INTRODUCTION. The three general approaches to the solution of kinetics problems are:

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2 INTRODUCTION According to Newton s lw, prticle will ccelerte when it is subjected to unblnced forces. Kinetics is the study of the reltions between unblnced forces nd the resulting chnges in motion. The three generl pproches to the solution of kinetics problems re: ) Direct ppliction of Newton s lw (clled the forcemss-ccelertion method) b) Work nd energy principles c) Impulse nd momentum methods

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4 The bsic reltion between force nd ccelertion is found in Newton s second lw, the verifiction of which is entirely experimentl. Newton s second lw cn be stted s follows: If the resultnt force cting on prticle is different from zero, the prticle will hve n ccelertion proportionl to the mgnitude of this resultnt nd the ccelertion will be in the direction of this resultnt force.

5 Verifiction of Newton s Second Lw: If we subject mss prticle to the ction of single force F 1 nd mesure the ccelertion 1 of the prticle, the rtio F 1 / 1 of the mgnitudes of the force nd the ccelertion will be some number C 1. We then repet the experiment by subjecting the sme prticle to different force F nd mesuring the corresponding ccelertion. The rtio F / of the mgnitudes will gin produce number C. The experiment is repeted s mny times s desired.

6 We drw two importnt conclusions from the results of these experiments. First, the rtios of pplied force to corresponding ccelertion ll equl the sme number, provided the units used for mesurement re not chnged in the experiments. Thus, F 1 1 F... F n n C, constnt

7 We conclude tht the constnt C is mesure of some invrible property of the prticle. This property is the inerti of the prticle, which is its resistnce to the rte of chnge of velocity. For prticle of high inerti (lrge C), the ccelertion will be smll for given force F. On the other hnd, if the inerti is smll, the ccelertion will be lrge. The mss m is used s quntittive mesure of inerti, nd therefore, we my write the expression C F km where k is constnt introduced to ccount for the units used. Thus, we my express the reltion obtined from the experiments s, F km

8 where F is the mgnitude of the resultnt force cting on the prticle of mss m, nd is the mgnitude of the resulting ccelertion of the prticle. The second conclusion is tht the ccelertion is lwys in the direction of the pplied force. F km (Eqution of Motion) In SI unit system, k = 1.

9 Primry Inertil System (Asl (Birincil-Temel) Atlet (Eylemsizlik) Sistemi) Although the results of the idel experiment re obtined for mesurements mde reltive to the fixed primry inertil system, they re eqully vlid for mesurements mde with respect to ny nonrotting reference system which trnsltes with constnt velocity with respect to the primry system.

10 From reltive motion nlysis, we know tht the ccelertion mesured in system trnslting with no ccelertion is the sme s tht mesured in the primry system. Thus, Newton s second lw holds eqully well in nonccelerting system, so tht we my define n inertil system s ny system in which the eqution of motion is vlid.

11 If the idel experiment described were performed on the surfce of the erth nd ll mesurements were mde reltive to reference system ttched to the erth, the mesured results would show slight discrepncy from those predicted by the eqution of motion, becuse the mesured ccelertion would not be the correct bsolute ccelertion. The discrepncy would dispper when we introduced the corrections due to the ccelertion components of the erth.

12 These corrections re negligible for most engineering problems which involve the motions of structures nd mchines on the surfce of the erth. In such cses, the ccelertions mesured with respect to reference xes ttched to the surfce of the erth my be treted s bsolute, nd the eqution of motion my be pplied with negligible error to experiments mde on the surfce of the erth.

13 An incresing number of problems occur, prticulrly in the fields of rocket nd spcecrft design, where the ccelertion components of the erth re of primry concern. For this work it is essentil tht the fundmentl bsis of Newton s second lw be thoroughly understood nd tht the pproprite bsolute ccelertion components be employed.

14 Before 1905, the lws of Newtonin mechnics hd been verified by innumerble physicl experiments nd were considered the finl description of the motion of bodies. The concept of time, considered n bsolute quntity in Newtonin theory, received bsiclly different interprettion in the theory of reltivity nnounced by Einstein in The new concept clled for complete reformultion of the ccepted lws of mechnics.

15 The theory of reltivity ws subjected to erly ridicule, but hs been verified by experiment nd is now universlly ccepted by scientists. Although the difference between the mechnics of Newton nd Einstein is bsic, there is prcticl difference in the results given by the two theories only when velocities of the order of the speed of light (300x10 6 m/s) re encountered. Importnt problems deling with tomic nd nucler prticles, for exmple, require clcultions bsed on the theory of reltivity.

16 Eqution of Motion nd Solution of Problems When prticle of mss m is subjected to the ction of concurrent forces F 1, F, F 3,.. whose vector sum is F, the eqution of motion becomes F m The eqution of motion gives the instntneous vlue of the ccelertion corresponding to the instntneous vlues of the forces which re cting.

17 When pplying the eqution of motion to solve problems, we usully express it in sclr component form with the use of one of the coordinte systems developed in kinemtics of prticles. The choice of n pproprite coordinte system depends on the type of motion involved nd is vitl step in the formultion of ny problem.

18 Solution of Problems We encounter two types of problems: 1) The ccelertion is either specified or cn be determined directly from known kinemtic conditions. We then determine the corresponding forces which ct on the prticle by direct substitution into the eqution of motion. F m

19 Solution of Problems ) The forces cting on the prticle re specified nd we must determine the resulting motion. If the forces re constnt, the ccelertion is lso constnt nd is esily found from the eqution of motion. When the forces re functions of time, position or velocity, or combintion of them, the eqution of motion becomes differentil eqution which must be integrted to determine the velocity nd displcement.

20 Constrined nd Unconstrined Motion (Kısıtlnmış ve Serbest Hreket) Degree of Freedom (Serbestlik Derecesi) There re two physiclly distinct types of motion: The first type is unconstrined motion where the prticle is free of mechnicl guides nd follows pth determined by initil motion nd by the forces which re pplied to it from externl sources. An irplne or rocket in flight nd n electron moving in chrged field re exmples of unconstrined motion.

21 The second type is constrined motion where the pth of the prticle is prtilly or totlly determined by restrining guides. A mrble is prtilly constrined to move in the horizontl plne. A trin moving long its trck nd collr sliding long fixed shft re exmples of more fully constrined motion.

22 Some of the forces cting on prticle during constrined motion my be pplied from outside sources, nd others my be the rections on the prticle from the constrining guides. All forces, both pplied nd rective, which ct on the prticle must be ccounted for in the eqution of motion.

23 The choice of n pproprite coordinte system is frequently indicted by the number nd geometry of the constrints. Thus, if prticle is free to move in spce, s is the center of mss of the irplne or rocket in free flight, the prticle is sid to hve three degrees of freedom since three independent coordintes re required to specify its position t ny instnt.

24 All three of the sclr components of the eqution of motion would hve to be integrted to obtin the spce coordintes s function of time. The mrble sliding on the surfce hs two degrees of freedom. Collr sliding long fixed shft hs only one degree of freedom.

25 Free-Body Digrm (FBD) Serbest Cisim Diygrmı (SCD) When pplying ny of the force-mss-ccelertion equtions of motion, we must ccount correctly for ll forces cting on the prticle. The only forces which we my neglect re those whose mgnitudes re negligible compred with other forces cting, such s the forces of mutul ttrction between two prticles compred with their ttrction to celestil body such s the erth. The best wy to do this is to drw the prticle s free body digrm (FBD).

26 F F m The vector sum in mens the vector sum of ll forces cting on the prticle. Likewise, the corresponding sclr force summtion in ny one of the component directions mens the sum of the components of ll forces cting on the prticle in tht prticulr direction. The only relible wy to ccount ccurtely nd consistently for every force is to isolte the prticle under considertion from ll contcting nd influencing bodies nd replce the bodies removed by the forces they exert on the prticle. The resulting free-body digrm is the mens by which every force, known nd unknown, which cts on the prticle is represented nd thus ccount for.

27 The free body digrm serves the sme key purpose in Dynmics s it does in Sttics. This purpose is simply to estblish thoroughly relible method for the correct evlution of the resultnt of ll ctul forces cting on the prticle or body in question. In Sttics, this resultnt equls zero, wheres in Dynmics it is equted to the product of mss nd ccelertion.

28 In Sttics the resultnt equls zero F 0 wheres in Dynmics it is equted to the product of mss nd ccelertion. F m

29 Rectiliner Motion If we choose the x-direction, for exmple, s the direction of the rectiliner motion of prticle of mss m, the ccelertion in the y- nd z- directions will be zero nd the sclr components of the eqution of motion become F x m x F y 0 F z 0

30 Plne Curviliner Motion 1) Crtesin Coordintes: F 1) ) x x m x v x x Fy m y y v y y F F x Fy x y ) Norml nd Tngentil Coordintes : F 1) ) t t m t v s Fn n v m n s F F t Fn t n

31 3) Polr Coordintes : F 1) r m r ) r r r F m r r F F F r r

32 Spce Curviliner Motion 1) Crtesin Coordintes : 1) x m x ) y m y 3) z z F x v x x F y v y y F m z v z z F F x Fy Fz x y z ) Cylindricl Coordintes : 1) F ) F m r r m r r r r r 3) Fz m z z v z z F F r F Fz r z

33 3) Sphericl Coordintes : F F 1) R m R ) 3) R F R Rcos R cos m F R R cos Rsin R R Rcos sin F F R F R m