Chapt. 9 Systems of Particles and Conservation of Linear Momentum
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1 Chapt. 9 Systes o Patcles ad Coseato o Lea oetu 9. Lea oetu ad Its Coseato 9. Isolated Syste lea oetu: P F dp d( d a solated syste F ext 0 dp dp F F dp dp d F F 0 0 ( P P P tot cost p p p p the law o coseato o lea oetu Exaple: Ca we eally goe the ketc eegy o the Eath? K E E E EE bb K b bb E b b E K K E E b b b E b E 9. Nosolated Syste (Ipulse I F t P Exaple: How good ae the bupes? I a cash test a autooble o ass 00 kg colldes wth a wall. The tal ad al eloctes o the autooble ae - /s ad.6 /s. I the collso lasts o 0. s d the pulse due to the collso ad the aeage oce exceted o the autooble.
2 ( I N t I F Collsos Oe Deso What s a collso? A collso s a solated eet whch two o oe bodes (the colldg bodes exet elately stog oces o each othe o a elately shot te. Do t eed eal touch. Rules: The lea oetu ust be coseed whethe t s elastc o elastc collso. Elastc collso: the total ketc eegy ust be coseed. Ielastc collso: the total ketc eegy o the syste s ot the sae beoe ad ate the collso. Peectly Ielastc Collsos: (lea oetu s coseed ( Elastc Collsos: (both lea oetu ad eegy ae coseed ( ( ( ( ( ( ( ( Specal cases: whe 0
3 . Equal asses: 0. A asse taget: <<. A asse pojectle: >> Saple Exaple: Two etal sphees suspeded by etcal cods tally just touch. Sphee wth ass 0 g s pulled to the let to heght h 8.0 c ad the eleased o est. Ate swgg dow t udegoes a elastc collso wth sphee whose ass 7 g. What s the elocty o sphee just ate the collso? gh gh ( ( ( Saple Exaple: A block o ass.60 kg tally og to the ght wth a speed o 4.00 /s o a ctoless hozotal tack colldes wth a spg attached to a secod block o ass.0 kg tally og to the let wth a speed o.0 /s. The spg costat s 600 N/. (a Fd the eloctes o the two blocks ate the collso ( ( (b Dug the collso at the stat block s og to the ght wth a elocty o.00 /s detee the elocty o block ( (c Detee the dstace the spg s copessed at that stat. kx
4 (d What s the axu copesso o the spg dug the collso? whe two blocks ae og at the sae speed. Exaple: The Ballstc Pedulu The ballstc pedulu s a appaatus used to easue the speed o a ast-og pojectle such as a bullet. A bullet o ass s ed to a lage block o wood o ass suspeded o soe lght wes. The bullet beds the block ad the ete syste swgs though a heght h. How ca we detee the speed o the bullet o a easueet o h? Obta the speed o ad edately ate collso: gh Apply the coseato o oetu: ( gh A Saple Poble: A kaate expet stkes dowwad wth hs st (o ass 0.70 kg beakg a 0.4 kg boad. He the does the sae to a. kg cocete block. The spg costats k o bedg ae N/ o the boad ad N/ o the block. Beakg occus at a delecto d o 6 o the boad ad. o the block. (a Just beoe the object (boad o block beaks what s the eegy stoed t? Boad: Block: 4 U kd 4. 0 (6 0. J 6 U kd.6 0 (. 0. 6J (b What s the lowest st speed st equed to beak the object (boad o block? Assue the ollowg: The collsos ae copletely elastc collsos o oly the st ad the object. Bedg begs just ate the collso. echacal eegy s 4
5 coseed o the begg o the bedg utl just beoe the object beaks. The speed o the st ad object s eglgble at that pot. st ( ( U st U ( ( ( U 9. Collsos Two Desos Fo coseato o oetu: Checked by the copoets p p p p p x px p x p x p y py p y p y Fo coseato o ketc eegy: K K K K p p p p p x p y px p > y Spled questos: cosθ cosφ sθ sφ Exaple: A poto colldes elastcally wth aothe poto that s tally at est. The cog poto has a tal speed o.x0 /s ad akes a glacg collso wth the secod poto. Ate collso oe poto oes o at a agle o 7 o to the ogal decto o oto ad the secod delects at a agle o φ to the sae axs. Fd the al speeds o the two potos ad the agle φ. cos7 o cosφ s 7 o sφ
6 cosφ cos7 sφ s 7 o o o o ( cos7 ( s 7 o cos sφ s 7 o 4 4 Saple Exaple: Two skates collde ad ebace a copletely elastc collso. Thus they stck togethe ate pact whee the og s placed at the pot o collso. Aled whose ass A s 8 kg s ogally og east wth speed A 6. k/h. Babaa whose ass B s kg s ogally og oth wth speed B 7.8 k/h. a What s the elocty V o the couple ate they collde? P ˆ 8 6.ˆ 4.6ˆ A A A P B ˆ j 7.8 ˆj 49 ˆj B B 4.6ˆ 49 ˆj P ( A B PA PB.7ˆ.ˆj k / h 9.6 The Cete o ass x x x C x C x x x x Thee Deso: dscete ass pots: 6
7 C cotuous ass: x C xd y C yd z C zd x C xdv y C V ydv z C V zdv V Saple Exaple: Thee patcles o asses. kg. kg ad.4 kg o a equlateal tagle o edge legth a 40 c. Whee s the cete o ass o ths thee-patcle syste? (00 (400 (700. (00. (400.4 (700 (...4 (88 Saple Exaple: Fgue shows a uo etal plate P o adus R o whch a dsk o adus R has bee staped out (eoed a assebly le. Usg the xy coodate syste show locate the cete o ass co P o the plate. x 4 0 ( ( R R R c 4 ( x 4 0 ( ( R R R c 4 ( Gatatoal Potetal Eegy o a Syste Fdg the Cete o ass by Itegato Rod: Exaple: Show that the cete o ass o a od o ass ad legth L les dway betwee ts eds assug the od has a uo ass pe ut legth. X C x L x ( λdx 0 λl L Suppose the od s o-uo wth ts ass desty aes as O λ αx. Fd the O 7
8 cete o ass. xλdx L α L dx L 0 λ α X C L 0 Seccula Hoop: d λds λdθ cosθ xˆ sθyˆ L αl C πλ C π 0 ( cosθxˆ sθyˆ λdθ C yˆ π 9.7 oto o a Syste o Patcles a a C C C C a a a... a F F F... F Saple Exaple: The thee patcles ae tally at est. Each expeeces a exteal oce due to bodes outsde the thee-patcle syste. The dectos ae dcated ad the agtudes ae F 6.0 N F N ad F 4 N. What s the acceleato o the cete o ass o the syste ad what decto does t oe? ( (4 ( c F ( 60 F (6 6 F (40 F (
9 a F (6 8 ˆ 6 6 ˆj Ketc Eegy o a Syste K u K C K ( C u ( C u C u C Kel The Cete-o-ass Reeece Fae I the et exteal oce o a syste eas zeo the elocty o the cete o ass eas costat. F d total 0 ( C F total 0 C Deoable Systes Pushg o a Spg: As show the ght gue two blocks ae at est o a ctoless leel table. Both blocks hae the sae ass ad they ae coected by a spg o eglgble ass. The sepaato dstace o the blocks whe the spg s elaxed s L. Dug a te teal t a costat oce o agtude F s appled hozotally to the let block og t though a dstace x. Dug the te teal the ght block oes though a dstace x. At the ed o ths te teal the oce F s eoed. (a Fd the esultg speed c o the cete o ass o the syste. F t (. C. x ( L x 0 L t ( x x ( 0 C ag x x F ( C C ( x x F C C ag C 9
10 9.9 Rocket Populso P el el P d U ( d ( d j : elate speed betwee the ocket ad the exhaust poducts d U d U d d 0 U d el d ( d el d 0 el d d d R Rel a d el d d d d el d el el l el l Saple Poble: A ocket whose tal ass s 80 kg cosues uel at the ate R. kg/s. The speed el o the exhaust gases elate to the ocket ege s 800 /s. a What thust does the ocket ege pode? T Rel b What s the tal acceleato o the ocket? a T s 80 / (c Suppose stead that the ocket s lauched o a spacecat aleady deep space whee we ca eglect ay gatatoal oce actg o t. The ass o the ocket whe ts uel s exhausted s 80 kg. What s ts speed elate to the spacecat at that te? Assue that the spacecat s so asse that the lauch does ot alte ts speed s 0 el l 800 l 400 / 80 0
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