VIII Dynamics of Systems of Particles
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1 VIII Dyacs of Systes of Patcles Cete of ass: Cete of ass Lea oetu of a Syste Agula oetu of a syste Ketc & Potetal Eegy of a Syste oto of Two Iteactg Bodes: The Reduced ass Collsos: o Elastc Collsos R whee: o Ielastc Collsos Collsos Cete-of-ass Coodate Syste X x Z z x Y y z y
2 Velocty of Cete of ass: v R v v x z x z x z Acceleato of Cete of ass: A R a v y y y Lookg at: oto of cete of ass coodate oto cete of ass coodates Cosevato Laws: Cosevato of lea oetu Cosevato of agula oetu Cosevato of eegy Cosevato Laws syetes Syety: Taslatoal cosevato of lea oetu hoogeety of space Rotatoal cosevato of agula oetu sotopy of space A A X X Z x Z z A Y Y y Ay cosevato law s a stateet of vaace of soe physcal popety dug all physcal pocesses. 3 4
3 P Cosevato of Lea oetu: P P v v P P P v v The lea oetu of a syste of patcles s equal to the velocty of the cete of ass ultpled by the total ass of the syste. Equato of oto: oces actg o the espectve patcles: o Exteal foces:,, 3,, o Iteal foces: j ( 0) The equato of oto of a patcle, : whee: j j P total exteal foce o patcle o patcles: j j P j (3 d law) j P P a The acceleato of the cete of ass of a syste of patcles s the sae as that of a sgle patcle havg a ass equal to the total ass of the syste ad acted o by the su of the exteal foces. I: 0 0 a v costat p p v costat Whch s Pcple of Cosevato of Lea oetu. The ate of chage of total lea oetu s equal to the total exteal foce. d p 5 6
4 If the total foce s zeo, the total oetu wll be costat. 0 p cost. Covesato of Agula oetu: o a patcle: L p v Z L p L vs y o lea oetu: dp p x p v o Agula oetu: dp P Toque (oet of exteal foce) dl d P dp P P o a syste of patcles: L dl dl dl P dv a The te ate of chage of agula oetu of a syste s equal to the total oetu of all the exteal foces actg o the syste. 7 8
5 If : 0 d L 0 L cost. L v cost. Ths s Pcple of cosevato of Agula oetu. o a solated, the total toque 0 wll be zeo. Agula oetu tes of the oto of Cete of ass: v v v L v L v v v velocty of each pot wth espect to cate of ass. L L v v The agula oetu of a syste ca be expessed tes of a obtal pat (oto of the cete of ass) ad a sp pat (oto about the cete of ass). Ketc Eegy of a Syste, T: o a syste of patcles: T ( v. v ) v v v v () whee v s the velocty of patcle espect to cete of ass. Substtute () (): whee v T v () The ketc eegy of taslato of the whole syste. 9 0
6 The ketc eegy of oto elatve to the ass cete. oto of Two Iteactg Bodes: Two bodes teact ude a cetal foce: Potetal Eegy: If the total foce actg o a patcle the syste s a fucto of posto,,..., ) ( If the foce satsfes: cul k k 0 V V (,,..., ) x V x Total Eegy: ca be show that: d ( K V ) cost. E y V y ( K V ) 0 z V z R o Assue : Isolated syste Cete of ass oves wth costat velocty Cete of ass s og ( ) 0 0 posto vecto of patcle espect to patcle. (I) C
7 o patcle, the equato of oto elatve to cete of ass: d f ( ) ( ) f ( ) ˆ (II) o Assue : have exteal foce (ot solated syste) e e Cete of ass s og (fo C syste) whee f () the agtude of the utual foce betwee the two patcles. Usg Eq. (I), ( ), Eq. (II): d f ( ) f ( ) ˆ (III) whee; educed ass (III) oto of patcle elatve to o vce vesa. oto of a patcle of ass ovg a cetal feld of foce of f (). f Equato of otos Lab syste: () e e () * Equato of oto C syste: The cete of ass coodate: R elatve coodate: R R 3 4
8 Addg two equatos () ad (): ) R ( R The () + () e e ( ) ( ) ( ) f f If e e 0 The: (3) e e o Also, cetal ass coodate: f (3) Collsos: Two bodes udego a collso If two bodes ae cosdeed as a sgle syste, cotact foce s a teal foce. The total lea oetu eas uchaged. o p p p p befoe collso afte collso v v v Eegy: o (geeal) p ) p Sys f ) Sys p p p p Q Lea oetu: p R Agula oetu: L ( R R ) ( ) Ketc eegy: k R 0 Q 0 0 elastc collso exoegc collso edoegc collso Q 5 6
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