, 457, 487
(See also Kinematics; Vibrations)
' N
Nozzles, 1, 8, 39, 159, 308
O
Ohm's law, 131, 252
Optics, 111, 306
(See also Illumination)
Orifices, 13, 38, 41, 160, 427, 428, 471
Parabolas, 12, 166-178, 272, 278,
279, 440, 441, 461, 475
Pendulum, 94, 487
Physical chemistry, 24, 133, 315
Pipes, 57, 73, 239, 254, 478, 479
Pistons (see Cranks)
Plates, 354, 480, 481
Power, electrical, 3, 77, 95, 97, 101,
197, 201, 302, 402, 419, 423, 424
mechanical, 267, 271, 384
R
Radiation, 216, 351, 490
Radio (see Electric circuits; Vacuum
tubes)
Railroads, curves, 220, 346, 493
train resistance, 126, 152, 172
Reflectors, 442
Resistance, 130
electrical (see Electric circuits)
mechanical, 126, 152, 172
Reynolds number, 239
Ropes and belts, 27, 206, 494
Rotating vectors, 102, 112-119
S
Simple harmonic motion (see Elec-
tric circuits; Vibrations)
Slide rule, 44, 337
Snell's law, 111
Sound, 161, 188, 357
Specific gravity, 9, 53, 60
Specific heat, 135, 263, 340, 367, 409,
414, 429
Spillways (see Weirs)
Stability, 7, 75
Strength of materials (see Beams;
Columns; Plates; Stress-strain
analysis)
Stress-strain analysis, materials, 122,
147, 231, 232
miscellaneous, 125, 142, 162, 179,
187, 253, 256, 258, 313, 348,
353, 362, 363, 398, 407, 439
(See also Beams; Columns;
Plates)
Structural design, 181, 245, 248, 257
Temperature of explosion, 14, 15
Thermodynamics, entropy, 473
160
ENGINEERING PROBLEMS
Thermodynamics, fluid flow, 1, 8, 10, Vectors (see Force analysis; Rotating
39, 44, 159, 308 vectors)
gas laws, 2, 21, 22, 29, 44, 137, 138, Velocity (see Motion)
151, 157, 158, 185, 251, 338, Vibrations, general, 16, 20, 51, 62,
356, 359, 360, 36&-371, 432, 63> 71, 74j 75j 94j 2 07, 218, 224,
225, 321
simple harmonic motion, 35, 103,
104, 125, 144, 145, 191, 195,
196, 322, 349
492, 495
heat conduction, 136, 140, 150,
164, 474
radiation, 490
specific heat, 135, 263, 340, 367,
409, 414, 429
temperature scales, 123
work, 23, 134, 208, 355, 413, 468-
470
Tides, 104
Tunnels, 247, 249, 250
Vacuum tubes, 3, 64, 127, 153, 178,
184, 228, 234, 262, 276
(See also Electric circuits)
Volume, 61, 358, 443-445
W
Weirs, 36, 40, 58, 59, 156, 235, 297,
471
Work, mechanical, 107, 459, 465-
467, 477
thermodynamical, 23, 134, 208,
355, 413, 468-470
ANSWERS
2. vi = tf 2 (p2/?>i) 1/n . 4a.
6. w = 8.51 in. 6. 15.1 ft.
10. UA* UB Z 2g(p B VB
lla. m = -5 jlOO; &. m = - ~
12a. 12.19ft.; 6. 12.24ft.
4AW , 4At \/hl , I o/vi /r-^
13a. y = -^- + ^ ; &. I/ == 1-01 4- 2.01 V/u ft.
14. J = 3350C., !T = 3350 + 273 = 3623abs.
16. * = 2180C. 16. / = 1/(1 + u).
17a. 20.8 - y5.6; 18. Z = 4 + j2 ohms.
b. 96 +^66; 19. G = R/(H 2 + X 2 ), # = X/(JK 2 -f Z 2 )
c. 3 + j*3. 21. T = 412.
22. As = 0.3972. 23. vi 1.71 cu. ft., W = -4,840 ft.-lb.
24. $ = 78.2M - 398 - 393, M 141.
26. About 1 to 40. 26. F = 18.0 Ib.
28a. = 24.9 \/Pd; b. 181 Ib. per sq. in.
30. RK 2 - DK + R - 0. 31. ? =
32. Mi = M & = 0, Ms = ~^I/ 2 /14, 33. IF = 850.8 Ib., 8 = 90.8 Ib.,
M 2 = Mi = -3w;L 2 /28. JV 58.4 Ib.
34. Li = 6.82 - j9.06 amp., 7 2 = 6.51 - jl.60 amp.
36. r = 10.5 in., o> == 9.22 radians per second.
36. Q = 450 cu. ft. per sec.
37. L == 0.0035AF 2 . 39a. 0.9877; 0.9943.
44. 1.064; 0.957; 0.922; 1.008. 46. 8 = 2SiSi/(Si + *S 2 ).
47. 192 miles; 2 hr. 24 min. 48a. 105 miles per hour;
49. i = 4e~ zt amp. 6. 420 miles and 450 miles.
60. Distance during t sec. = 5.47^ 2 ft.
61. Logarithmic decrement = CTT/W.
52. 7,840 miles; l,760x 2 -f 399a;. 63. h = -3.41 in.
54. -249,000; -749 + J10,050. 66. x - 0.422L.
56. x - 5.20 ft. 67. d - 3.40 in.
161
162 ENGINEERING PROBLEMS
68. 2.04 ft. 59. h = 1.48 ft.
60. t = 0.089 in. 61. r = 2.69 in., h = 5.51 in.
62. F = M, (2 \/2)/2.
63a. a; = 9.664; 104.8; 292.6; 6. p 2 = 2/3, (2 \/2)/3;
c. p 2 = 25,600; 80,400; d. co = 1,080; 2,180; 5,630.
66a. /i = 3.46 amp., 7 2 = 1.17 amp., 7 3 = 1.50 amp.;
c. 7i = 0.3#i + 0.06# 2 + O.OSEs, 7 2 = 0.06#i + 0.092E 2 -f 0.056723,
7 8 = 0.08#i -f 0.056^2 + 0.208# 3 .
676. I g = 0.000,003,24 amp.
68. I a = 12.9 H- j'10 amp., 7& = 6.6 -f jS.2 amp., 7 C = 3.9 + j'10.8 amp.
69. M 2 = -ll,3001b.;M 3 == -15,200 lb.;
M 4 == -13,500 lb.; M 6 = -7,600 Ib.
70. P 51 lb.
71. Fx = 1.707, 2/! = 1, 2/ 2 = 1.414, 2/3 = 1;
^ 2 = 0.500, 7/1 = 1, 7/2 = 0, 7/3 = -1;
F 3 = 0.293, T/! = 1, 7/2 = -1.414, 7/3 == 1.
73. x = n 2, T/ = n 1, z = 2 n.
75&. Roots are -1, -2, (-1 i \/3)/2. 80. P = 137 lb., 7 7 - 146 lb.
81. Resultant = 127.7 lb., = 16258 / .
82. Resultant = 18.1 lb., = 189. 83. 6 39 r .
84. 8,960 lb., 785 lb. 85. d = 1.9 meters.
86. = v> = cos- 1 (%) 2857'; m = 30.2 ft.; p = 17.8 ft.; P = 413 lb.
88. Z * 13.42 ohms; = 2634'; 7 = 7.45 amp.; P = 667 watts.
89a. e = 812 / ; 90a. 6 = S ^';
b. 9812' track; 6. 26148 / track;
c. 130 miles per hour. c. 78 miles per hour.
91. t 1 hr. 29 min.; = 10.9; heading = 76.6 track.
94a. 1.000,076; 1.0161; 1.054.
96. p 85 - 30 cos 240^ - 50 cos 480^ - 5 cos 720^.
96. e = 62.8 sin (120^ - 0.124).
98. e = 100 sin 4,000,000^ + 35 sin 4,004,000* + 35 sin 3,996,000*
- 15 sin 4,008,000* - 15 sin 3,992,000*.
99d. x = 0.98992L + 0.2Lcos0 + 0.010102L cos 20 - 0.000, 025,8L cos 40;
r 2 cos 20
( T r*
L " 4L
C S
= 0.99L + 0.2L cos -f 0.01L cos 20.
100. 51.5 volts, 95.8 volts.
105&. N - 2 COB-I -; r == 7 9 , 106. r = 30/19 in.
r + &' 1 cos
107. = 3526'. 108. - 41 49'.
110. 0; 6015'; 8921'; 8947'; 8954'; 8958'.
111. 0; 045'; 730'; 225'; 328'; 4039 / ; 4846 / .
1126. 245 watts; c. 103.5 watts. 113d. P = El cos 0.
114a. (1) 10 +yO; (2) 48.2 /5129 / = 30.0 +J37.6 - 48.
(3) 0+;'16;(4) - j'2,500;
6. (2) / - 2.08 /-51 W, etc.
ANSWERS
163
115. Fi = 12.4 + J12.4; F w - +16.9 - J4.55; Fci = -4.55 +J16.9.
116. Fao = (H)(F + V b + F c ); Fai = (H)(F a + oF b + a 2 F c );
F 2 - (M)(F + a 2 F 6 -f aF c ).
117. /i = 55.0 A^105; 7 2 = 110 /-105; / 3 = 530 /-117.
118a. (1) 4 + j*8; (2) 96.6 + J25.9; (3) 200/156.9; (4) 10 /2634^
(5) 2.303 + /L.047; (6) 0.729 + #.685; (7) -8 + ./13.9;
6. (1) 10 /60; c. # = 34.9, = 32.5.
119a. a = 0, = 0.357 radian; b. + j$ = 0.020 + jl.98.
121. (4.75, 3.25).
122. 30,000,000 Ib. per sq. in.
123. Slope = %.
120.
Member
Slope
Inclination
DE
EF
-0.1
-0.4
17417'
15812'
124a.
6.
c.
125.
128a.
. P(ti, aiti), Q(ti + <2,
. 2A = a^i? 7 + 01^/2
. (01 + a a )(ai + a 3 )^i 2
300 in.-lb.
129a. x
adit* -
a 3 )^i 2 - 2a 8 Ti(oi + a 2 )
126. 2,000 Ib.
; b. F = 400 - 40a;; c. W eeM d
ai = 500x - 20z 2 .
to x = 3: F = 6,600 -
3toz = 6:F = 4,600 -
7: F = 600 -
12: F = -5,400 -
- a 2 ) + aaa
F intercept.
400o; - 20z 2 ;
0.
6 to x
7 to x
130.
132.
x
x
0;
x = to x
x = 3 to x
x = 6t<5 x
a? = 7 to a;
3: A = 6,600 - 50x 2 ;
6: A = 6,000 + 4,600z - 50z 2 ;
7: A = 30,000 + 600o: - 50x 2 ;
12: A = 72,000 - 5,400* - 50z 2 .
R - 0.0203 + G.000,0837 7 . The resistance of this wire is always
equal to its resistance at zero temperature plus 0.000,083 times the
temperature in degrees centigrade.
R = 1.00 is the consumption when no alcohol is added; the slope,
0.0060, is the rate at which the consumption increases per unit of
added alcohol.
w = 1,840 ft.-lb. 136. 1.66 calories. 136. T = 10 -f-
. EF: T = 2.0S3H - 1,002; CD: T = H - 70; b. 12,000.
660 tons.
R 3.5 + 0.0055S 2 .
134.
138a.
139.
152.
161. 4.33; 3.40; 2.93.
167.
141. (0.085, 0.496).
155. x - 0.42.
163. RIS - Eh + RIJ = 0.
3 \/3
6
X intercepts: x =*
vertex: x = L/2, M = mL 2 /24.
168a. y = -0.104 ft. 1696. t - 5 sec.
L; Y intercept: M =
12
170. 18.3 ft.
164
ENGINEERING PROBLEMS
16L 2 )Mi.
172. L = 6.0 Ib. per ton, S = 10 miles per hour.
173. AC - (p 2 + L 2 )^; C7> = 3p/4; >
174. 136ft.
176a. y - 9z/8 - 3z 2 /8,000, ^ y i n feet; 6. 844 ft.; c. 6 = 47.4.
176. 11.3 radians per second. 178. Vertex at (-40,0).
1806. a = 4.39; c. Error = 0.08, or a 6 per cent error;
d - = 3
if 6 > 0.193/c.
1816. A = (7r/82,944)(144 - 3) (108 - a;).
182a. 10.8 ft.; 13.3 ft.; 14.7 ft.; 6. 54.3 cu. yd.
183a. / = 1,010,000; 6. / = 20.2.
185.
Curve
(a)
(6)
AB
V = 0.4
y = 0.4
BC
pV = 15
pF = 2,160
CD
p 15
p = 2,160
DE
V = 2
V = 2
EF
pV = 80
^57 = 11,520
AF
p = 100
p = 14,400
1876. An asymptote: St = st\ c. an asymptote: /S< = 0.
1886. (z 2 /1.06) - (2/V5.19) = 1.
189. Coordinates may be obtained from x 3 + 0.866^' 0.5?/';
y 5 + 0.5x' -f 0.866?/', where a; 7 is along AH and ?/ along BA.
190. L = N cos a T sin a; D = ]V sin a + T cos a.
191c. ?/' = y, ut' = co^ + a.
192. f = I cos (a> - ^), where P = 7i 2 + 7 2 2 + 2/i/ 2 cos d and
tan = 7 2 sin 0/(/i + ^2 cos 6).
193. t'a 0. 194. i z == 0.5187 cos (2*ft - 4.45).
196. 30,170 Ib., 13,830 Ib. 1996. e max = 150 volts.
202. Plot l/E m in terms of ^ ij 0.74; E m 68.
204. ^ - and the wires touch. 2056. r = 0.37#, RE m = 2.72 F.
2066. Ti = 2,640 Ib. 2076. A = 0.5, = 0.643 radian; c. t = 230 sec.
208c. W = 181 ft.-lb. 209. c = 33.8; / = 26.6.
211a. -0.835 +j0.355;
6. -1.27 +J0.233.
210a.
L
E 8 volts
Is amp.
10
142
0.224
100
1,006
1.115
214c. B = 73.3. For any angle between 73.3 and 90 the block will
not move.
219. x - 10 cos 0.4J 2 ; y 10 sin 0.4* 2 .
222c. Slope = ~6.
ANSWERS
165
223d. x =
224a. x = 6 cos ut, y =* a cos (co a) ; 6. sin a = c/a = d/6.
2266. 22.7 ft. per sec.
227. P: x = 10 cos atf, ?/ = 10 sin a>;
M: x = 15 cos a>, ?/ = 5 sin a>2.
229. Mi: a; 2 ?/ 2 - (a - ?/) 2 (a 2 - ?/ 2 ); M 2 : 4s V = (a - ?/) 2 (a 2 - 4?/ 2 ).
231. s = 35 + 3,960, OOOe - l,255,000,000e 2 ;
Smax = 3,150 lb. per sq. in. when = 0.00158 in. per in.
232a. s = 2,880,000e; 6. s -100 + 4,000,000e - 94,900,000c 2 ;
s m ax = 42,000 lb. per sq. in. when e = 0.021 in. per in.
233a. e = 4.59 F 2 + 9.027 + 0.50.
234. i b = 4.18 + 0.274e c + 0.00351e c 2 . 235. C = 3.31, n = 1.52.
236. i - 26,500,000 V2V 55 ' 500/r ; i = 81.4TV- 53 ' 000 / 27 .
237. a = 15.5, 6 = 1.75.
2416. a = 11940', = 7222', 7 = 14355 / ;
c . ^ x = -40.4 lb., F y = 24.4 lb., F t = -64.7 lb.
242c. 29.2 lb., a = 54.3, ft = 65.9, T = 62.2; d. 27.3.
2436. ^ = 193 lb., a. = 42.8, = 53.9, 7 = 110.4.
244. 2 T = 389 lb., P - 625 lb., B x = 97.2 lb., B y = 583 lb., J5, 1,083 lb.,
J5 = 1,230 lb. at a = 85.5, /3 = 62.3, 7 = 28.5.
Member
Length
Direction cosines
AO
10
1;0;0
BE
43.9
-0.570; -0.455; +0.683
CO
32.2
-0.186; 4-0.311; +0.932
DO
50.4
-0.119; -0.795; +0.596
EO
5
1;0;0.
245.
246. 88 53'.
2476. (1) 359ft.; (2) N.26 37 ; W.; (3) 1.7 per cent; (4) 1.0.
Wire
Length, ft.
Direction angles
CB
157
112.3; 104. 7; 27. 3
DB
180
63.6; 96. 4; 27. 3
EB
152
93.8; 62. 7; 27. 6
248.
x is positive in east direction, y in north, and z upward.
2496. 82s + SOy + 57* 28,500; c. 302 ft.; 50.1, 51.3, 62.9.
250e. E on AB: (333, 250, -333); F on CD: (420, 474, -333).
263. When t = 20C., dQ/dt = 0.5662.
266. i (E Q /r)e-v Cr in both cases.
267. P - dW/dt.
268. F - wL/2 - wx.
166
ENGINEERING PROBLEMS
269. For < x < L/2: V = P/2; L/2 < x < L: V -P/2;
discontinuous at x = L/2.
270, F = k/r z . 271. P = 16 ft.-lb. per sec. 272. 10.6 ft.
2736. P/A = p - (2Lp/&rr)(p/3E)K. 274. 6 2 = 4am.
__ , t /fc sinh 1\ 276a. -3.18;
275. cot -i^__). Z, * = 0.2* -f 9.
277. 27y = 704fcs 6,400/c.
278. a 4, 6 = 2, and ?/ = x 1 is the common tangent.
279. Join points (1,0) and (9.5,8-5) to obtain the common tangent line.
280. y = -
281. y = x + 3 \/2 ft.
286b. F = (f//P)rc; c. x = (H - P)xJ/P.
287. x = 1.068 in., y =* 1.141 in.
288c. a; = and x = 12; d. x = 6, M 31,800 Ib.-ft.
289a. At the wall M = ^L 2 /12. Notice that though dM/dx = at
x = L/2, the corresponding value of M is numerically less than the
value for M at the wall. 6. x = 0.21 1L and x = 0.789L.
2906. s 0.707a; c. a? = 0, 1.225a.
/
10
20
30
40
50
60
70
ds/dt
31
59
80
93
96
92
81
292.
293. y = 0.29690 Vx - 0. 12600s - 0.35160s 2 + 0.28430s 3 - 0.10150s 4 .
295. d = (5B/2A)H. 297. D = 2/7/3.
302. 77 =
when Po
304. 72 = cr, E m
306. 3.54ft.
E +2-
305. a = 0.851D, t = 0.528D.
308. pz = Pi (jg-jTi) / t - 528 -
309. Pmax = 100 Ib. when = 90; P m in = 19.6 Ib. when = 11.3.
310a. * = 5.16 ft. 311. x = 0.577L, y = -0.005, 42wL*/EL
312. 10 ft. 313. 3d = D.
314a. tan 20 * 2P XV /(I V - /*) ;
b. the angles differ by an odd multiple of 45.
315. t = 3.9C. 3166. x = 0.417, v = 0.100.
0.20
317. A n
1 - a
318. 949 articles per day.
[ -a/(a-i) _ -i/(a-i)] > w here a =
ANSWERS
167
t
dz/cft
dx/dt
d*x/dt*
d*x/dt*
exact
approx.
exact
approx.
0.30
0.30
7T/4
0.40406
0.40355
0.27730
0.27678
7T/2
0.5
0.5
-0.0510
-0.0500
1T
-0.20
-0.20
e.
3206.
321a.
<
323.
324a.
6.
325.
e.
d.
326a.
6.
d.
e.
327a.
328.
329a.
Maximum force = 0.30m.
-3401b.;c. 400 Ib.
t = 0.063 sec.; V = -3.56 ft. per see.; y = 0.0683 ft.;
y*/yi = -0.9908.
1 71 = 0.307 ft. per sec. at 130.7.
\a\ = 0.254 ft. per sec. per sec. at 190.4.
VA = 16 ft. per sec., VB = 12 ft. per sec.;
\VA/B\ = 20 ft. per sec. at -36.9.
5 r.p.s.; 6. \V\ = lOOyr |sin 20^j ft. per sec.;
\a\ = SOOvr 2 \/26 10 cos 407ri ft. per sec. per sec.;
t n/40 where n is any positive or negative integer.
x = 10 cos 6 2 cos 50; y = 10 sin d 2 sin 50, where = QAirt;
x 2 - 40 cos l.GTrf - 20z cos 0.4^ + 4x cos 2irf = 2,396; c. V = 0;
a 63.2 in. per sec. per sec. at 0;
V 0, a = 63.2 in. per sec. per sec.
5 r.p.s.; b. \V\ = 60?r |sin 207r/| ft. per sec.; c. \a\ = 3 c*> 2
ft. per sec. per sec.
|Fmax| = 12 ft. per sec.;|a max l == 9 ft. per sec. per sec.
\V\ = 8 ft. per sec. at 210;
\a\ = 20 ft. per sec. per sec. at 2638';
a x = 4 cos d (dd/dt) 2 + 4 sin (d 2 6/dt z );
a y = -4 sin (dO/dt} 2 + 4 cos (d 2 d/dt 2 );
6 cos 40
7 ITTI ^ 1rt r ,
' ' = ~' at = = ~
330.
332.
333.
334.
335.
a n = 16 ft. per sec. per sec.
0)A = 4 radians per second, a
\Vc\ = 8 ft. per second at 135;
\ac\ = 32 ft. per second per second at 225;
1 7*| =8 ft. per second at 225;
\as\ = 36 ft. per second per second at 72.6.
x - 6/2 ft.
|7| - (r 2 o> 2 + g 2 t*}y>] \a\ = (r 2 co* + 2 )'/i
47TV 2 2^ , M \X , , /167TV 2
= 8.36 radians per second per second.
# == 250^, y
y axis positive
y - 2,000 ft.
336. Ap pv Av.
sin 2 ~y 4- ]
4- 250 \/3 with axes through the bomber and
downward; when t = 4 sec., x = 1,000 ft., and
337. Ay = 0.264 in.
168 ENGINEERING PROBLEMS
338o . dv = *>*-* d b ^ = /nfl\
pv* at> -f ab "' \ P /
339. 6.25 in., 113 cu. in. 340. dQ = 0.16 + 60e.
341. cty 0.000,59. 343. 2,560 ft. per sec.
344. Relative error 1 [1 + (dy/dx) 2 ]~M; largest percentage error
is 0.375 per cent.
345. When x = L/2, R = 320 ft. 346. Not defined; 600 ft., 254 ft.
347. R = a 2 /2 = 0.0441. 349. Y = -(#/2) cos co*.
350. Lim (t) = Ift/L. 351. Limit in each case is zero.
352. e = 0: x - 0, y = ( - A)(l - 2/ir);
= 7T/2: y = 0, # = B(l - 2/ir) + 2/V.
353. A - (P/cr)e-y*/.
364. At r = 0: w = Pa 2 /16irN, dw/dr = 0, M does not exist;
at r = a: M = -P/47T.
355&. TF = pi^i In (v-z/vi), when pi^i ^2^2.
356. piViln (wa/wi). 357. y = (yj*.
358. dFM = 2.4/cD 2 . 359. dF/di == -0.034 cu. in. per sec.
360. dp/dt = -1.26 Ib. per sq. ft. per sec.
361a. 10.2 ft. per sec.*; b. 0.041 ft. per sec. per sec.
o* n 2My 2Mx
362a. Zero; c. r xz = - _^, rw . -_.
3636. 0-3: = 2(x a), o-y = 2m?m + 2ny 2m2/ 6mnx,
T XV 2mx 2nx 2y ma;
c. a x curvature in y direction, <r y curvature in ic direction.
364a. a/r 2 = a/(x 2 + i/ 2 ); 6. V x - (2 -
367a. 0.4228 B.t.u. per Ib.; 6. 0.2988 B.t.u. per Ib.
*
3686. (1) -2fc/a; (2) -2fc/6. 3696. dl 7 - (l/fc)(Fdp + P dF).
370. Six different first partial derivatives.
371. (6<3/F)r - (P/R)(c f - c,); (BQ/8P) T
(SQ/8P) V
373. 13.3 ft. per sec. 374. = OiOOSi 3 - 0.4/ 2 + Wt.
375. 200 sec. 376. 51.6 ft. per sec.; 1.71 sec. 377. 5.47 sec.; 96.2 ft.
378c. (1) As = 22 miles, Ay = 5 miles; (2) Ax = 10 miles, Ay = 7 miles.
379a. ce = 17 radians per sec. per sec.;
6. 104 revolutions; c. a increasing.
380. i = 1.326(1 - cos 120^), * = 1.99 amp.
381. gfo - 1/VXC, i? sin~i [(/2/2) VC/]; fc - (^o/g) cos rj.
382. ^240 13,5 ft. per sec.; s 2 4o = 4,730 ft.
ANSWERS
169
hit.
a; ft.
Angle
Impact speed,
miles per hour
10,000
11,000
28.8
625
20,000
15,500
21.1
833
30,000
19,000
17.5
995
383.
386. The total energy is constant.
f\r p r
386. For < x < -: Ely = ~ (9* 2 - 8L 2 ),
- 350,000*;
-0.49 in.
387. ISEIy = 5,000s 3 -
388. 6.25 ft.
389. y = 0.000,34s 3 - 0.000,014,2s 4 - 0.024,5s ft.;
when s = 6 ft., y = -0.092 ft.
3916. < * < 10: s = 0.04* 2 miles; 10 < * < 100: s = 0.8* - 4;
100 < * < 105: s = 16.8* - 0.08* 2 - 804; 105 < * < 120: s = 78.
392. kt = In
where A
and C =
1
(6 a)(c a)
1
(a -c)(6-c)
393. a? - 10.0(1 - e-' 00154t ).
394. T = (Zn
396. k - 0.002,397 if * is in minutes; T = 289 min.
396. 2/ 2 = 2pt.
398a.
399. 246 Ib.-sec.
400a. yn * ~ Fn ^
'; 6. A(12) = 2.30 sq. in.
6. A = 182 + 0.251* + 0.000,187* 2 ; c. V
where * 3 = w and A; 8 = gr;
c. 0(s) = 0.819e- 14 627a; + 0.
. (61.6 - C e )(63.6 - C,) , 62.6
J* /r*c\ /> f~i \ o ^n
(62.6 - Ce) 2
Co
0.0559;
1 , C e
TV 2 - m
62.6(62.6 - C e )(62.6 - c )
101. i = 0.402 amp.
404, P.E, - tf
62.6 - Co ^ (62.6 - C,) 2 " C e - c
, W = J
403c. Q
406. 9,610 Ib.-sec,
170 ENGINEERING PROBLEMS
406. 4,528 units, 3,600 units. 407. 8.73 ft.-lb.
408. 0.385. 409. 1,750 calories; 1,590 calories.
4lOa.
412. Results by Simpson's rule: 6.359; 5.816; 6.045.
413. Simpson's rule: 6 subdivisions: 0.749; 12 subdivisions: 0.783.
414a. H T = 12.04124, H s = 12.04113; b. H - 12.000,00;
t c. s = 1.006,389 - 0.000,4860, // = 12.041,71.
416. 0.722 416. 3,480 cu. yd.
4176. WT = 15.5; w s = 15.0; B == ///(0.0508// + 0.208); w = 13.9.
418. 24 miles. 419a. 277 watts.
420a. 2ir/E, 0; b. 0.707#; c. 170 volts.
4216. (1) 75.5 volts; (2) 0; c. 79.4 volts.
422a. (2/7T)(/i + /s/3 + 7 5 /5 +
6. 0.707 (/i 2 + / 3 2 + - - +/ fc 2
423. P = 0.5(^i/i -i- # 3 / 8 + - . .
424a. 0.5/V; b. 0.5(/i 2 + J 3 2 )r; c. 0.5(/i 2 + 7 3 2 )r;
426a. 40 ft. per sec.; b. 66.2 ft.
427. /w = (H)(/^2 -f
428. r = (vr*/Ac)(2h/g)K.
429. c mea n - + (0/2) (!T a + TO + (7/3)(? 7 2 2 + T,T, +
430. *
rj
431. Fmean == (7T/6)d a v 3 = (ir/24) (D + d)(Z) 2 4- d 2 ); da v = 0.0403 mm.
432. n = 1: p av = In - 2
^ ^2 - i
4336. 0.014 per cent; 1.62 per cent.
434a. 0.0785 in.; 6. s = 75,000 Ib. per sq. in. and F = (s)(area) = 37,500 Ib.
436(a) 0.0785 in.; (6) 0.0851 in.; (c) 0.0634 in.
436. (a) 2.74 ft.; (6) 2.74 ft.; (c) 2.82 ft.; (d) 3.16 ft.; (e) 2.80 ft.
437. 87.0 Ib.; 79.5 Ib. 438. SADB = 81.4 ft., SACB = 65.9 ft.
439. fcL 2 /2. 440. 100.425 ft. 441. 100 + 0.4267 - 0.0016.
442. 81.5 sq. in. 443. 20 cu. ft. 444. 0.43 cu. ft.
446. 83.0 cu. yd. 446. S = 59.7a 2 = 14.96 2 . 447. 6 by 8.33 in.
448a 4V^fi3^ r s 4V2 2V2
448a * 37T R* - r 2 ' ' 3* H > C ' TT Kt __
449. Fig. 128: (21/3, h/3)' } Fig. 129: (36/5, 3 VaS/6).
461a. (w/2, h/2), R x ,z - ^ \/3, ^^,2 - ^ V3;
6. JRx.3
462. P - i
ANSWERS 171
464a. 2T 2 ; b. 2!T 3 /3; c. :F 2 /2 + 2 T3 /12; d. 10T 3 /3 -
455a. vkLRW/4gtiAb.;
b. r = 0.707E = radius of gyration with respect to the geometrical
axis of the disk.
466a. 0.293 slug; 6. 4.98 ft.-lb.; c. 5.00 ft.-lb.; d. 5.78 ft.-lb.
467. 47.5 radians per second per second. 461. 6d 3 /24, approximately.
462. 145,000 Ib. 463. 22,100 Ib.
464a. 3,120 Ib.; b. 3,260 Ib.; c. 4,520 Ib. at 46.3 with horizontal.
465. 7,200 ft.-lb. 466. 10,400 ft.-lb.
468. 2,330 ft.-lb. 470. 3,550 ft.-lb.
472. M f = ^ -T-IT71' $> Wr *' 473 ' 4 - 41 Rt - u -
474. Q = ?!!^5-~y. 477. W = 4w7.
478. g = 27r f a r/(r) dr.
a/0
479a. g = y F m a 2 ; 6. q = 7rF m a 2 .
480. w = ^7 ( 4c 4 In - + 4a 2 c 2 - 3c 4 ).
64, \ a /
483a. L =
484
i. Error term is (^) I (dy/dx}*dx.
J JCl
486. Error term is (ERW/6L*). 487. t = 0.136 sec.
488. P = 0.0797. 489. 0.92690.
491. m
^-^ii4..__
M ~ 72 < 12 V# 2 A 2 / 80
494. L = (7r/2)(D + d) + 2C + 0(& - d) - C(9 2 ; error term is + CO*/12.
497. 24.5 miles. 498. 380 ft.
499. 8.7 in. per mile. 500. // = 100 + 50gr 2 , where g = 0.05,
601a. 67rft.;6. 1,280 ft. ; c. 0.0220 radian.
603a. e L = (Q/Ck)e~ at (k cosh kt - a sinh kt); b. q L = Q;
c. ec = (Q/Ck)e- at (a sinh fci + k cosh Art); * == (Q/CLp)e' at sin p^.
505. E, = 127,000 +J16, 100 volts.
506a. r = 1.76;
5. T = y( 0.723 -f- 2n7r), where n is a positive or negative integer;
c. T = 0.962 + jnir.
172 ENGINEERING PROBLEMS
607&. E 8 = 57,311 4- 1,147(L - 200) + 11.5(L - 200) 2 + ;
/. = 57.35 + 1.15(L - 200) + 0.0115(L - 200) 2 + ;
c . /(L) = -17 + 0.42(L - 200) - 0.01 15(L - 200) 2 + ;
g(L) = -0.0175 - 0.0012(L - 200) + 0.000,004(L - 200) 2 + -
508. c - 33.8, / = 26.5.
609a. 8455'; 6. 801.1 ft.; c. 98 ft.; d. 18 ft.;
e . y = 80 + z 2 /9,000, error less than 0.02 ft.
610. - 5628'.