Theoretical Mechanics: A Short Course

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n ieoretical _>>^ -€ b n s tra in ls and T h eir Reactions &. Axiom of C o n s tr a in ts .................................................... C h a p t e r 2. Concurrent Force Systems ^ 6. Geometrical Method of Com position of Concurrent-. Forces . . . 23 7. Resolution of F o r c e s ....................................................' ...............................25 8. P rojection of a Force on an A x is and on a P la n e .............................. 28 9. A n a lytica l Method of D efining a F o r c e ..................................................30 10. A na lytica l .Method for the Com position of F o r c e s ..............................31 11. E q u ilibrium of a System of C oncurrent Forces .................................32 12. Problems S ta tic a lly D eterm inate and S ta tic a lly Indeterm inate 34 13. Solution of Problems of S t a t i c s ...................................................................35 14. Moment of a Force About an A x is (or a P o i n t ) ..................................43 15. Var.ignon’ s Theorem of the .Moment of a Resultant ......................... 45 16. Equations of Moments of Concurrent F o r c e s .........................................46 C h a p t e r 3. P arallel Forces and Couples in a Plane 17. - Com position and Resolution of P arallel F o r c e s ...........................47 18. Force Couples. Moment of a C o u p le ........................................................50 19. E quivalent C o u p le s ........................................................... 51 2U. Composition ol Coplanar Couples. C onditions for the E q u ilib riu m of C o u p le s............................................................................................................ 53 C lt a p t e r 4. General Case of Forces in a Plane 21. Theorem of the T ra n sla tio n of a Force lo a P a ra lle l P osition 22. Reduction of a Coplanar Force System to a G ive n Centre . . . 23. Reduction of a C oplanar Force System to the S im plest Possible • F o r m ................................................................................................................ 24. Conditions for the E q u ilib riu m of a Coplanar Force System. , . 25. E q u ilib riu m of a Coplanar System of P a ra lle l F o r c e s ................. 26. S olution of P r o b le m s ................................................................................. 27. E q u ilib riu m ol Systems of B o d ie s ............................ ’ ' ’ 28. D istribu ted Forces ....................................................................

55 56 58 61 63 63 70 74

Contents

C h a p t e r 5. Elements ol G raphical Statics 29. Force and S trin g Polygons. Reduction ot a Coplanar Force Sys* t i in to Two F o r c e ' ........................................................................................ 78 3". G raphical D e le m iin a tio n ol a R e s u lta n t ................................................ 80 31. Graphical D e term in atio n ot a Resultant C o u p l e ..............................80 32. G raphical C onditions o l E q u ilib riu m of aCoplanar Force System 81 ................................81 33. D e te rm in a tio n ol the Reactions ol C onstraints 34. Graphical A nalysis of Plane T ru s s e s ........................................................ 82 35. The .Maxwell D ia g r a m .................................................................................... 85 Chapt 36. 37. 38. 39. 40.

e r 6. Friction Laws of S ta tic F r i c t i o n ................................................................................ 86 Reactions ot Rough Constraints. A ngle of F ric tio n ...........................88 E q u ilib riu m w ith F r i c t i o n ............................................................................ 89 B e ll F r i c t i o n .................................................................................................... 92 R o llin g F ric tio n and P ivo t F r i c t i o n .................................................... 94

C It a p I e r 7. Couples and Forces In Space 41. Moment of a Force A bout a P o in t as a V ector ..............................95 42. .Moment of a Force w ith Respect to an A xis .......................................97 43. R elation Between the Moments of a Force about a P oint and an A xis ...........................................................................................................100 44. V ector Expression ot the M om ent o f a C o u p le ...................................101 45. C om position ol Couples in Space. C onditions ol E q u ilib riu m of C o u p le s ............................................................................................................101 46. Reduction ol a Force System in Space to a G iven Centre . . . 104 47. Reduction of a Force System in Space to the Sim plest Possible F o r m ...................................................................................................................106 48. C ondition of E q u ilib riu m of an A rb itra ry Force System in Space. The Case of P arallel F o rc e s .......................................................................108 49. V arignon's Theorem of Ihc M om ent of a R e sulta n t w ith Respect to an A x i s ....................................................................................................... 109 50. Problems on the E q u ilib riu m of Bodies Subjected to the Action of Force Systems in S p a c e ....................................................................... NO 51. C onditions ol E q u ilib riu m ol a Constrained R ig id B ody. Concept of S ta b ility of E q u ili b r i u m ....................................................................... 117 C h a p t e r 8. Centre ol G ravity 52. Centre of P a ra lle l F o r c e s ...........................................................................116 53. Centre ol G ra v ity of a R ig id B o d y .......................................................120 54. Coordinates ol Centres ol G ra v ity of Homogeneous Bodies . . . 122 55. Methods o l D e te rm in in g (he Coordinates o l the Centre o l G ra v ity ol B o d ie s ...........................................................................................................122 56. Centre ol G ra v ity of Some Homogeneous Bodies ...........................125 PAR T l l . Chapt 57. 58. 59. 60. 61. 62.

KIN EM ATICS OF A PARTICLE AND A RIG ID BODY

e r 9. R ectilinear M otion of a P article In tro d u ctio n to K in e m a t ic s .......................................................................128 E quation ol R e ctilin e a r M o t i o n ............................................................... 129 V e lo city and A cceleration o l a P a rtic le In R e c tilin e a r M otion 130 Some Exam ples of R e c tilin e a r M otion of a P a r t i c l e ....................... 132 Graphs of D isplacem ent. V e lo c ity and A ccelera tio n ol a P a rticle 134 S olution ol Problems ...................................................................................135

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Contents C h a p t e r 10. Curvilinear Motion of a Particle 63. 64. 63. 66. 67. 68. 69. 70. 71. 72.

Vector Method of Describing M otion of a P a r t i c le ........................... 1/7 V elocity Vector of a P a r t i c l e ...................................................................1*18 Acceleration Vector of a P a r t i c l e ...........................................................130 Theorem of the Projection of the D e riv a tiv e of a Vector . . . Ml Coordinate Method of Describing M otion. D eterm ination of the Path, V elocity and Acceleration of a P a r t i c l e ................................... M2 Natural Method of Describing M otion. D eterm ination of Hie Veloc­ ity of a P a r t i c l e ...........................................................................................M7 Tangential and Normal Acceleration of a P a r t i c l e ........................... 148 Some Special Cases of ('a rticle M o t i o n ................................................ 1SI V elocity in Polar C o o r d in a te s ...................................................................136 Graphical Analysis of P article M o t i o n ................................................... 136

C h a p t e r I I . Translalory and Rotational Motion of a Rigid Body 73. M otion of T r a n s la tio n ...................................................................................160 74. Rotational Motion of a R igid Body. A ng u lar V elocity and A ng u ­ la r Acceleration ...........................................................................................162 75. U niform and U n ifo rm ly Variable R otation ..........................................164 76. Velocities and Accelerations of thePoints of a R o tatin g B">dy 166 C h a p t e r 12. Plane Motion of a Rigid Body 77. Equations of Plane M otion. Resolution of M otion in to T ransla­ tion and R otation ....................................................................................... 170 78. Determination of Ihe Paths of the Points of aB o d y ......................... 172 79. Determination of the V elocity of A ny P oint of aBody . . . . 173 80. Theorem of the Projections of the Velocities of Two Points of a B o d y ......................................... ....................... .................... l74 81. Determ ination ol Ihe V elocity of A ny P oint of a Body Using Ihe Instantaneous Centre of Zero V e lo city ................................................175 ........................ 170 82. Solution of Problems 83. V elocity Diagram . ! . . ' . ! 182 84. D eterm ination of the Acceleration of Any P oint ofca Body . ! ! 184 . Instantaneous Centre of Zero Acceleration . . . |9 i C h a p t e r 13. Motion ol a Rigid Body Having One Fixed Point and M o­ tion ol a Free Rigid Body 86. Motion of a R ig id Body H a vin g One Fixed P oint . . . . 00 X « y . ai ! d Acceleration of A n y P oint of a Body . . . Nt. Ih e Mosl General M olion of a Free R ig id Body

193 195 196

C h a p t e r 14. Resultant Molion ol a P article 89. Relative, Transport, and Absolute M o lio n 90. Composition of Velocities . . . .

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r f i T Pi° V n ° n ,of A*:c* ,eraliorw * C o rio iis th e o re m o?* £ * {c“ ,at,0l2 of Coriolis A cceleialion ..................... . . . . 93. Solution of Problems qo

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200 203 207 209

C h a p t e r 15. Resultant M olion ol . R ie,d Body OS S H E 0'-!'.01’ 0f. T ra n slo lo ry M o l i o n , ................

96! Toolhed"spur°Gearlng°n> Ab°Ul Tw° Par......

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2IS 215 218 221 223

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Confetti* P A R T TH R E E . PARTICLE DYNAMICS

C li .1 p l c r IG. Introduction to Dynamics. Laws ol Dynamics 90. Basic Concepts anil D e f in i t i o n * ............................................................. 226 I HO. The Law * ol D y n a m ic s .................................... . . . . 227 101. S y 'tim s of U n i t s .......................................................................... 230 |02. The Problem* of D ynam ic* for a Free and a Constrained P article 230 10.1. Solution ol the F ir * l Problem ol D y n a m ic s ...................................... 231 Chapter 104. 105. I0G. |07. 106.

R ic tilin e a r M otion of a P article ........................................................ 233 S tlu lio n of P r o b le m * .................................................................................. 236 Body F a llin g in a Resisting Medium (in A ir) . . . . . . 241 C u rvilin e a r Motion of a P article .......................................................... 244 Motion of a P article Thrown at an Angle to the Horizon in a U niform G ra v ita tio n a l Fie ld ...................................................................245

Chapter 109. 110. 111. 112. 113. 114. 115. 116.

17. D ifle re n tia l Equations o l Motion for a P article and Their Integration

18. General Theorems of P article Dynamics

Momentum and K in e tic Energy of aP a r t i c l e .................................... 248 Impulse of a F o r c e ...................................................................................... 249 Thecrim of the Change in the Momentum of a P article . . . . 250 W ork Done by a Force. P o w e r ...............................................................251 Examples of C alcu latio n of W o r k ...........................................................254 Theorem of the Change in the K in e tic Energy of a P article. . . 256 Solution of P r o b li m s .................................................................................. 258 Theorem c l the Change in the A ng u lar Momentum of a P article (the P rin cip le of M o m e n t s ) ...................................................................... 264

Chapter

19. Constrained M otion of a P a rticle and D 'A le m b ert’ s Principle

117. Equations of M otion of a P article A long a G iven Fixed Curve 268 11 8 -X H e rm in a lio n of the Reactions of C o n s t r a in t s ...............................270 D 'A h m b e rl’s P r in c ip le ...............................................................................272 Chapter

20. Relative Motion of a P article

120. Equaticns of R elative M otion and Rest of a P a r t i c l e ...................... 275 121. E ffe ct of the R otation of Hie E arth on the E q u ilib riu m and M otion of B o d ie s ...........................................................................................278 122. Deflection of a F a llin g P article from the V ertical by the E arth's R o t a t io n ...........................................................................................................281 C h a p L e r 21. V ibration of a P article Free H arm onic M otion . . . 4#4. The Sim ple Pendulum . . . 125. Damped V ib ra tio n s . . . 126. Forced V ibrations. Resonance

. , . ,

284 268 289 291

C h a p t e r 22. Motion of a Body In the Earth's G ravitational Field 127. M otion of a P article Thrown al an Angle to (he Horizon In the .................................................................. 299 E a rth ’s G ra v ita tio n a l F ie ld 128. A r tific ia l E arth S atellites. E llip tic a l P a t h s .......................................304

Contents

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P A R T iV . DYNAMICS OF A SYSTEM AND A R IG ID BODY C h a p t e r 23, Introduction to the Dynamics of a System. Moments nl Inertia ol Rigid Bodies 129. Mechanical Systems. External and In te rn a l Forces .......................3 g 130. Mass ol a System. Centre ol Mass . . . . . . . . . • • • • • 131. Moment ol in ertia ol a Body About an A xis. Radius ol G yra tio n J iu 132 Moments ol Inertia ol Sonic’ Homogeneous B o d ie s .................... • 3H 133. Moments ol Inertia ol a Body About Parallel Axes.The ParallelAxis (Huygens’ ) Theorem .......................................................................... 3,3 C h a p t e r 24. Theorem ol the Motion of the Centre of Mass of a System 131. The D ilfe re n lia l Equations ol Motion ol a S y s te m ...........................315 135. Theorem ol the Motion of Centre of M a s s ...........................................3|b 136. The Law' ol Conservation ol M otion ol Centre ol Mass . . . 317 137. Solution ol P ro b le m s ................................................................................ C h a p t e r 25. Theorem ol the Change in the Linear Momentum ol a System 139. Linear Momentum ol a S y s te m .............................................................. 323 139. Theorem ol the Change in Linear Momentum ...................................324 HO. The Law ol Conservation ol Linear Momentum ...............................325 141. S olulion ol P ro b le m s .................................................................................. 326 H2. Bodies Having V ariable Mass. M otion ol a R o c h e t .......................329 C h a p t e r 26. Theorem ol (he Change In the Angular Momentum of a System 143. Total A ngular Momentum ol a S y s t e m ...............................................332 144. Theorem of the Change in Ihe Angular Momentum of a System (Ihe P rin cip le of M o m e n t s ) .......................................................................333 145. The Law of Conservation of the Total A ng u lar Mom entum . . . 334 146. Solulion of P ro b le m s ...................................................................................337 C h a p t e r 27. Theorem o l the Change In the K in elic Energy of a System 147. K in e lic Energy of a System ................................................................. 339 146. Theorem of Ihe Change in (lie K in e tic Energy of a System . . . 344 149. Some Cases of Com putation ol W o r k ................................................... 316 150. Solution of P r o b le m s ...................................................................................3t8 151. Fie ld of Force. P otential E n e r g y ........................................................... 353 162. The Law ol Conservation ol MechanicalE n e r g y .................................355 C h a p t e r 26. Some Cases o f Rigid-Body Motion 153. Rolatlon of a Rigid Body .......................................................................355 154. The Compound P e n d u lu m ...........................................................................359 155. Determ ination of Moments ol Inertia by E xperim ent ................... 3>1 156. Plane Atotion of a R ig id B o d y ............................................................... 351 157. Approximate Theory of Gyroscopic A ction ....................................... 368 C h a p t e r 29. D’ Alembert's P rinciple. Forces Acting on the A xis o l a yr Rotating Body V d w . D 'A lem bert's P rin cip le lo r a System ................................................. 373 159. The Principal Vector and the P rin cip a l Moment of the Ine rt ia Forces of a R ig id Body ............................................. 374 160. S olulion of P r o b le m s ........................................ 376

Ctiti tents

101. D yn am ica l Pressures on the A xis t il a R o tatin g B o d y ................ 380 162. The P rin cip al Axes of Inertia of a Body. D ynam ic Balancing of M a s s e s ............................................................................................................ 382 Ch a p I er 163. 16-1. 105. 106. 107.

30. The P rinciple of V irtu al W ork and Ihe General Equation of Dynamics V irtu a l Displacements of a System. Degrees ol Freedom . , . 386 Idea] C o n s tr a in ts ........................................................................................ 388 The P rin cip le of V irtu a l W o r k ............................................................ 368 S olution of P ro b le m s ........................................................................... « 390 The General Equation of D y n a m ic s ................................................... * 395

C h a p t e r 31. The Theory of Impact ................ 106. The Fundam ental E quation of Ihe Theory of Impact 169. General Theorems of Ihe Theory of I m p a c t .................................... 170. C oefficient of R e s t it u t io n ........................................................................ 171. Impact of a Body Against a Fixed O b s ta c le ........................... . . 172. D irect Central Impact of Two Bodies (Impact of Spheres) . . . 173. Loss of K in e tic Energy in P erfectly Inelastic Impact. C arnot’s T h e o r e m ........................................................................................................ 174. Im pact w ith a R o tatin g Body ............................................................ Name Index

398 400 401 403 405 407 409

........................................................................................................ 414

Subject I n d e x ........................................................................................................ 414

PREFACE Tlii> book is designed Tor students of higher ami secondary technical schools. II (reals ol the basic methods o( theoretical mechanics and their spheres o( application. It also briefly examines motion in a gra vitatio na l Held (a rtific ia l satellites and e llip tica l trajectories), the motion ol bodies w ith variable mass (c.g.. rockets), (he elementary theory ol gyroscopic motion, and other topics. These questions are ol such importance today that no course ol mechanics, even a short one. can neglect them altogether. The organisation of this book is based on Ihe profound conviction, borne out by many years o l experience, that the best way of presenting study material, especially when it is contained in a short course, is to proceed from the particular to the general. Accordingly, in this book, plane statics comes before three-dimensional statics, particle dynamics before systems dynamics, rectilinear motion before curvilinear motion, etc. Such an arrangement helps the student to understand and digest the material belter and faster, and the teaching process itself is made more graphic and consistent. Alongside of Ihe geometrical and analytical methods of mechanics ttie book makes wide use of Die veclor method as one of the main generally accepted methods which, furthermore, possesses a number of indisputable advantages. As a rule, however, only those vector operations are used which are s im ila r to corresponding operations w ith scalar quantities and which do not require addi­ tional acquaintance with many new concepts. Considerable space—more than one-third of the total volume of the book— is given to examples and worked problems. They were chosen w ith an eye to ensure a clear comprehension or Ihe relevant mechanical phenomena, and they embrace a ll (lie main types of problems and methods of problem solution. The problem solutions contain instructions designed to assist the student in his independent *work on the course. In (his respect, the book should prove useful to students from many Helds interested in advancing the ir knowledge of the subject.

INTRODUCTION The progress of technology confronts the engineer w ith a wide varie ty of problems connected wi t h structural design (buildings, bridges, canals, dams. etc.), the design, manufacture and opera­ tion of various machines, motors and means of locomotion, such as automobiles, steam engines, ships, aircraft and rockets. Despite the diversity of problems that arise, (heir solution, at least in part, is based on certain general physical principles common to all of them , nam ely, the laws governing the m otion and equ ilib­ rium of m aterial todies. The science which treats ol the general laws of m otion and eq u ilib riu m of m aterial bodies and of th e ir resulting mutual in­ teractions is called theoretical mechanics. This science constitutes one of the scientific bedrocks of modern engineering*. By motion in mechanics wc mean any change in the relative positions of m aterial bodies in space which occurs in the course of time. By mechanical interaction between bodies is meant such reciprocal action which changes or tends to change (be state of m otion or the shape of the bodies involved (deformation). The physical measure of such mechanical interaction is called force. Theoretical mechanics is p rim a rily concerned w ith the general laws of m otion and equ ilibrium of m aterial bodies under the ac­ tion of forces to which they are subjected. According to the nature of the problems treated, mechanics is divided into statics, kinematics, and dynamics. Statics studies the laws of composition of forces and the conditions of equilibrium of m aterial bodies subjected to the action of forces. Kinematics deals w ith the general geometrical properties of the m otion of bodies. F in a lly , dynamics studies the laws of m otion of m aterial bodies under (he action of forces. • Mechanic-, in Ihe broa which has the meaning of “slruciure", "m achine'', “device". * * Subsequent scientific developments have revealed (hat a( velocities approaching that of lig h t the motion of bodies is governed by the mechanical laws of the theory of re la tiv ity , w h ile the motion ot "elem entary" particles (clecIrons, positrons, etc.) is described by the laws of quantum mechanics. These discoveries, however, o nly served to delinc more accurately the spheres of application of classical mechanics and reaffirm tlie v a lid ity o f its laws lo r the motion of all bodies oilier than “elem entary" particles at a ll velocities not approaching the velocity ol lig h t, i.e., for molions w ith which engineering and celestial mechanics are p rim a rily concerned.

12

In tro d u c tio n

In the I8 th century, analytical methods, i.e.. methods based on the application of dilTcrenlial and integral calculus, began to develop rapidly in mechanics. The methods of solving problems of particle and rigid body dynamics by the integration of differ­ ential equations were elaborated by the great m athem atician and mechanic Leonhard Euler (1707-1783). Among the other m a­ jor contributions to the progress of mechanics were the works of the outstanding French scientists Jean d ’Alem bert (1717-1783), who enunciated his famous principle for solving problems of dy­ namics, and Joseph Lagrange (1736-1813), who evolved the gener­ al analytical method of solving problems of dynamics on the basis of d'A le m b ert’s principle and the principle of virtu a l work. Today an a lytic al methods predominate in solving problems of dy­ namics. Kinem atics emerged as a special branch of mechanics only in the first half of the 19th century under pressure from the grow­ ing machine-building industry. Today kinematics is essential in studying the m otion of machines and mechanisms. In Russia, the study of mechanics was greatly influenced by the works of the great Russian scientist and thinker Atikhail Lomonosov (1711-1765) and of Leonhard Euler, who for many years lived and worked in St. Petersburg. Prominent among the galaxy of Russian scientists who contributed to the develop­ ment of different divisions of theoretical mechanics were M . V. Ostrogradsky (1801-1861), the author of a number of im ­ portant studies in an a lytic al methods of problem solution in mechanics; P. L. Chebyshev (1821-1894), who started a new school in the study of the motion of mechanisms; S. V. K o va­ levskaya (1850-1891), who solved one of the most difficult prob­ lems of rigid body dynamics; A. M . Lyapunov (1857-1918), who elaborated new methods of studying the sta b ility of motion; I. V. Meshchersky (1859-1935), who laid the foundations of the mechanics of bodies of variable mass; K- E. Tsiolkovsky (1857-1935), who made a number of fundamental discoveries in the theory of jet propulsion; and A. N. K rylov (1863-1945), who elaborated the theory of vessels and contributed much to the development of Ihe theory of gyroscopic instruments. O f tremendous importance to Ihe further study of mechanics were the works of N. E. Zhukovsky (1847-1921), the “ Father of Russian aviation”, and his best pupil S. A. Chaplygin (1869-1942). Zhukovsky’s special contribution was in the field of applying methods of mechanics to the solution of actual engineering prob­ lems. Zhukovsky's ideas have greatly influenced the teaching of theoretical mechanics in Soviet higher technical educational estab­ lishments.

PART

ONE

STATICS OF RIGID BODIES C h a p te r

1

BASIC CONCEPTS AN D P R IN C IPLE S I. The Subject of Statics. Statics is the branch of mechanics which studies the laws of composition of forces and the condi­ tions of equilibrium of m aterial bodies under the action of forces. Equilibrium is the state of rest of a body relative to other m aterial bodies. If the frame of reference relative to which a body is in equilibrium can be treated as fixed, the given body is said to be in absolute equilibrium ; otherwise it is in relative equilibrium . In statics we shall study only absolute equilibrium . In actual engineering problems equilibrium relative to the earth or to bodies rigidly connected w ith the earth is treated as abso­ lute equilibrium . The justification of this premise w ill be found in the course of dynamics, where the concept of absolute eq u ilib ­ rium w ill be defined more stric tly . And there, loo, we shall exam ine the concept of relative equilibrium . Conditions of equilibrium depend on w hether a given body is solid, liquid or gaseous. The equilibrium of liquids and gases is studied in the courses of hydrostatics and aerostatics respectively. General mechanics deals essentially w ilh the equilibrium of solids. All solid bodies change their shape to a certain extent when subjected to external forces. This is known as deform ation. The amount of deformation depends on the m aterial, shape and dimensions of the body and the acting forces. In order to ensure the necessary strength of engineering structures and elements, the m aterial and dimensions of various parts are chosen in such a way that the deformation under specified loads would remain to l­ erably sm all*. This makes it possible, in studying the general conditions of equilibrium , to treat solid bodies as undeformable or absolutely rigid, ignoring the small deformations that a ctu ally occur. An absolutely rig id body is said to be one in which the For example, the material and the diameter of rods in various structures are so chosen (hat, under the working loads, they would extend (or contract) oy no more than one-thousandth of their original length. Deformations of a similar order are tolerated in bending, torsion, etc.

1I

It a lic C on ce p t and P rin c ip le «

|Ch. 1

distance between any pair of particles is always constant. In solv­ ing problems of slutics. in this book we shall always consider bodies as absolutely rigid, and w ill simply refer to them as rigid bodies. It w ill be shown at the end of § 3 that the laws of equilibrium of absolutely rigid bodies can be applied not only to solid bodies w ith relatively small deform ation, but to any deformable bodies a> w ell. Thus the sphere of application of rig* id body statics is extrem ely wide. Deform ation is of great importance in calculating the strength of engineeringstruclures and machine parts. These questions are studied in the cours-s of strength of m aterials and theory of elasticity. For a rigid body to be in equilibrium (at rest) when subject­ ed to the action of a system of forces, the system must satisfy certain condition* of equilibrium . The determ ination of these con­ ditions is one of the principal problems of statics. In order to find out the equilibrium conditions for various force systems and to solve other problems of mechanics one must know the prin­ ciples of the composition, or addition, of forces acting on a rigid body, the principles of replacing one force system by another and. p a rtic u la rly , the principles of reducing a given force system to as simple a form as possible. Accordingly, statics of rigid bodies treats of two basic problems: ( I ) the composition of forces and reduction of force systems acting on rigid bodies to as simple a form as possible, and (2) determ ination of the conditions for the equilibrium of force systems acting on rigid bodies. The problems of statics m ay be solved either by geometrical constructions (the graphical method) or by m athem atical calculus (the analytical method). The present course discusses both m eth­ ods, but it should always be borne in mind that geometrical constructions are of special importance in solving problems of mechanics. 2. Force. The state of equilibrium or m otion of a given body depends on its mechanical interactions w ith other bodies, i.e ., on the loads, attractions or repulsions it experiences as a result of such interactions. In mechanics, the quanlitative meas­ ure of the mechanical interaction of material bodies is called force. Q uantities employed in mechanics are either scalar, i.e., pos­ sessing magnitude alone, or vector, i.e ., quantities which besides magnitude are also characterised by direction in space. Force is a vector quantity. Its action on a body is character­ ised by (1) its magnitude, (2) its direction, and (3) its point of application. The magnitude of a force is expressed in terms of a standard force accepted as a unit. In statics, the unit of force is the

15

F -rre

Sec. 2]

kilogram (1 k g * ) . S ialic forces are measured w ith dynamometers, which are described in the course of physics. The direction and the point of application of a force depend on the nature of the interaction between the given bodies and their respective posi­ tions. The force of gravity acting on a body, for exam ple, is always directed vertically downwards; the forces w ith which two smooth contacting spheres act on each other are normal to both their surfaces at the points of contact and are applied at those points, etc. Force is represented graphically by a directed straight line segment w ith an arrowhead. The length of the line (AH in Fig. I) denoles the magnitude of the force to some scale, the direction of the line shows the direction of the force, its initial point (point A in Fig. 1) usually indicating the point of application of the force, though sometimes it may be more convenient to depict a force as ‘'pushing'' a body w ith its tip (as in Fig. 4c). The line D E along which Hg. 1. the force is directed is called the line of action of the force. W e shall denote force, as all vector quantities bv a_boldiace-lype letter (F ) or by overlined standard-type letters (.48 ). The absolute value of a torce is represented by the symbol | F (two vertical lines •■Itanking" a vector) or sim plv by a standard-type letter (F). (In handwriting overlined letters are used ) VVe shall call any set of (orces acting on a rigid body a force system. W e shall also use the follow ing definitions’ 1. A body not connected w ith other bodies and which from caM ed'V/ree°bol/g1

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I*

1 '° 3 f° rCe SyS,e'" iS 3t

3. If a force system acting on a free rigid body can be replaced ? L i " ° _ her. (orce sysle!? "'ithoul disturbing the body's In it ia l condition of rest or motion, the twoo systems systeilSfare M id are sa ‘ ‘°to be ivalent to a sir «

o

f ,0

/le flA a

—f

_

,

,

, loms

m

”.

a

in

* nK

K

If,

Basic Concepts a nd P rin cip le s

(Ch.

5. Forces acting on a rigid body can be divided into two groups: external and internal forces. External forces represent the action of other m aterial bodies on the particles of a given body. Inter­ nal forces are those w ith which the particles of a given body act on each other. 6. A force applied to one point of a body is called a concen­ trated force. Forces acling on all the points of a given volum e or given area of a body are called distributed forces. A concentrated force is a purely notional concept, insofar as it is actually impossible to app ly a force to a single point of a body. Forces treated in mechanics as concentrated are in fact the resultants of corresponding systems of distributed forces. Thus, the force of g ravity acting on a rigid body, as conven­ tio n a lly treated in mechanics, is the resultant of the g ravitation­ al forces acting on its particles. The line of action of this resul­ tant force passes through the body’s centre of g r a v ity * . 3. Fundamental Principles. A ll theorems and equations in statics are deduced from a few fundamental principles, which are accepted w ithout m athem atical proof, and are known as the prin­ ciples. or axioms, of statics. The principles of statics represent general formulations obtained as a result of ^ a vast number of experiments w ith, and obserA ^ a/ \ ' valions of, the equilibrium and motion of bodf >7 ' / ies and which, furtherm ore, have been con/ ,1 / sistently confirmed by actual experience. Some of these principles are corollaries of the fundam enlal laws of mechanics, which w ill be examined in the course of dynamics. Fig- 2. 1 s t P r i n c i p l e . .4 free rigid body subjected to the action of two forces can be in equilibrium if. and only if. the two forces are equal in magnitude (F , = F g), colt inear. and opposite in direction (Fig . 2). The 1st principle defines the simplest balanced force system, since we know from experience that a free body subjected to the action of a single force cannot be in equ ilibrium . 2 n d P r i n c i p l e . The action of a given force system on a rig ­ id body remains unchanged if another balanced force system is added to, or subtracted from, the original system. This principle establishes that two force systems differing from each other by a balanced system are equivalent. * The d eterm ination of Chapter 6. Meanwhile it a centre of symmetry (e.g.. centre of g ra vity is in (he

the centre of g ra vity of bodies w ill be discussed in may be noted tlia t if a homogeneous body lias a rectangular beam, a cylin d er, a sphere, etc.) its centre of symmetry.

Sec. 3|

Fundamental Principles

17

Corollary of the Isl and 2nd P r i nc i p i e s . The point of application of a force acting on a rigid body can be trailsferred lo any other point on tine of action of the force without altering its effect This is known as the principle of Iransm issibility. Consider a rigid body w ilh a force F applied al a point A (Fig. 3). Now take an arb itrary point B on the line of action of the lorce and apply to that point two equal and opposite forces F , and F a such that F t = F and F t — — F . This operation does not affect the action of F on the body. From the 1st principle it follows that forces F and F t also form a balanced system and thus cancel each o th e r * , leaving force F x. equal lo F in magnitude and direction, w ith the point of application shifted lo point B. Thus the vector denoting force F can be regarded as applied a l any point along the line of action (such a vector is called a sliding vector). This principle holds good only for forces acting on absolutely rigid bodies. In engineering problems it can be used only when we determine the conditions of equilibrium o fa s tru c lu re w ilh -. F, out taking into account the ri internal stresses experienced A c 3 — » by its parts. ~ y " > For example, the rod A B t ' * 9 in Fig. 4a w ill be in eqyilib. » r-rr-- ----------- ------------- u — — a rium if F x= F f . It w ill rerj main in equilibrium if both Fig. 4. forces are transferred to point C (Fig. 4b) or if force F x is transferred to point B and force F , to point A (Fig. 4c). The stresses in the rod. however, differ in each case. In the first case the rod is in tension, in the second there is no internal stress, and in the third it is in compres­ s io n **. Consequently, the principle of transm issibility cannot be employed in determ ining internal stresses. * Wc shall denote cancelled or transferred forces in diagrams by a dash across (lie respective vectors. ** For the rod to be slrcld icd (or compressed) w ith a f6rce F x. the lorce should be applied al one end of Die rod. w ill* “the other end supported rig id ly or constrained by a force F t ~ — F x, as in Fig. 4. The tension (or thrust) is Ine same in both cases and is equal to F x. and not lo 2F. as is sometimes erroneously supposed.

Basic C fiice p i* and P rin cip le s

(Ch. 1

3 rd Principle (the Parallelogram Law) . Tivo {frees applied at one point of a body have as their resultant a force Qpplied at ihe same point and represented by the diagonal of a par­ allelogram constructed u:ith the tivo given forces as its sides. Vector /?, which is (he diagonal of the parallelogram with vectors F t and F t as its sides (Fig. 5). is called the geometrical sum of the vectors F x and F t : R = F, + F t. Hence, the 3rd principle can also be form ulated as follows: The resultant of tivo forces applied a l one point of a body is the geometrical (vector) sum of those forces and is applied at that Fig. 5. point. It is im portant to discrim inate between the concepts of a sum of forces and their resultant. Consider, say. a body to which two forces F t and F t are applied at points A and B (Fig. 6). The sum of the two forces can be found by laying them off from any point in the body. In Fig. 6. force Q is the sum of forces F x and F t (Q — F y -^ -F f, as the diagonal of the corresponding parallelog ram ). But Q is not the resultant of the two for­ ces. for it w ill be readily observed that Q alone cannot replace the action of F t and F t on the body. Moreover, forces F , and F t , as we shall learn la te r in § 47, have no resultant at a ll. 4 th P r i n c i p l e . To any action of one material body on another there is altvays an equal und op­ posite reaction, or, reaction is always equal to action. The law of action and reaction is one of the fundam ental laws of mechanics. It follows from it th a t when a body A acts on a body B w ith a force F , body B sim ul­ taneously acts on body A w ith a force F " Fig. 7. equal in magnitude, collinear w ith , and op­ posite in sense to force F (F ' = — F ) (Fig. 7). Forces F and F . however, do not form a bajanced system as they are applied to d i f f e r c n t bodies. I n t e r n a l F o r c e s . It lollows from the 4lh principle that any two particles of a rigid body act on each other w ith forces

Spc. -I|

Cm ys'cm is in equilibrium , the state of equilibrium w ill nol be disturbed if the body solidifies (becomes r ir id ). The idea expressed in this principle is self-evidi.nl. for, obvi­ ously, Ihe equilibrium of a chain w ill not be disturbed if its links are welded together, and a flexible string w ill remain in equilib­ rium if it turns into a bent rigid rod. Since the same force sys­ tem acts on a reposing body before and after solidification, (lie 5th principle can also be formulated as follows: I f a freely deform­ able body is in equilibrium . Ihe forces acting on il satisfy (he conditions for Ihe equilibrium of a rigid body; for a deformable body, however, the*e conditions, though necessary, may not be suf­ ficient. For example, for a flexible string w ith two forces applied at its ends to be in equilibrium , Ihe same condilions are necessary as for a rigid rod (the forces must be of equal m agnitude and directed along the siring in opposite directions). These conditions, though, are not sufficient. For the siring to be in equilibrium the forces must be lensile, i.c., they must stretch the body as shown in Fig. 4a. The principle of solidification is widely employed in engineer­ ing problems. It makes it possible to determine equilibrium conditions by treating a flexible body-(a b elt, cable, chain, etc.), or a collapsible structure as a rigid body and to apply to it the methods of rigid-body statics. If the equations obtained for the solution of a problem prove insufficient, additional equations must be derived which take into account either the conditions for the equilibrium of separate parts of the given structure or their Reformation (problems requiring consideration of deformation are studied in the course of strength of m aterials). -^ C o n s tra in ts and Their Reactions. As has been defined above, aHfody nol connected w ith other bodies and capable of displace­ ment in any direction is called a f r e e b o d y (e.g., a balloon floating in the air). A body whose displacement in space is re­ stricted by other bodies, either connected to or in contact w ith

20

B asic Concepts and P rin cip le s

ICh. 1

it, is called a c o n s t r a i n e d b o d y . We shall call a constraint anything that restricts the displacement of a given body in space. Exam ples of constrained bodies are a weight lyin g on a table or a door swinging on its hinges. The constraints in these cases are the surface of the table, which prevents the weight from fa ll­ ing, and the hinges, which prevent the door from sagging from fts jamb. r A body acted upon by a force or forces whose displacement Is restricted by a constraint acts on th at constraint w ith a force which is custom arily called the load or pressure acting on that constraint. At the same tim e, according to the 4lh principle, the constraint reacts w ith a force of the same magnitude and opposite sense. The force with which a constraint acts on a body, thereby restricting its displacement, is called the force of reaction of the constraint, (force of constraint), or simply the reaction of the constraint. ) The procedure w ill be to call all forces which are not the reactions of constraints (e.g., gravitatio n al forces) applied or active forces. C haracteristic of active forces is th at their magni­ tude and direction do not depend on the other forces acting on a given body. The difference between a force of constraint and an active force is that the magnitude of the former always depends on the active forces and is not therefore im m ediately apparent: if there are no applied forces acting on a body, the forces of constraint vanish. The reactions of constraints are determined by solving corresponding problems of statics. The reaction of a conslraint points away from the direction in which the given constraint prevents a body's displacement. If a constraint prevents the dis­ placement of a body in several directions, the sense of the reac­ tions is not im m ediately apparent and has to be found by solving the problem in hand. The correct determ in ation-of the direction of forces of constraint is of great importance in solving problems of statics. Let us therefore consider the direction of the forces of constraint (reac­ tions) of some common types of constraints (more examples are given in § 26). 1. S m o o t h P l a n e ( Sur f a c e ) or S u p p o r t . A smooth surface is one whose friction cun be neglected in the first approx­ im ation. Such a surface prevents the displacement of a body perpendicular (norm al) to both contacting surfaces at their point of contact (F ig . 8a)*. Therelore, the reaction N of a smooth sur* * In Figs 8-11. Hie applied forces acting on Hie bodies are not shown. In Hie cases illustrated in Figs 8 and 9, the reactions act in the indicated directions regardless of the applied (orccs and irrespective of whether the bod­ ies are at rest or in motion.

S«c. 41

21

Constraint* and Their React ion*

contacting surfaces is a p normal to the other surface.

Fig. 'J.

Fig. 8.

2. S t r i n g . A constraint provided by a flexible inexlensible siring (Fig. 9) prevents a body M from receding from the point of suspension of the string in the direction A M . The reaction T of the siring is thus directed along the string towards the point of suspension. 3. C y l i n d r i c a l P i n ( B e a r i n g ) . W hen two bodies are joined by means of a pin passing through holes in them the connection is called a joint or hinge. The axial line of the pin 2

Fig. 10.

is called the axis of the joint. Body A B in Fig. 10a is hinged to support D and can rotate freely about the axis of the joint (i.e., in the plane of the diagram ); at the same tim e, po in t A cannot be displaced in any direction perpendicular to the axis. Thus, the reaction R of a pin can have any direction in the plane perpendicular to the axis of the jo in t (plane Axy in Fig. 10a). In th is case neither the magnitude R nor the direction (angle a ) of force R are imm ediately apparent.

22

B ti'ic Concept* an

|Ch. 1

4. B a I l-a n d-S o c k e I a n d S t e p B e a r i n g . This type of constraint prevents displacement in any direction. Examples of such a constraint is a ball-pivot w ith which a camera is attached to a tripod (Fig . 106) and a slep bearing (Fig. 10c). The reac­ tion R of a hall-and-socket joint or a pivot can have any direction in space. Neither its magnitude R nor its angles w ith the x, y, and z axes are im m ediately apparent. 5. R o d . If a thin rod w ith two force applied at its tips is in equilibrium , the forces, according to the 1st principle, must be collincar and directed along the axis of the rod (see Figs 4a and 4c). Conse­ quently. a rod subjected to forces applied £ at its lips, where the weight of the rod as Fig. ||. compared w ith the magnitude of the forces can be neglected, can be only under tension or under thrust. Hence, if in a structure such a rod AB is used as a constraint (Fig. 11) the reaction N w ill be directed along its axis *. 5. Axiom of Constraints. The equilibrium of constrained bod­ ies is studied in statics on the basis of the following axiom: A n y constrained body can be treated as a free body detached from its constraints, provided the latter are represented by their reactions.

For exam ple, the beam A B of weight P in Fig. surface O E , support D, and cable KO are constraints, ed as a free body (Fig . 126) in equilibrium under the given force P and the reactions N A, N D> and straints. The magnitudes of these reactions, which

12o. for which can be regard­ the action of T of the con­ are unknown,

• S tric tly speaking, this is po'sible o n ly if the ends o' the rod arc fas­ tened w ith pins. A ctu a lly, however, the reaction of a rod may he regarded as directed along its axis even if the ends arc welded (e.g., in trusses composed of triangles).

Composition of Concurrent fi-rc a

Sec. 6)

23

ran be determined Iroin the conditions for the equilibrium of the forces acting on the now free body. This is Ihe basic method of solving problems of slalics. The determination of the reactions of constraints is of practical importance because, from the 4th principle, if we know them , we shall know the loads acting on the constraints, i.e .. the bas­ ic information necessary to calculate the strength of structural elements. Chapter

2

CONCURRENT FORCE SYSTEM S 6. Geometrical Method of Composition of Concurrent Forces. In studying statics we shall proceed from simple systems to complex systems. Let us commence, then, w ith systems of concur­ rent forces. Forces whose lines of action intersect at one point are called concurrent (see Fig. 16a). A system of concurrent forces acting on a rigid body can be replaced by an equivalent force system applied at the same point (A in Fig. 16a). The problem of determining the resultant of concurrent forces is reduced, according to the 3rd principle of statics, to the com­ position of the given forces. 1) C o in p o s i t i o n of T w o F o r c e s . The resultant R of two concurrent forces F x and F t is determined either by the par­ allelogram rule (Fig. 13a) or by constructing a force triangle (Fig. 13b), which is in fact one-half of the p aralle lo ­ corresponding parallelogram. To construct a force triangle lay off a vector denoting one of the forces Fig . 13. from an arbitrary point A t, and from its lip lay olT a vector denoting the second force. The vector R joining the in ilia l point of the first vector with the term inal point of the second vector denotes the resultant force. The magnitude of the resultant is expressed by the form ula R* = F*i + F \ — 2 F tF t cos (180° — a ), where a is the angle between th e .tw o forces. Hence, ^

^

+

+

cos a .

(1)

21

Concurrent Force Systems

|Ch. 2

The angles |3 and y which the resullanl makes w ith the com­ ponent forces can be determined by the law of sines. As sin (1 8 0 ' — a)=^sincc, we have Fj_ _ _ _Fj_ _ _ R s i l l y ' sin ft stna

( 2)

If |he point of intersection o f (lie forces F x and F t is outside the body (Fig. H u), I lie resultant can lie found by m e n ta lly extending the body as shown hy the dallied lin o . Now we can transfer the forces to point 0 . con­ struct a force parallelogram, and apply the resultant force A at any poin t C of the body on Die lin e of action of R. If Ihc lines of action ol the component forces F x and F t intersect outside the dia­ gram (Fig. 14b). a point through which their resultant pa->ses can be found by applying at points A and B two balanced forces P, and P t (P, = — P,) directed along AB. According to the 2nd prin cip le, this does not afTccl I lie action ol forces F , and F t on the body. B y separately compounding F , and P t and F t and Pt we obtain a force system equivalent to (he F ,. F t force system and consisting of (wo forces whose linos ol action intersect at C. Consequently, the resultant R also passes through that point. The vector R can now be determined by constructing a force triangle (Fig. 146).

2) C o m p o s i t i o n of T h r e e N o n - C o p l a n a r Forces. The resultant R of three concurrent non-coplanar forces F t , F t , F t is represented by the diagonal of a parallelepiped w ith the given forces for ils edges (the parallelepiped laid). This rule can be verified by successively applying the parallelogram law (F ig . 15). 3) C o m p o s i t i o n o f a S y s t e m of F o r c e s . By consecutively applying the par­ allelogram law we can draw the follow ing conclusion: The resultant of any number of concurrent forces is the geometrical sum of those Kig. isforces and is applied at their point of intersec­ tion. The geometrical sum R of all the forces of a system is called the principal vector of that system. The principal vector of a force system can be determ ined either by suc­ cessively compounding the forces of the system according to the parallelogram law . or by constructing a force polygon. The la t­ ter method is simpler and more convenient. In order to find Ihe resultant of forces F (. F t . F ............ F „ (Fig. 16a) lay_ofT to scale from an arb itrary point 0 (Fig . 16£>) a vector Oa denoting

Sec. 7)

25

RetolutifM of Fora

force F t. Now from poinl u_1ay off vector ab denoting force F ,, from poinl b lay off vector be denoting force F ,. and so on; from the lip m of the penultim ate vector lay ofT vector mu denoting force F„. Vector On = R . laid off from the in itia l poinl of the first vector to the lip of the last vector, represents the geomet­ rical sum. or the principal vector, of the component forces: y? = F , + /rI + - - - - l - / rfl

or

R = 'Z F „ .

(3)

The magnitude and direction of R do not depend on the order in which the vectors are laid o(T. It w ill be noted that the construction carried out is in effect a consecutive applica­ tion of the triangle law. The figure constructed in Fig. 166 is called a force polygon (or. generally speak­ ing, a vector polygon). Thus. the geometrical sum. or the principal vector, of a set of forces is represented by the clos­ ing side of a force polygon constructed with the given forces as its sides (the polygon law ). In constructing a vector polygon, care should be taken to arrange all the component vectors in one sense along the periphery of the polygon, w ith vector R being drawn in the opposite sense. If the force R found by constructing a force polygon is applied at point A of the body in Fig. 16a, it w ill replace the action of all the given forces; j.e ., it is their resultant. 7 7. Resolution of Forces. T o resolve a force into two or more components means to replace it by a force system whose result­ ant is the original force. This problem is indeterm inate and can * be solved uniquely only iT additional conditions are stated. Tw o cases are of p a rtic u la r interest; I) Resolution of a Force I n t o T w o C o i n p o n e n t s of G i v e n D if h . >7. L ' . . 1.!.? "Consider the *' resolution of force F in „ , . ,. , Fig. 17 into tw o compo­ nents whose lines of action are parallel to A B and A D (th e lorces and the lines are coplanar). The task is to construct a par-

20

Concurrent Force System*

[Cl. 2

allelogram w ith fo rc e /7 a? its diagonal and ils sides respectively parallel lo AB and A D . The problem is solved by drawing through the beginning and Up of F lines parallel to A B and A D . Forces P and Q are the respective components, as P - \ - Q = F . The resolution can also be carried out by applying the t r i ­ angle law (Fig . 17b ). For this, the force F is laid ofT from an a rb itra ry point and lines parallel lo A B and A D are drawn through ils in itia l and term inal points to th e ir point of inter­ section. Forces P and Q replace F if applied at point A or at any other point along the line of action of F . 2) R e s o l u t i o n of a F o r c e I n t o T h r e e C o m p o n e n t s o f G i v e n D i r e c t i o n . If the given directions are not coplanar the problem is determ inate and is reduced to the construc­ tion of a parallelepiped w ith the given force F as its diagonal and its edges parallel to the given directions (see F ig . 15). The student is invited to consider for himself the resolution of a given force F into tw o components P and Q coplanar w ith F if th eir magnitudes are given and P - \ - Q ^ F . The problem has tw o solutions. S o l u t i o n of P r o b l e m s . The method of resolution of forces is useful in determ ining the pressure on constraints induced by applied forces. Loads acting on rigid constraints are deter­ mined by resolving the given forces along the dicpclions of the reactions of the constraints as, according to the 4th principle, a force acting on a constraint and ils reaction have the same line of action. It follows, then, th at this method can be applied only if the directions of the reactions of the respective con­ straints are im m ediately apparent. Problem I. Members AC and BC ol the bracket in Fie. 18a are joined together and attached to (lie wall w ith pins. Neglecting me weight of the members, determine (he thrust in BC if (lie suspended load weighs P, £ B A C = 9 f\ \ and £ A B C = a. S o l u t i o n . Force P acts on both members, and the reactions are directed along them. The unknown thrust is determined by applying force P at point C and resolving it along AD and DC. Component 5, is the required force. From triangle CDE we obtain:

s 1, = -cos ? -a . From the same tria n gle we find that member AC is under a tension ol St = P Ian a. The larger the angle o, the greater the load on both members, which increases rapidl y as a approaches 9 0 For example, at P = 1 0 0 kg and a = 85=. S, = 1,150 kg and $ , = 1,140 kg. Thus, to lessen Ihe load angle a should be made smaller. , We see Irom these results d ia l a small applied force can cause very large stresses in structural elements (see also Problem 2). The reason tor this is that

R C 'o liilio n o f Force*

Sec. 7f

27

force* are compounded and resolved according lo the parallelogram law: a diag­ onal of a parallelogram can be very much smaller llia il i t ' 'ide*.. If. Ihereforc. in solving a problem you Hud Dial Ihe load* or reactions >eem loo big 3 ' compared wilh tile applied forces, this does not necessarily mean that your solution is Rf°F?nally. beware ol a mistake frequently made in applying the method of force revolution. In Problem 1 we have lo determine the force of thrust

Fig

Ifl.

acting on member BC. If we were lo apply force P at C (Fig. 18(>) and resolve it into a component