Ninth E CHPTER VECTOR MECHNICS OR ENGINEERS: DYNMICS edinnd P. ee E. Russell Johnston, J. Lectue Notes: J. Wlt Ole Texs Tech Univesity Kinetics of Pticles: Newton s Second Lw The McGw-Hill Copnies, Inc. ll ights eseved.
Vecto Mechnics fo Enginees: Dynics Contents Intoduction Newton s Second Lw of Motion Line Moentu of Pticle Systes of Units Equtions of Motion Dynic Equilibiu Sple Poble. Sple Poble.3 Sple Poble.4 Sple Poble.5 Sple Poble.6 ngul Moentu of Pticle Equtions of Motion in Rdil & Tnsvese Coponents Consevtion of ngul Moentu Newton s Lw of Gvittion Sple Poble.7 Sple Poble.8 Tjectoy of Pticle Unde Centl oce ppliction to Spce Mechnics Sple Poble.9 Keple s Lws of Plnety Motion The McGw-Hill Copnies, Inc. ll ights eseved. -
Vecto Mechnics fo Enginees: Dynics Intoduction Newton s fist nd thid lws e sufficient fo the study of bodies t est (sttics) o bodies in otion with no cceletion. When body cceletes (chnges in velocity gnitude o diection), Newton s second lw is equied to elte the otion of the body to the foces cting on it. Newton s second lw: - pticle will hve n cceletion popotionl to the gnitude of the esultnt foce cting on it nd in the diection of the esultnt foce. - The esultnt of the foces cting on pticle is equl to the te of chnge of line oentu of the pticle. - The su of the oents bout O of the foces cting on pticle is equl to the te of chnge of ngul oentu of the pticle bout O. The McGw-Hill Copnies, Inc. ll ights eseved. - 3
Vecto Mechnics fo Enginees: Dynics Newton s Second Lw of Motion Newton s Second Lw: If the esultnt foce cting on pticle is not zeo, the pticle will hve n cceletion popotionl to the gnitude of esultnt nd in the diection of the esultnt. Conside pticle subjected to constnt foces, 3 constnt ss, 3 When pticle of ss is cted upon by foce, the cceletion of the pticle ust stisfy cceletion ust be evluted with espect to Newtonin fe of efeence, i.e., one tht is not cceleting o otting. If foce cting on pticle is zeo, pticle will not ccelete, i.e., it will ein sttiony o continue on stight line t constnt velocity. The McGw-Hill Copnies, Inc. ll ights eseved. - 4
Vecto Mechnics fo Enginees: Dynics Line Moentu of Pticle Replcing the cceletion by the deivtive of the velocity yields dv dt d dl v dt dt L line oentu of the pticle Line Moentu Consevtion Pinciple: If the esultnt foce on pticle is zeo, the line oentu of the pticle eins constnt in both gnitude nd diection. The McGw-Hill Copnies, Inc. ll ights eseved. - 5
Vecto Mechnics fo Enginees: Dynics Systes of Units Of the units fo the fou piy diensions (foce, ss, length, nd tie), thee y be chosen bitily. The fouth ust be coptible with Newton s nd Lw. Intentionl Syste of Units (SI Units): bse units e the units of length (), ss (kg), nd tie (second). The unit of foce is deived, N kg s kg s U.S. Custoy Units: bse units e the units of foce (lb), length (), nd tie (second). The unit of ss is deived, lb lb 3. ft s lb slug ft s lbs ft The McGw-Hill Copnies, Inc. ll ights eseved. - 6
Vecto Mechnics fo Enginees: Dynics Equtions of Motion Newton s second lw povides Solution fo pticle otion is fcilitted by esolving vecto eqution into scl coponent equtions, e.g., fo ectngul coponents, i j k i j k x x x y x x z y y y y x o tngentil nd nol coponents, t t t dv dt n n n v y z z z z z The McGw-Hill Copnies, Inc. ll ights eseved. - 7
Vecto Mechnics fo Enginees: Dynics Dynic Equilibiu The McGw-Hill Copnies, Inc. ll ights eseved. ltente expession of Newton s second lw, inetil vecto With the inclusion of the inetil vecto, the syste of foces cting on the pticle is equivlent to zeo. The pticle is in dynic equilibiu. Methods developed fo pticles in sttic equilibiu y be pplied, e.g., copln foces y be epesented with closed vecto polygon. Ineti vectos e often clled inetil foces s they esue the esistnce tht pticles offe to chnges in otion, i.e., chnges in speed o diection. Inetil foces y be conceptully useful but e not like the contct nd gvittionl foces found in sttics. - 8
Vecto Mechnics fo Enginees: Dynics Sple Poble. SOLUTION: Resolve the eqution of otion fo the block into two ectngul coponent equtions. -lb block ests on hoizontl plne. ind the gnitude of the foce P equied to give the block n cceletion o ft/s to the ight. The coefficient of kinetic fiction between the block nd plne is k.5. Unknowns consist of the pplied foce P nd the nol ection N fo the plne. The two equtions y be solved fo these unknowns. The McGw-Hill Copnies, Inc. ll ights eseved. - 9
Vecto Mechnics fo Enginees: Dynics Sple Poble. y O x W g lbs 6. ft N k.5n lb 3.ft s SOLUTION: Resolve the eqution of otion fo the block into two ectngul coponent equtions. x : y P cos3.5n : 6.lbs ft ft s 6.lb N Psin3 lb Unknowns consist of the pplied foce P nd the nol ection N fo the plne. The two equtions y be solved fo these unknowns. N Psin3 lb P cos3.5 Psin3 lb 6.lb P 5lb The McGw-Hill Copnies, Inc. ll ights eseved. -
Vecto Mechnics fo Enginees: Dynics Sple Poble.3 SOLUTION: Wite the kinetic eltionships fo the dependent otions nd cceletions of the blocks. Wite the equtions of otion fo the blocks nd pulley. The two blocks shown stt fo est. The hoizontl plne nd the pulley e fictionless, nd the pulley is ssued to be of negligible ss. Deteine the cceletion of ech block nd the tension in the cod. Cobine the kinetic eltionships with the equtions of otion to solve fo the cceletions nd cod tension. The McGw-Hill Copnies, Inc. ll ights eseved. -
Vecto Mechnics fo Enginees: Dynics Sple Poble.3 O y x SOLUTION: Wite the kinetic eltionships fo the dependent otions nd cceletions of the blocks. Wite equtions of otion fo blocks nd pulley. x T kg : y y y : 3kg 9.8 s T 3kg T T x g T 94N - C C T : 3kg The McGw-Hill Copnies, Inc. ll ights eseved. -
Vecto Mechnics fo Enginees: Dynics Sple Poble.3 O y x Cobine kinetic eltionships with equtions of otion to solve fo cceletions nd cod tension. y T T T x kg 94N - 3kg 94N - T T 3kg T 94 N 5kg kg 8.4 kg T s 4. 68 N s 84 N The McGw-Hill Copnies, Inc. ll ights eseved. - 3
Vecto Mechnics fo Enginees: Dynics Sple Poble.4 SOLUTION: The block is constined to slide down the wedge. Theefoe, thei otions e dependent. Expess the cceletion of block s the cceletion of wedge plus the cceletion of the block eltive to the wedge. The -lb block stts fo est nd slides on the 3-lb wedge, which is suppoted by hoizontl sufce. Wite the equtions of otion fo the wedge nd block. Solve fo the cceletions. Neglecting fiction, deteine () the cceletion of the wedge, nd (b) the cceletion of the block eltive to the wedge. The McGw-Hill Copnies, Inc. ll ights eseved. - 4
Vecto Mechnics fo Enginees: Dynics Sple Poble.4 y x SOLUTION: The block is constined to slide down the wedge. Theefoe, thei otions e dependent. Wite equtions of otion fo wedge nd block. : x x y N.5N sin3 W N x W sin3 y g cos3 : W g cos3 cos3 g sin3 sin3: W cos3 3 W g sin The McGw-Hill Copnies, Inc. ll ights eseved. - 5
Vecto Mechnics fo Enginees: Dynics Sple Poble.4 Solve fo the cceletions..5n W g N W cos3 W g sin3 W g W cos3 W g gw cos3 W W sin3 3. ft s lbcos3 3lb lbsin3 5.7ft sin3 s cos3 g sin3 5.7 ft s cos3 3. ft s sin3.5ft s The McGw-Hill Copnies, Inc. ll ights eseved. - 6
Vecto Mechnics fo Enginees: Dynics Sple Poble.5 SOLUTION: Resolve the eqution of otion fo the bob into tngentil nd nol coponents. Solve the coponent equtions fo the nol nd tngentil cceletions. The bob of - pendulu descibes n c of cicle in veticl plne. If the tension in the cod is.5 ties the weight of the bob fo the position shown, find the velocity nd cceletion of the bob in tht position. Solve fo the velocity in tes of the nol cceletion. The McGw-Hill Copnies, Inc. ll ights eseved. - 7
Vecto Mechnics fo Enginees: Dynics Sple Poble.5 SOLUTION: Resolve the eqution of otion fo the bob into tngentil nd nol coponents. Solve the coponent equtions fo the nol nd tngentil cceletions. : g sin3 t t n n : t g sin3.5g n g t t.5 cos3 4.9 g cos3 n n s 6.3 Solve fo velocity in tes of nol cceletion. n v v n 6.3 s v 5.66 s s The McGw-Hill Copnies, Inc. ll ights eseved. - 8
Vecto Mechnics fo Enginees: Dynics Sple Poble.6 SOLUTION: The c tvels in hoizontl cicul pth with nol coponent of cceletion diected towd the cente of the pth.the foces cting on the c e its weight nd nol ection fo the od sufce. Deteine the ted speed of highwy cuve of dius = 4 ft bnked though n ngle q = 8 o. The ted speed of bnked highwy cuve is the speed t which c should tvel if no ltel fiction foce is to be exeted t its wheels. Resolve the eqution of otion fo the c into veticl nd nol coponents. Solve fo the vehicle speed. The McGw-Hill Copnies, Inc. ll ights eseved. - 9
Vecto Mechnics fo Enginees: Dynics Sple Poble.6 SOLUTION: The c tvels in hoizontl cicul pth with nol coponent of cceletion diected towd the cente of the pth.the foces cting on the c e its weight nd nol ection fo the od sufce. Resolve the eqution of otion fo the c into veticl nd nol coponents. y : R cosq W W R cosq n n : Rsinq W g W sinq cosq Solve fo the vehicle speed. v g tnq n W g 3. ft s 4ft tn8 v v 64.7ft s 44.i h The McGw-Hill Copnies, Inc. ll ights eseved. -
Vecto Mechnics fo Enginees: Dynics ngul Moentu of Pticle The McGw-Hill Copnies, Inc. ll ights eseved. H O V oent of oentu o the ngul oentu of the pticle bout O. H O is pependicul to plne contining nd V H O V sin i j k vq HO x y z q v v v Deivtive of ngul oentu with espect to tie, H O V V V V M O It follows fo Newton s second lw tht the su of the oents bout O of the foces cting on the pticle is equl to the te of chnge of the ngul oentu of the pticle bout O. x y z -
Vecto Mechnics fo Enginees: Dynics Eqs of Motion in Rdil & Tnsvese Coponents Conside pticle t nd q, in pol coodintes, q q q q q This esult y lso be deived fo consevtion of ngul oentu, H O q q q d q dt q q q q The McGw-Hill Copnies, Inc. ll ights eseved. -
Vecto Mechnics fo Enginees: Dynics Consevtion of ngul Moentu When only foce cting on pticle is diected towd o wy fo fixed point O, the pticle is sid to be oving unde centl foce. Since the line of ction of the centl foce psses though O, M O H O nd V H constnt O Position vecto nd otion of pticle e in plne pependicul to. H O Mgnitude of ngul oentu, H O V sin constnt o H O H O V sin q q h constnt ngul oentu unit ss The McGw-Hill Copnies, Inc. ll ights eseved. - 3
Vecto Mechnics fo Enginees: Dynics Consevtion of ngul Moentu Rdius vecto OP sweeps infinitesil e d dq Define d dt dq dt q el velocity Recll, fo body oving unde centl foce, h q constnt When pticle oves unde centl foce, its el velocity is constnt. The McGw-Hill Copnies, Inc. ll ights eseved. - 4
Vecto Mechnics fo Enginees: Dynics Newton s Lw of Gvittion Gvittionl foce exeted by the sun on plnet o by the eth on stellite is n ipotnt exple of gvittionl foce. Newton s lw of univesl gvittion - two pticles of ss M nd ttct ech othe with equl nd opposite foce diected long the line connecting the pticles, M G G constnt of gvittion 66.73 3 kg s 34.4 9 ft 4 lbs o pticle of ss on the eth s sufce, W MG R g ft g 9.8 3. s s 4 The McGw-Hill Copnies, Inc. ll ights eseved. - 5
Vecto Mechnics fo Enginees: Dynics Sple Poble.7 block of ss cn slide feely on fictionless O which ottes in hoizontl plne t constnt te q. Knowing tht is elesed t distnce fo O, expess s function of ) the coponent v of the velocity of long O, nd b) the gnitude of the hoizontl foce exeted on by the O. SOLUTION: Wite the dil nd tnsvese equtions of otion fo the block. Integte the dil eqution to find n expession fo the dil velocity. Substitute known infotion into the tnsvese eqution to find n expession fo the foce on the block. The McGw-Hill Copnies, Inc. ll ights eseved. - 6
Vecto Mechnics fo Enginees: Dynics Sple Poble.7 SOLUTION: Wite the dil nd tnsvese equtions of otion fo the block. : q : q q q q Integte the dil eqution to find n expession fo the dil velocity. v v v v dv dv dv dt v q q dv d dv dt d d d dt dv d q d v q v dv d d dt v dv d Substitute known infotion into the tnsvese eqution to find n expession fo the foce on the block. q The McGw-Hill Copnies, Inc. ll ights eseved. - 7
Vecto Mechnics fo Enginees: Dynics Sple Poble.8 SOLUTION: Since the stellite is oving unde centl foce, its ngul oentu is constnt. Equte the ngul oentu t nd nd solve fo the velocity t. stellite is lunched in diection pllel to the sufce of the eth with velocity of 88 i/h fo n ltitude of 4 i. Deteine the velocity of the stellite s it eches it xiu ltitude of 34 i. The dius of the eth is 396 i. The McGw-Hill Copnies, Inc. ll ights eseved. - 8
Vecto Mechnics fo Enginees: Dynics Sple Poble.8 SOLUTION: Since the stellite is oving unde centl foce, its ngul oentu is constnt. Equte the ngul oentu t nd nd solve fo the velocity t. vsin H v v v O v constnt 396 4 88 i h 396 34 i i v 55i h The McGw-Hill Copnies, Inc. ll ights eseved. - 9
Vecto Mechnics fo Enginees: Dynics Tjectoy of Pticle Unde Centl oce o pticle oving unde centl foce diected towds foce cente, q q q q Second expession is equivlent to q h h d nd dq q h constnt, fo which, fte substituting into the dil eqution of otion nd siplifying, d u u whee u dq h u If is known function of o u, then pticle tjectoy y be found by integting fo u = f(q), with constnts of integtion deteined fo initil cons. The McGw-Hill Copnies, Inc. ll ights eseved. - 3
Vecto Mechnics fo Enginees: Dynics ppliction to Spce Mechnics Conside eth stellites subjected to only gvittionl pull of the eth, d u u dq d u u dq h h GM u whee constnt u Solution is eqution of conic section, GM u h GM GMu cosq eccenticity Ch GM Oigin, locted t eth s cente, is focus of the conic section. Tjectoy y be ellipse, pbol, o hypebol depending on vlue of eccenticity. The McGw-Hill Copnies, Inc. ll ights eseved. - 3
Vecto Mechnics fo Enginees: Dynics ppliction to Spce Mechnics Tjectoy of eth stellite is defined by GM h cosq eccenticity Ch GM hypebol, > o C > GM/h. The dius vecto becoes infinite fo GM cosq q cos cos C h pbol, = o C = GM/h. The dius vecto becoes infinite fo cosq q 8 ellipse, < o C < GM/h. The dius vecto is finite fo q nd is constnt, i.e., cicle, fo <. The McGw-Hill Copnies, Inc. ll ights eseved. - 3
Vecto Mechnics fo Enginees: Dynics ppliction to Spce Mechnics Integtion constnt C is deteined by cons t beginning of fee flight, q =, =, GM Ch h GM GM C h cos GM v Stellite escpes eth obit fo v esc o v C GM GM h GM Tjectoy is elliptic fo v < v esc nd becoes cicul fo = o C =, v cic GM v The McGw-Hill Copnies, Inc. ll ights eseved. - 33
Vecto Mechnics fo Enginees: Dynics ppliction to Spce Mechnics Recll tht fo pticle oving unde centl foce, the el velocity is constnt, i.e., d q h constnt dt Peiodic tie o tie equied fo stellite to coplete n obit is equl to e within the obit divided by el velocity, b b h h whee b The McGw-Hill Copnies, Inc. ll ights eseved. - 34
Vecto Mechnics fo Enginees: Dynics Sple Poble.9 stellite is lunched in diection pllel to the sufce of the eth with velocity of 36,9 k/h t n ltitude of 5 k. Deteine: ) the xiu ltitude eched by the stellite, nd b) the peiodic tie of the stellite. SOLUTION: Tjectoy of the stellite is descibed by GM h C cosq Evlute C using the initil cons t q =. Deteine the xiu ltitude by finding t q = 8 o. With the ltitudes t the peigee nd pogee known, the peiodic tie cn be evluted. The McGw-Hill Copnies, Inc. ll ights eseved. - 35
Vecto Mechnics fo Enginees: Dynics Sple Poble.9 v GM 637 5 6.87 7.4 398 6.5 6 3 6.87.5 s 9 6 9.8 s 6.37 3 s k k /k 369 h 36s/h h v gr 3 s s SOLUTION: Tjectoy of the stellite is descibed by GM C cosq h Evlute C using the initil cons t q =. GM C h 6.87 65.3 6 9-398 7.4 s 3 s The McGw-Hill Copnies, Inc. ll ights eseved. - 36
Vecto Mechnics fo Enginees: Dynics Sple Poble.9 Deteine the xiu ltitude by finding t q = 8 o. GM h C 66.7 6 398 7.4 s 3 667 k x ltitude s 65.3 9 667-637k 63 k With the ltitudes t the peigee nd pogee known, the peiodic tie cn be evluted. b 6.87 66.7 b h 6.87 66.7 6 6 36.8.4 7.4 9 6.4 7.3 3 s 9h 3in 6 s 36.8 6 6 The McGw-Hill Copnies, Inc. ll ights eseved. - 37
Vecto Mechnics fo Enginees: Dynics Keple s Lws of Plnety Motion Results obtined fo tjectoies of stellites ound eth y lso be pplied to tjectoies of plnets ound the sun. Popeties of plnety obits ound the sun wee deteined stonoicl obsevtions by Johnn Keple (57-63) befoe Newton hd developed his fundentl theoy. ) Ech plnet descibes n ellipse, with the sun locted t one of its foci. ) The dius vecto dwn fo the sun to plnet sweeps equl es in equl ties. 3) The sques of the peiodic ties of the plnets e popotionl to the cubes of the seijo xes of thei obits. The McGw-Hill Copnies, Inc. ll ights eseved. - 38