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DO 64985 CD OSMANIA UNIVERSITY LIBRARY Call No./^/ J S/K,^ L ^' / t Accession No, rv<f \uthor > \ I . ' This book should be returne(|/n or beJg^ the date ^sJL marked belgw, 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'.