DYNAMICS. Systems of Particles VECTOR MECHANICS FOR ENGINEERS: Seventh Edition CHAPTER. Ferdinand P. Beer E. Russell Johnston, Jr.

Transcription

1 Seeth Edto CHPTER 4 VECTOR MECHNICS FOR ENINEERS: DYNMICS Ferdad P. eer E. Russell Johsto, Jr. Systes of Partcles Lecture Notes: J. Walt Oler Texas Tech Uersty 003 The Mcraw-Hll Copaes, Ic. ll rghts resered.

2 Seeth Edto Vector Mechacs for Egeers: Dyacs Cotets Itroducto pplcato of Newto s Laws: Effecte Forces Lear ad gular Moetu Moto of Mass Ceter of Syste of Partcles gular Moetu bout Mass Ceter Coserato of Moetu Saple Proble 4. Ketc Eergy Work-Eergy Prcple. Coserato of Eergy Prcple of Ipulse ad Moetu Saple Proble 4.4 Saple Proble 4.5 Varable Systes of Partcles Steady Strea of Partcles Steady Strea of Partcles. pplcatos Streas ag or Losg Mass Saple Proble The Mcraw-Hll Copaes, Ic. ll rghts resered. 4 -

3 Seeth Edto Vector Mechacs for Egeers: Dyacs Itroducto I the curret chapter, you wll study the oto of systes of partcles. The effecte force of a partcle s defed as the product of t ass ad accelerato. It wll be show that the syste of exteral forces actg o a syste of partcles s equpollet wth the syste of effecte forces of the syste. The ass ceter of a syste of partcles wll be defed ad ts oto descrbed. pplcato of the work-eergy prcple ad the pulseoetu prcple to a syste of partcles wll be descrbed. Result obtaed are also applcable to a syste of rgdly coected partcles,.e., a rgd body. alyss ethods wll be preseted for arable systes of partcles,.e., systes whch the partcles cluded the syste chage. 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. 4-3

4 Seeth Edto Vector Mechacs for Egeers: Dyacs pplcato of Newto s Laws. Effecte Forces Newto s secod law for each partcle P syste of partcles, F r j j ( r f ) F exteral force a effecte force F f j a j r a f j teral forces a The syste of exteral ad teral forces o a partcle s equalet to the effecte force of the partcle. The syste of exteral ad teral forces actg o the etre syste of partcles s equalet to the syste of effecte forces. 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. 4-4

5 Seeth Edto Vector Mechacs for Egeers: Dyacs pplcato of Newto s Laws. Effecte Forces Sug oer all the eleets, F f a j ( ) ( ) r F r f ( r a ) j j j Sce the teral forces occur equal ad opposte collear pars, the resultat force ad couple due to the teral forces are zero, F a r F r a ( ) ( ) The syste of exteral forces ad the syste of effecte forces are equpollet by ot equalet. 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. 4-5

6 Seeth Edto Vector Mechacs for Egeers: Dyacs Lear & gular Moetu Lear oetu of the syste of partcles, L L a Resultat of the exteral forces s equal to rate of chage of lear oetu of the syste of partcles, F L gular oetu about fxed pot O of syste of partcles, H r H O O ( ) ( r ) ( r ) ( r a ) Moet resultat about fxed pot O of the exteral forces s equal to the rate of chage of agular oetu of the syste of partcles, M O H O 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. 4-6

7 Seeth Edto Vector Mechacs for Egeers: Dyacs Moto of the Mass Ceter of a Syste of Partcles Mass ceter of syste of partcles s defed by posto ector whch r satsfes r r Dfferetatg twce, r r a L F L The ass ceter oes as f the etre ass ad all of the exteral forces were cocetrated at that pot. 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. 4-7

8 Seeth Edto Vector Mechacs for Egeers: Dyacs gular Moetu bout the Mass Ceter a a a Cosder the cetrodal frae of referece x y z, whch traslates wth respect to the Newtoa frae Oxyz. The agular oetu of the syste of partcles about the ass ceter, H H ( r ) ( r a ) ( r ( a a )) ( r a ) ( r a ) ( r F ) M r a The cetrodal frae s ot, geeral, a Newtoa frae. The oet resultat about of the exteral forces s equal to the rate of chage of agular oetu about of the syste of partcles. 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. 4-8

9 Seeth Edto Vector Mechacs for Egeers: Dyacs gular Moetu bout the Mass Ceter gular oetu about of the partcles ther oto relate to the cetrodal x y z frae of referece, H ( r ) gular oetu about of partcles ther absolute oto relate to the Newtoa Oxyz frae of referece. H H ( r ) ( r ( )) H r M ( r ) gular oetu about of the partcle oeta ca be calculated wth respect to ether the Newtoa or cetrodal fraes of referece. 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. 4-9

10 Seeth Edto Vector Mechacs for Egeers: Dyacs Coserato of Moetu If o exteral forces act o the partcles of a syste, the the lear oetu ad agular oetu about the fxed pot O are cosered. L F 0 H O M O 0 L costat H costat O Cocept of coserato of oetu also apples to the aalyss of the ass ceter oto, L F 0 H M 0 L costat costat H costat I soe applcatos, such as probles olg cetral forces, L F 0 H O M O 0 L costat H costat O 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. 4-0

11 Seeth Edto Vector Mechacs for Egeers: Dyacs Saple Proble 4. SOLUTION: Sce there are o exteral forces, the lear oetu of the syste s cosered. Wrte separate copoet equatos for the coserato of lear oetu. 0 kg projectle s og wth a elocty of 30 /s whe t explodes to.5 ad 7.5 kg fragets. Iedately after the exploso, the fragets trael the drectos θ 45 o ad θ 30 o. Detere the elocty of each fraget. Sole the equatos sultaeously for the fraget eloctes. 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. 4 -

12 Seeth Edto Vector Mechacs for Egeers: Dyacs Saple Proble 4. SOLUTION: Sce there are o exteral forces, the lear oetu of the syste s cosered. Wrte separate copoet equatos for the coserato of lear oetu. (.5) ( 7.5) ( 0) 0 0 y x x copoets:.5 cos cos30 y copoets:.5 s s ( 30) Sole the equatos sultaeously for the fraget eloctes. 6./s 9.3 s 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. 4 -

13 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. Vector Mechacs for Egeers: Dyacs Seeth Edto 4-3 Ketc Eergy Ketc eergy of a syste of partcles, ( ) T Expressg the elocty ters of the cetrodal referece frae, ( ) ( ) [ ] T Ketc eergy s equal to ketc eergy of ass ceter plus ketc eergy relate to the cetrodal frae.

14 Seeth Edto Vector Mechacs for Egeers: Dyacs Work-Eergy Prcple. Coserato of Eergy Prcple of work ad eergy ca be appled to each partcle P, T U T where U represets the work doe by the teral forces ad f j the resultat exteral force actg F o P. Prcple of work ad eergy ca be appled to the etre syste by addg the ketc eerges of all partcles ad cosderg the work doe by all exteral ad teral forces. lthough f j ad f jare equal ad opposte, the work of these forces wll ot, geeral, cacel out. If the forces actg o the partcles are coserate, the work s equal to the chage potetal eergy ad T V T V whch expresses the prcple of coserato of eergy for the syste of partcles. 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. 4-4

15 Seeth Edto Vector Mechacs for Egeers: Dyacs Prcple of Ipulse ad Moetu L F t t L Fdt t t L Fdt L L H M t t M O H O t dt t M O H O dt H H The oeta of the partcles at te t ad the pulse of the forces fro t to t for a syste of ectors equpollet to the syste of oeta of the partcles at te t. 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. 4-5

16 Seeth Edto Vector Mechacs for Egeers: Dyacs Saple Proble 4.4 SOLUTION: all, of ass,s suspeded fro a cord, of legth l, attached to cart, of ass, whch ca roll freely o a frctoless horzotal tract. Whle the cart s at rest, the ball s ge a tal elocty 0 gl. Detere (a) the elocty of as t reaches t axu eleato, ad (b) the axu ertcal dstace h through whch wll rse. 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. Wth o exteral horzotal forces, t follows fro the pulse-oetu prcple that the horzotal copoet of oetu s cosered. Ths relato ca be soled for the elocty of at ts axu eleato. The coserato of eergy prcple ca be appled to relate the tal ketc eergy to the axu potetal eergy. The axu ertcal dstace s detered fro ths relato. 4-6

17 Seeth Edto Vector Mechacs for Egeers: Dyacs Saple Proble 4.4 y x SOLUTION: Wth o exteral horzotal forces, t follows fro the pulse-oetu prcple that the horzotal copoet of oetu s cosered. Ths relato ca be soled for the elocty of at ts axu eleato. t L Fdt L, x copoet equato: t, 0 Veloctes at postos ad are,,,,,, 0, ( ) 0,, (elocty of relate to s zero at posto ),, The Mcraw-Hll Copaes, Ic. ll rghts resered. 4-7

18 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. Vector Mechacs for Egeers: Dyacs Seeth Edto 4-8 Saple Proble 4.4 The coserato of eergy prcple ca be appled to relate the tal ketc eergy to the axu potetal eergy. V T V T Posto - Potetal Eergy: Ketc Eergy: Posto - Potetal Eergy: Ketc Eergy: gl V 0 T gh gl V ( ), T ( ) gh gl gl, 0 g h 0 0 0, 0 g g g g h g g h 0 0

19 Seeth Edto Vector Mechacs for Egeers: Dyacs Saple Proble 4.5 all has tal elocty 0 3 /s parallel to the axs of the table. It hts ball ad the ball C whch are both at rest. alls ad C ht the sdes of the table squarely at ad C ad ball hts oblquely at. ssug perfectly elastc collsos, detere eloctes,, ad C wth whch the balls ht the sdes of the table. SOLUTION: There are four ukows:,,x,,y, ad C. Soluto requres four equatos: coserato prcples for lear oetu (two copoet equatos), agular oetu, ad eergy. Wrte the coserato equatos ters of the ukow eloctes ad sole sultaeously. 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. 4-9

20 Seeth Edto Vector Mechacs for Egeers: Dyacs Saple Proble 4.5 SOLUTION: There are four ukows:,,x,,y, ad C. j, x, y j C C The coserato of oetu ad eergy equatos, L Fdt L H T O, V 0 0 M dt H T O V, x O, C 0 ( ), x, y C, y ( 0.6 ) 0 (.4 ) (.), y ( 0.9 ) C y x Solg the frst three equatos ters of C, 3, y 3 C 6, x Substtutg to the eergy equato, ( 6) ( 3 ) 3 0 C C 78 C C C 7 0. s 4 j C 9 ( ) s.34 s C.4 s 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. 4-0

21 Seeth Edto Vector Mechacs for Egeers: Dyacs Varable Systes of Partcles Ketcs prcples establshed so far were dered for costat systes of partcles,.e., systes whch ether ga or lose partcles. large uber of egeerg applcatos requre the cosderato of arable systes of partcles, e.g., hydraulc turbe, rocket ege, etc. For aalyses, cosder auxlary systes whch cosst of the partcles stataeously wth the syste plus the partcles that eter or leae the syste durg a short te teral. The auxlary systes, thus defed, are costat systes of partcles. 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. 4 -

22 Seeth Edto Vector Mechacs for Egeers: Dyacs Steady Strea of Partcles Syste cossts of a steady strea of partcles agast a ae or through a duct. Defe auxlary syste whch cludes partcles whch flow ad out oer t. The auxlary syste s a costat syste of partcles oer t. L [ ( ) ] F t [ ( ) ] t t Fdt L F d dt ( ) 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. 4 -

23 Seeth Edto Vector Mechacs for Egeers: Dyacs Steady Strea of Partcles. pplcatos Flud Strea Derted by Vae or Duct Fa Flud Flowg Through a Ppe Jet Ege Helcopter 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. 4-3

24 Seeth Edto Vector Mechacs for Egeers: Dyacs Streas ag or Losg Mass Defe auxlary syste to clude partcles of ass wth syste at te t plus the partcles of ass whch eter the syste oer te teral t. The auxlary syste s a costat syste of partcles. L t t Fdt L d d F u dt dt d a F u dt [ ( ) a ] F t ( )( ) F t ( ) ( ) a 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. 4-4

25 Seeth Edto Vector Mechacs for Egeers: Dyacs Saple Proble 4.6 ra falls oto a chute at the rate of 0 kg/s. It hts the chute wth a elocty of 6 /s ad leaes wth a elocty of 4.5 /s. The cobed weght of the chute ad the gra t carres s 3 kn wth the ceter of graty at. Detere the reactos at C ad. SOLUTION: Defe a syste cosstg of the ass of gra o the chute plus the ass that s added ad reoed durg the te teral t. pply the prcples of coserato of lear ad agular oetu for three equatos for the three ukow reactos. 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. 4-5

26 Seeth Edto Vector Mechacs for Egeers: Dyacs Saple Proble 4.6 SOLUTION: Defe a syste cosstg of the ass of gra o the chute plus the ass that s added ad reoed durg the te teral t. pply the prcples of coserato of lear ad agular oetu for three equatos for the three ukow reactos. H L C, C x 0.9 t Fdt ( ) cos0 ( ) ( C W ) t ( ) s0 M C dt L y H C, ( ) (.W 3.6) t.8( ) cos0 3.6( ) s0 Sole for C x, C y, ad wth t 0kg s 3. ft s 7.45slug s 0 N C j ( ) N 003 The Mcraw-Hll Copaes, Ic. ll rghts resered. 4-6