VECTOR MECHANICS FOR ENGINEERS: Vector Mechanics for Engineers: Dynamics. In the current chapter, you will study the motion of systems of particles.

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1 Seeth Edto CHPTER 4 VECTOR MECHNICS FOR ENINEERS: DYNMICS Fedad P. ee E. Russell Johsto, J. Systems of Patcles Lectue Notes: J. Walt Ole Texas Tech Uesty 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed. Seeth Edto Vecto Mechacs fo Egees: Dyamcs Itoducto I the cuet chapte, you wll study the moto of systems of patcles. The effecte foce of a patcle s defed as the poduct of t mass ad acceleato. It wll be show that the system of exteal foces actg o a system of patcles s equpollet wth the system of effecte foces of the system. The mass cete of a system of patcles wll be defed ad ts moto descbed. pplcato of the wok-eegy pcple ad the mpulse-mometum pcple to a system of patcles wll be descbed. Result obtaed ae also applcable to a system of gdly coected patcles,.e., a gd body. alyss methods wll be peseted fo aable systems of patcles,.e., systems whch the patcles cluded the system chage. 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed. 4-3

2 Seeth Edto Vecto Mechacs fo Egees: Dyamcs pplcato of Newto s Laws. Effecte Foces Newto s secod law fo each patcle P a system of patcles, F + f m a j j F + ( f ) j F exteal foce m a effecte foce j m a f j teal foces The system of exteal ad teal foces o a patcle s equalet to the effecte foce of the patcle. The system of exteal ad teal foces actg o the ete system of patcles s equalet to the system of effecte foces. 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed. 4-4 Seeth Edto Vecto Mechacs fo Egees: Dyamcs pplcato of Newto s Laws. Effecte Foces Summg oe all the elemets, F + f m a j j ( F ) + ( f ) ( m a ) j j Sce the teal foces occu equal ad opposte collea pas, the esultat foce ad couple due to the teal foces ae zeo, F ma ( F ) ( m a ) The system of exteal foces ad the system of effecte foces ae equpollet by ot equalet. 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed. 4-5

3 Seeth Edto Vecto Mechacs fo Egees: Dyamcs Lea & gula Mometum Lea mometum of the system of patcles, L m L & m& ma Resultat of the exteal foces s equal to ate of chage of lea mometum of the system of patcles, F L & gula mometum about fxed pot O of system of patcles, H ( m ) H & O O ( m ) + ( m ) & ( m a ) Momet esultat about fxed pot O of the exteal foces s equal to the ate of chage of agula mometum of the system of patcles, M O H & O & 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed. 4-6 Seeth Edto Vecto Mechacs fo Egees: Dyamcs Moto of the Mass Cete of a System of Patcles Mass cete of system of patcles s defed by posto ecto whch satsfes m m Dffeetatg twce, m& m & m m L ma L & F The mass cete moes as f the ete mass ad all of the exteal foces wee cocetated at that pot. 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed

4 Seeth Edto Vecto Mechacs fo Egees: Dyamcs gula Mometum bout the Mass Cete a a + a Cosde the cetodal fame of efeece x y z, whch taslates wth espect to the Newtoa fame Oxyz. The agula mometum of the system of patcles about the mass cete, H ( m ) H & ( m a ) ( m ( a a )) ( m a ) M m a ( m a ) ( F ) The cetodal fame s ot, geeal, a Newtoa fame. The momet esultat about of the exteal foces s equal to the ate of chage of agula mometum about of the system of patcles. 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed. 4-8 Seeth Edto Vecto Mechacs fo Egees: Dyamcs gula Mometum bout the Mass Cete gula mometum about of the patcles the moto elate to the cetodal x y z fame of efeece, H + ( m ) gula mometum about of patcles the absolute moto elate to the Newtoa Oxyz fame of efeece. H H ( m ) ( m ( + )) m H M + ( m ) gula mometum about of the patcle mometa ca be calculated wth espect to ethe the Newtoa o cetodal fames of efeece. 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed

5 Seeth Edto Vecto Mechacs fo Egees: Dyamcs Coseato of Mometum If o exteal foces act o the patcles of a system, the the lea mometum ad agula mometum about the fxed pot O ae coseed. L & F 0 H & O M O 0 L costat H costat O Cocept of coseato of mometum also apples to the aalyss of the mass cete moto, L & F 0 H & M L m costat costat H costat 0 I some applcatos, such as poblems olg cetal foces, L & F 0 H & O M O 0 L costat H costat O 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed. 4-0 Seeth Edto Vecto Mechacs fo Egees: Dyamcs Sample Poblem 4. Sce thee ae o exteal foces, the lea mometum of the system s coseed. Wte sepaate compoet equatos fo the coseato of lea mometum. 0-lb pojectle s mog wth a elocty of 00 ft/s whe t explodes to 5 ad 5-lb fagmets. Immedately afte the exploso, the fagmets tael the dectos θ 45 o ad θ 30 o. Deteme the elocty of each fagmet. Sole the equatos smultaeously fo the fagmet eloctes. 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed. 4-5

6 Seeth Edto Vecto Mechacs fo Egees: Dyamcs Sample Poblem 4. Sce thee ae o exteal foces, the lea mometum of the system s coseed. Wte sepaate compoet equatos fo the coseato of lea mometum. m + m m0 5 g + 5 g 0 g ( ) ( ) ( ) 0 y x x compoets: ( ) 5 cos cos y compoets: 5 s 45 5 s30 0 Sole the equatos smultaeously fo the fagmet eloctes. 07 ft s 97.6ft s 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed. 4 - Seeth Edto Vecto Mechacs fo Egees: Dyamcs Ketc Eegy + Ketc eegy of a system of patcles, T m ( ) m Expessg the elocty tems of the cetodal efeece fame, T [ m ( + ) ( + )] m m + + m m + m Ketc eegy s equal to ketc eegy of mass cete plus ketc eegy elate to the cetodal fame. 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed

7 Seeth Edto Vecto Mechacs fo Egees: Dyamcs Wok-Eegy Pcple. Coseato of Eegy Pcple of wok ad eegy ca be appled to each patcle P, T + U T whee U epesets the wok doe by the teal foces f j ad the esultat exteal foce F actg o P. Pcple of wok ad eegy ca be appled to the ete system by addg the ketc eeges of all patcles ad cosdeg the wok doe by all exteal ad teal foces. lthough f j ad f j ae equal ad opposte, the wok of these foces wll ot, geeal, cacel out. If the foces actg o the patcles ae coseate, the wok s equal to the chage potetal eegy ad T + V T + V whch expesses the pcple of coseato of eegy fo the system of patcles. 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed. 4-4 Seeth Edto Vecto Mechacs fo Egees: Dyamcs Pcple of Impulse ad Mometum F L & t Fdt L L t t L + Fdt L t M H & O O t M Odt H H t t H + M Odt H t The mometa of the patcles at tme t ad the mpulse of the foces fom t to t fom a system of ectos equpollet to the system of mometa of the patcles at tme t. 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed

8 Seeth Edto Vecto Mechacs fo Egees: Dyamcs Sample Poblem 4.4 all, of mass m,s suspeded fom a cod, of legth l, attached to cat, of mass m, whch ca oll feely o a fctoless hozotal tact. Whle the cat s at est, the ball s ge a tal elocty 0 gl. Deteme (a) the elocty of as t eaches t maxmum eleato, ad (b) the maxmum etcal dstace h though whch wll se. Wth o exteal hozotal foces, t follows fom the mpulse-mometum pcple that the hozotal compoet of mometum s coseed. Ths elato ca be soled fo the elocty of at ts maxmum eleato. The coseato of eegy pcple ca be appled to elate the tal ketc eegy to the maxmum potetal eegy. The maxmum etcal dstace s detemed fom ths elato. 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed. 4-6 Seeth Edto Vecto Mechacs fo Egees: Dyamcs Sample Poblem 4.4 y x Wth o exteal hozotal foces, t follows fom the mpulse-mometum pcple that the hozotal compoet of mometum s coseed. Ths elato ca be soled fo the elocty of at ts maxmum eleato. t L + Fdt L, t x compoet equato: m + m m + m,,,, Veloctes at postos ad ae, 0,, + 0, ( m m ) m 0 +,, (elocty of elate to s zeo at posto ) m,, 0 m + m 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed

9 Seeth Edto Vecto Mechacs fo Egees: Dyamcs Sample Poblem 4.4 The coseato of eegy pcple ca be appled to elate the tal ketc eegy to the maxmum potetal eegy. T + + V T V Posto - Potetal Eegy: Ketc Eegy: Posto - Potetal Eegy: Ketc Eegy: V m gl T m 0 + V m gl m gh ( m m ), T + ( m + m ) + m gl m gh, m 0 + mgl + 0 m + m, 0 m + m m h 0 g m g g g m m + m m h 0 g m + m 0 g m h m + m 0 g 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed. 4-8 Seeth Edto Vecto Mechacs fo Egees: Dyamcs Sample Poblem 4.5 all has tal elocty 0 0 ft/s paallel to the axs of the table. It hts ball ad the ball C whch ae both at est. alls ad C ht the sdes of the table squaely at ad C ad ball hts oblquely at. ssumg pefectly elastc collsos, deteme eloctes,, ad C wth whch the balls ht the sdes of the table. Thee ae fou ukows:,,x,,y, ad C. Soluto eques fou equatos: coseato pcples fo lea mometum (two compoet equatos), agula mometum, ad eegy. Wte the coseato equatos tems of the ukow eloctes ad sole smultaeously. 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed

10 Seeth Edto Vecto Mechacs fo Egees: Dyamcs Sample Poblem 4.5 Thee ae fou ukows:,,x,,y, ad C. j, x +, y j C C The coseato of mometum ad eegy equatos, L + Fdt L m0 m, x + mc 0 m m, y H M dt H O, + O O, ( ft) m0 ( 8ft) m ( 7ft) m, y ( 3ft) mc T + V T + V m 0 m + ( + ) + m m, x, y C y x Solg the fst thee equatos tems of C,, y 3 C 0, x 0 C Substtutg to the eegy equato, ( 3C 0) + ( 0 C ) + C 00 0C 60C ft s 8ft s ( 4 j ) ft s 4.47ft s 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed. 4-0 C 0