Recall that there was ore to oton than just spee A ore coplete escrpton of oton s the concept of lnear oentu: p v (8.) Beng a prouct of a scalar () an a vector (v), oentu s a vector: p v p y v y p z v z Newton's n law tells us how forces cause acceleraton. Acceleratons are changes n oton; thus forces ust cause changes n oentu as well: p p (8.3) a F v v F Shoup 66 How can we use ths concept of oentu to analyze stuatons? We wll fn (as n Chapter's 6 & 7) that another conservaton law hols whch we can eploy Conser two nteractng partcles: Assue they for an solate syste (F et = ) Fro fgure: F Fro Newton's 3 r law: So: F F p F p p t p F F p p F p = v F p = v Shoup 67 f the te ervatve of a quantty s zero, then the quantty s constant over te, so: constant (8.4) or p (8.5) Ths s true for any nuber of partcles n the syste: syste p p f syste We can also break (8.5) nto coponents as well: p p f p y p fy p z p fz syste syste syste syste syste syste Ths s the concept of conservaton of lnear oentu: (8.6) The total lnear oentu of an solate syste s constant p p p f Shoup 68 Now lets look at case where neorce sn't zero fro (8.3): p F We (you!) can ntegrate ths: p f p p t p p f p t F F We efne the ntegral of the force over the te t s apple as the pulse (): t ΣF p = v acts for t = -t (before) F p (8.9) p p (after) p f = v f Shoup 69
Ths s also true for systes of partcles where an eternal force acts on the syste: t F et f the force s constant, then the ntegral s sply F t, so: F t Now lets see how to use ths very powerful tool of conservaton of oentu for solate systes Collsons are a large class of probles whch can be solve usng t Frst, soe concepts & assuptons about collsons: Bascally, they are nteractons between two or ore objects n the gven syste Soetes they nvolve "physcal contact", soetes not We can also copute the "constant" force that woul result n the sae pulse: So that: F F t t F (8.) t (8.) F t F areas equal t Shoup 7 F F F F p + ++ 4 He F Shoup 7 Concepts & assuptons about collsons (contnue): We can oel soe of the as perfectly nelastc collsons: partcles "stck" together collson oentu s conserve v v knetc energy s not conserve au aount of K s transfore collson f collson s hea-on v f (one ensonal) then: + Concepts & assuptons about collsons (contnue): We can oel soe of the as elastc collsons: partcles on't efor n collson oentu s conserve knetc energy s conserve f collson s hea-on (one ensonal), then: collson v v collson v v v f (8.4) v v v f (8.6) (oentu conserve) v f v v (8.5) Shoup 7 f f (8.7) (knetc energy conserve) Wth these two equatons we can solve probles wth two unknowns Shoup 73
Lets look at two ways to splfy these equatons v v f f v v f we are gven ntal veloctes an asses, we can solve for fnal veloctes because we have two equatons an two unknowns: f f v v v f v v v v f v f v v v ve v v (8.) v v & v v v v v f (8.) v f v v (8.) so relatve spee before collson equal neg. of relatve spee after Shoup 74 Shoup 75 Lets look at soe specal cases: f ( = ) then v v f s uch ore assve than v v f s uch ore assve than then partcles echange spees an s at rest then assve one contnues on as before, but lghter one goes off wth twce velocty of assve one Lets generalze to collsons n two ensons real-worl eaple: bllars f elastc, stll have conservaton of knetc energy f nelastc, on't have conservaton of knetc energy can break conservaton of oentu nto coponents: p p y v v y v v y conser "glancng" collson: y (for partcles) v f y v lghter one rebouns of assve one Shoup 76 Shoup 77
We have conservaton of oentu (always for solate systes): We have conservaton of oentu (always for solate systes): p p y v y y sn v f sn v p p y v p X v f y y sn vf sn Shoup 78 Shoup 78 We have conservaton of oentu (always for solate systes): f an elastc collson, we also have: f f (conservaton of knetc energy) v p p y v p p y X v f y y sn vf sn Lets epan our horzons an conser "etene" objects Approach s to conser a syste of screte pont partcles, then epan ths to a "contnuous eu",.e. sol objects So conser the followng syste of two partcles attache by a rg ro: rotates sn vf sn Y sn v f sn Shoup 78 "center of ass" Shoup 79
f we push at C, then syste oes not rotate, but oves as a "pont-partcle": so we can splfy f we use C Center of ass for two partcles s: ("weghte average") C For a larger syste of partcles: C...... n n n y c + C Shoup 8 X C s the coornate of the center of ass We can use slar equatons for y & z, so we have: C y C y z C z (8.3) These are the coornates of the Center of ass. The poston vector s: r C C y C z C y z r C k j k j y j z k r C r C r (8.3) Shoup 8 So, to copute the C of a collecton of partcles, su over all partcle poston vectors: r C r 3 4 r 3 r 4 z + C r r y As before, can put nto poston vector for: r C r ((8.34) f the followng boes are hoogeneous, where are there centers of ass? Now eten ths to a contnuous boy (etene boy): su over all sall ass eleents: take lt: C l C (fg 8.4) sae for y & z Shoup 8 +C + C sphere ro + + C C oughnut cone Shoup 83
Center of gravty: A up all the F g 's for each eleent of the boy Su s g, an the boy acts as f g ( acte at a pont whch s efne as the "center of gravty" ts the "average poston" of all the gravtatonal forces on each eleent n the boy Only f "g" s constant over full boy oes C = Center of Gravty Can eterne center of gravty by: "hang objecro two ponts an eterne "vertcal" lne for each. ntersecton of lnes s center of gravty a C The C allows us to "easfy" the escrpton of the oton of a syste of partcles (lke an etene object) Take the te ervatve of the C ( poston vector to get ts velocty: r v C r C v v C v p (just lke a sngle partcle of ass ) (8.36) Take the te ervatve of v C to get the acceleraton: v C v a v C a C F (8.38) F a Shoup 84 Shoup 85 Note: the F n (8.38) are s the neorce on the th eleent, both nternal an eternal to the syste However, the nternal forces cancel each other by Newton's 3 r law an only the eternal forces survve, so Newton's n law for a syste of partcles s: F et a C P tot (8.39) Shoup 86