2/24/2014. The point mass. Impulse for a single collision The impulse of a force is a vector. The Center of Mass. System of particles

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1 /4/04 Chapte 7 Lnea oentu Lnea oentu of a Sngle Patcle Lnea oentu: p υ It s a easue of the patcle s oton It s a vecto, sla to the veloct p υ p υ p υ z z p It also depends on the ass of the object, sla to the knetc eneg. Newton s nd Law fo a Sngle Bod. The te ate of the change of the oentu of a patcle s equal to the net foce actng on the patcle. υ ( υ) p a The oentu foulaton of Newton s law s oe nclusve as t s applcable to bodes and sstes wth vaable ass. p A few pontes: The lnea oentu of an object changes (and theefoe thee s a net foce actng on t) f The veloct changes n agntude The veloct changes n decton The ass of the bod changes Eaple- a ball hts a wall Collsons p p f p υ ) υ ) υ ) p p p f p p υ snθ υ snθ υ snθ p υ cosθ + υ cosθ 0 p p f p 0 υ ) υ ) p If the ball stcks to the wall the foce s salle than f t bounces. θ I II III Befoe and afte the collson the oentu of a bod changes. Thee s a net foce actng on the bod. p The collson takes te (although qute sall). The foce vaes dung the te of the collson. The total change of the oentu of the bod dung the collson s: p p f p F net Ipulse

2 /4/04 Ipulse fo a sngle collson The pulse of a foce s a vecto. F aea unde the cuve The aea below the two cuves s the sae! sae pulse F The pont ass The pont ass s a phscal appoaton of a eal object that has densons. How good s ths ths appoaton??? Can we appl Newton s laws and all we leaned so fa to a bg, etended object, o to a sste of objects that ae not even attached to each othe? F p p f p The change of the lnea oentu s equal to the pulse of the actng foce. Sste of patcles A bod can have a coplcated shape We can epesent the bod as a su of salle pats Appl Newton s law to each pat. F a n F F a F a, F, a The Cente of ass The cente of ass () of a sste of patcles s the pont that oves as though the sste s ass s concentated n ths pont and all etenal foces ae appled thee. n Ipotant note: We can substtute the ente sste wth a sngle pont! s a geoetcal pont (t ght be outsde of the bod). We can eplace the object wth pont ass at the cente of ass! The oton of the bat s descbed b the oton of the. Eaple: asses Two patcles have asses and and postons and espectvel. Fnd the cente of ass of the sste ) + ) >> 0.5( co + )

3 /4/04 Cente of ass of a Sste Sste of patcles z z If a bod has a set and t has a unfo denst then the s on the lne of set. The cente of set concdes wth the. A few pontes. The ght be outsde the object Eaple Fnd the of a 48 unfo sheet of plwood wth the uppe a ght quadant eoved (a4ft). a a a ;(, ) (, ) a a 3a a ;(, ) (, ) 4 + ( a / ) + (3a / ) 5a 0 ft ( a / ) + ( a / 4) 5a 5 ft Newton s Second Law fo a Sste of Patcles The su of all etenal foces actng on the sste s equal to the poduct of the total ass of the sste and the acceleaton of the cente of ass a. F net a - su of foces that ae etenal to the sste (?) s the total ass It does not gve nfoaton on the acceleaton of the ndvdual objects of the sste. If no etenal foces ae pesent the cente of ass wll sta at est o ove wth constant veloct! Eaple Poble A 0 long boat stas at est n a qute lake. A tutle wth a ass, s ognall standng at the ght end of the boat. The tutle walks to the othe end of the boat. Does the boat ove? If so how fa? Soluton Yes the boat wll ove, but the cente of ass of the boat+tutle sste wll sta at est. L + L befoe afte (0.5L + ) L/ L (0.5L + ) + 0.5L + L + L + L 0 L/ L 3

4 /4/04 Eaple Poble A cannon shell s fed wth an ntal veloct of 50 /s at an angle 45 o to the hozontal. When the shell s at the hghest pont of ts tajecto, t eplodes nto two peces, and. falls staght down fo the hghest pont afte the eploson. If, whee does land elatve to the ntal fng pont? 45 o Soluton The wll follow the tajecto of the ognal cannon shell and fnal c can be found wth the ange equaton. υ0 R sn θ 55 g 0.5R + R + R R + R 3 3R R R 50 Lnea oentu of a Sste of patcles The vecto su of the lnea oentu of all patcles n the sste! P p + p + p pn P υ + υ + υ υ Can show wth soe wok P υ Total ass 3 3 veloct n n υ Newton s nd Law fo a sste of patcles The net foce (the vecto su of all etenal foces) actng on the sste of patcles s equal to the ate of change of the the total lnea oentu of the sste. P The lnea oentu of a sste eans constant f no net etenal foce s actng on the sste. Consevaton of Lnea oentu If a sste s closed and solated the oentu of the sste s constant. P constant P 0 If no net etenal foce s actng on the sste then the oentu s conseved p f + p f p nf p + p If the net foce on a closed sste s zeo n a gven decton (as) the oentu n ths decton cannot change. p n Eaple A cannon fes a shell at 60 o angle to the hozontal. The shell ntal veloct s v s 00 /s. If the ato of the asses of the cannon and the shell s 00 fnd the veloct of the cannon afte the shot was fed. P P f 0 p c + p s p c p s s υ s s υ s cosθ p c c υ c υ c s c υ s cosθ υ c cos /s p c p s Etenal foces n : F g F N oentu s conseved n θ 4

5 /4/04 oentu and Knetc Eneg consevaton Laws echancal eneg consevaton Condtons: solated sste, (no etenal foces) closed sste, (do not lose bodes) consevatve foces. (no fcton, dag ) Lnea oentu consevaton Condtons: solated sste, closed sste. Elastc Collsons Elastc collsons: no defoatons, no heatng of the bodes, no eplosons,no cashng sound. Eaples: two balls collde and sepaate afte the collson (pool gaes). Consevaton laws: lnea oentu s conseved knetc eneg s conseved. Inelastc collsons Inelastc collsons: thee s defoaton, o heatng, o eploson, o sound Eaples: a bullet hts a taget and goes though the taget; Copletel nelastc collsons: two balls collde and stck togethe; a bullet hts a taget and eans nsde; a ball s thown n a cat and stas thee Consevaton laws: Lnea oentu s conseved Knetc eneg s NOT CONSERVED v v v f befoe afte Inelastc Collsons n D What happens befoe and afte the collson? Lnea oentu of the sste s conseved Knetc eneg of the sste s not conseved p + p p f + p f υ + υ υ f + υ f We have onl equaton. We have 6 paaetes. We need to know 5 of the to fnd the 6th. If we know all ntal condtons (asses of the bodes, veloctes befoe the collson) we stll cannot fnd both veloctes afte the collson! Copletel nelastc collson D What happens befoe and afte the collson? Lnea oentu of the sste s conseved Knetc eneg of the sste s not conseved Afte the collson the two bodes ove togethe wth a coon veloct V. Eaple Poble: ballstc pendulu If ou know the heght and the asses, fnd the ntal veloct of the bullet. h p + p p f + p f υ + υ ( + )V v v Dung the collson: oentu consevaton Afte the collson: oentu consevaton V υ + υ + v f υ 0 ( + )V υ 0 + V ( + )V ( + )gh V gh V V υ 0 + gh 5

6 /4/04 Elastc Collsons n D What happens befoe and afte the collson? Lnea oentu of the sste s conseved Knetc eneg of the sste s conseved Eaple: statona taget plang pool The taget does not ove v 0 p + p p f + p f υ υ f + υ f υ + υ υ f + υ f K + K K f + K f υ + υ υ f + υ f If we know the ntal veloctes and the asses then we can fnd both fnal veloctes. v v v f v f befoe afte υ υ f + υ f υ f + υ υ f + υ v f v v f Specal cases the two asses ae the sae υ f 0 υ f υ << assve taget (lke wall) v Eaple: ovng taget Both bodes ae ovng befoe the collson. Ths s the ost geneal case and ou can deve the est of the cases fo these foulas υ f υ υ f υ >> assve pojectle (cannon ball and png-pong ball) υ f υ v v υ f + υ + + υ υ f + υ + + υ υ f υ Elastc Collsons n D p + p p f + p f v K + K K f + K f v oentu equatons (fo and ) eneg equaton. We have total of 8 paaetes. We need to know 5 of the paaetes to detene the est of the. β v f v f α 6