I. V. NIESHCHERSKY COLLECTION OF PROBLEMS IN THEORETICAL MECHANICS Edited by Professor A. I LURIE, Dr. Tech. Sc. The Su...
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I. V. NIESHCHERSKY COLLECTION OF PROBLEMS IN THEORETICAL MECHANICS Edited by Professor A. I LURIE, Dr. Tech. Sc. The Supplement is written by V I. KUZNETSOV, Cand. Tech. Sc. me moman scnoor punusnmo House Moscow
Translated from the Russian by N. M. SINELNIKOVA Printed in the Union of Soviet Sobalist Republics
CONTENTS Editor’s note 5 Preface. 7 Part I. Statics of Rigid Bodies 1. Coplanar Force System 1. Collinear Forces 9 2. Concurrent Forces 10 3. Parallel Forces and Couples 19 4. Arbitrary Coplanar Force Systems 26 5. Graphical Statics. 40 11. Statics in Space 6. Concurrent Forces . . . 43 7. Reduction of a System of Forces to Its Simplest Possible Form 47 8. Equilibrium of an Arbitrary System of Forces 48 9. Centre of Gravity 56 Part I1. Kinematics III. Motion of a Particle 10. Equation of Motion and Path of a Particle 59 ll. Velocity of a Particle . 61 12. Acceleration of a Particle 63 IV. Simplest Motions of a Rigid Body 13. Rotation of a Rigid Body about a Fixed Axis 68 14. Conversion of Simplest Motions of a Rigid Body 70 V. Composition and Resolution of Motions of a Particle 15. Equations of Motion and Path of the Resultant Motion of a Particle 74 16. Composition oi Velocities of a Particle . . . . . 76 17. Composition of Accelerations of a Particle Undergoing Translatory Motion of Transport . . . . . . 79 18. Composition of Accelerations of a Particle Performing Rotational Motion of Transport about a Fixed Axis 82 VI. A Rigid Body Motion in a Plane 19. Equations oi Motion of a Body and Its Particles in a Plane . 87 20. Velocity of a Point of a Body Which Performs Motions in a Plane. Instantaneous Centres of Velocities 89 21. Space and Body Centrodes . . . . 96 22. Accelerations of a Point on a Body Which Performs Motions in a Plane. Instantaneous Centres of Accelerations 99 23. Composition of Motions of a Body in a Plane 102 VII. Motion of a Rigid Body about a Fixed Point 24. Rotation of a Rigid Body about a Fixed Point . 10525. Composition of Rotations of a Rigid Body about Intersecting Axes l07‘ 3
PREFACE This book is an abridged translation of the latest (28th) Russian edition of Meshchersky’s Collection of Problems in Theoretical Mechanics, published in the USSR in 1962. This Collection was compiled by a large group of Soviet professors and highly experienced instructors of the Leningrad Politechnical Institute, named after M. I. Kalinin, among whom are S. A. Sorokov (statics), N. N. Naugolnaya and A. S. Kelson (kinematics), A. S. Kelson (dynamics of a particle), M. I. Baty (dynamics of a system), G. J. Djanelidze (analytical statics, dynamics of bodies having variable masses, stability of motion). Professor A. 1. Lurie, Dr. Tech. Sc. a prominent Soviet scientist, is the general editor of this book. The first Collection of Problems, edited by I. V. Meshchersky, was published in 1914. With every new edition the book has been revised and supplemented taking into consideration the developments in science and engineering for the past period. The present edition embraces all basic principles of theoretical mechanics usually taught during the first two years of studies in higher and secondary technical schools. The material in this book is presented consistently, i. e., proceeding from the particular to the general. The same method is applied to the order of paragraphs and the arrangement of the text. Accordingly, the initial section of the book deals with fairly easy problems on basic concepts_ and principles of statics of rigid bodies, while the last section of the book embraces rather complicated problems on principles of stability of motion. It should be noted that most of the problems, chosen for this collection, are not only an illustration of the theoretical material, but are well in accord with the materials which serve as a bridge between theoretical mechanics and intermediate sciences. The primary objective of the book was to present to the reader problems of practical value by giving the examples in such fields as the operation of machines and mechanisms, hydrodynamics, resistance of materials, and other branches of science and engineering. All this has made the book widely popular among the Soviet students of technical schools. 7
The book Collection of Problems in Theoretical Mechanics is at present one of the basic text—books for Soviet students of theoretical mechanics. The material collected in this book will aid students in the practical application oi laws and methods of theoretical mechanics in their engineering practice. The present translation of the Collection of Problems is supplemented with solutions of certain typical problems for each paragraph to help the student to apply to specific situations the principles and theorems that he has learned. The book is well illustrated with drawings and diagrams.
Part I STATICS OF RIGID BODIES I. COPLANAR FORCE SYSTEM I. Collinear Forces l. Two weights of 10 kgf and 5 kgf, respectively, are suspended from a string at different points. The heavier weight is suspended lower than the lighter one. Find the tensions in the string. Ans. l0 kgf and 15 kgf. 2. A uniform vertical cylindrical column with the height h=5 m and weight Q=3000 kgf is mounted on a solid foundation. It carries a load P=4000 kgf. Determine the pressure that the column exerts on the foundation and the compressive forces in the sections located at distances l1=l2=0.5 m from the top and bottom ends of the column. Ans. N=7000 kgf; Nl=4300 kgf; N2:-6700 kgf. 3. The weight of a man standing at the bottom of a pit is 64 kgf. By means of a rope running over a fixed pulley the man lifts a load of 48 l=600 mm; the diameters of the piston rod are: d,=60 mm; 9
d3=l00 mm. The mean vapour pressure is pl=9.5 kgf/cm2; pz.-=2.5 kgf/cmg; p3=0.1 kgf/cm2. Ans. 12,100 kgf. == - ·—·~ Y-1 ”//”’ /”//” ”/”” ’ ///y,»..y,r...m//,r.. .. Fig. 1 2. Concurrent Forces 6. Concurrent forces of 1, 3, 5, 7, 9 and ll kgf applied at the centre of a rectilinear hexagon act towards the vertices. Determine the magnitude and direction of the resultant and the equilibrant. Ans. 12 kgf; the direction of the equilibrant is opposite to the direction of the given 9-kgf force. 7. Resolve a force of 8 kgf into two components, 5 kgf each. Is it possible to resolve the same force of 8 kgf into two components of 10 kgf each; 15 kgf each; 20 kgf each, or even two forces of 100 kgf each? Ans. The answer is positive if the directions of resolutions are not given. 8. A force Q=250 kgf acts in the direction of a rafter inclined at an angle oi=45° to the horizontal (Fig. 2). Compute the magnitudes of the force S which acts in the direction of the horizontal joining beam, and the force N which acts on the wall in the vertical direction. Ans. S=N=177 kgf. 9. The rings A, B and C of three spring __ balances are tightly fixed on the horizontal § A board. Three strings are tied up to the hooks of {Q A >"·r? c n' Fig. 18 Fig. 19 8 k 30 0 0 0 A W R W B- an _..i;mm4.:¢.a,, M/.~r Fig. 20 Fig.21 27. Fig. 20 represents an ABCD system of links, one side of which CD is fixed. A force Q= 10 kgf acts at a hinge A at an angle BAQ=45° Determine the force R which, acting at the hinge B at an angle ABR=30°, keeps the system in equilibrium. Angles CAQ=90° and DBR=60° Ans. R=16.3 kgf. 28. Fig. 21 represents a system consisting of four rods of equal length. The ends A and E are fixed pivots. The joints B, C and D are acted on by identical vertical forces Q. At equilibrium the angle of inclina- ..L,.. . tion of the extreme rods with the horizontal is ot-=60° Determine the angle “ of inclination of the middle rods to the _ _ horizontal. {li 6 / / A x A/; Ans. 3=30°. Fig.22 15
29. Find the reactions of the supports A and B when a horizontal folrce P is applied to a three-hinged arch (Fig. 22). Neglect the weig t. Ans. RA=RB=PA/Eg. 30. A system consists of three three-hinged arches (dimensions are given in Fig. 23). Express the reactions of the supports A, B, C and D in terms of the applied horizontal force P Ans. RA=P%z—; RB=P; RC=P; RFP?. P 0 - wen _.-__ -__-.;r__. @/%*2 5, , D Fig. 23 31. A derrick crane (Fig. 24) consists of the fixed tower AC and the movable truss BC which is hinged at C and is supported by a cable AB. A weight Q=40,000 kgf is held by a chain which runs over the pulley at B and from there it goes to the winch along the straight line BC. The length AC=BC. Determine (as functions of the angle ACB=q>) the tension T in the cable AB, and the force P which compresses the truss along the straight line BC. Neglect the weight of the truss and the friction on the pulley. Ans. T-:80,000 sin ig? kgf; P=80,000 kgf, independently of the angle rp. A 32. A pulley C with the weight ( P= 18 kgf can slide along a i 5 flexible 5-m long cable ACB A (Fig. 25) The ends ofthe cable are A A at 5 ‘ . Fig. 24 _ Fig-25 .15
fastened to the walls; the distance between the walls is 4 m. Find the tension in the cable when the pulley and the weight are in equilibrium. Neglect the weight of the cable and the friction on the pulley. H i nt. The tensions in the parts AC and CB are equal, and their magnitude can be determined by similarity between the force triangle and the isosceles triangle, one side of which is a straight line BCE and its base is the vertical BD. Ans. 15 kgf, independently of the height BF 33. Two small balls A and B weighing 0.1 kgf and 0.2 kgf, respectively, rest on a smooth circular cylinder with horizontal axis and radius OA=0.l m. The 6 balls are connected by a thread 5 AB=0.2 m long (Fig. 26). A ° 95. #*9;/ is !// A V0 ' l .0 ¤, .0 Fig.26 Fig. 27 Determine the angles rpl and rpg formed by the radii OA and OB and the vertical straight line OC when the system is in equilibrium. Determine the pressures Nl and Ng exerted by the balls on the cylinder at points A and B. Neglect the sizes of the balls. Ans. rp1=2——rpg; tan rpg=—E-2-; rp1=84°45’; rpg=29°50’· 2+cos2 NI=0.l cos rp1kgf=O.O092 kgf; Ng=—-0.2 cos rpg kgf=0.l73 kgf. 34. A smooth ring A can slide without friction on the fixed wire bent into a circular arch in a vertical plane (Fig. 27). A weight P is suspended from the ring; a cord ABC, also attached to it, runs over the fixed pulley B. The latter is suspended from the highest point of the wire. (Neglect the size of the block.) A weight Q is attached at the point C. Determine the central angle rp, subtended by the arch AB, for equilibrium, and find the position when equilibrium is possible. The weight of the ring and friction may be neglected. Ans. sin -?$=5%; rpg=n. The first state of equilibrium is possible when Q, ·:4;yy,··>A J000»r f /, .0 47 % vw 7 ii" ‘ 7% 0 QL- .t|