Lecture 16 Chapter 9 Physcs I 11.06.2013 Lnear oentu. Center of ass. Course webste: http://faculty.ul.edu/ndry_danylov/teachng/physcsi Lecture Capture: http://echo360.ul.edu/danylov2013/physcs1fall.htl
Outlne Chapter 9 2-D collsons Systes of partcles (extended objects) Center of ass
Collsons n 2D v v B B (1D) You know how to fgure out the results of a collson between objects n 1D: -use conservaton of oentu- and, -f collson s ellastc, conservaton of ech. energy. You can contnue to use the sae rules n 2D collsons as follows: If If ext F x ext F y 0, then oentu n x - drecton s conserved ntal fnal px px 0, then oentu n y - drecton s conserved ntal fnal p p ech. Energy s v 2 ( 2 ) y ntal ( y conserved n elastc collsons (not n each denson) v 2 2 ) fnal
Collsons n 2D: oentu Conservaton (I) projectle ( ) oves along the x-axs and hts a target ( B ) at rest. fter the collson, the two objects go off at dfferent angles. Snce net external forces n x and y drectons are zero, p x and p y are conserved
Collsons n 2D: oentu Conservaton (II) conservaton of x-oentu: v y v' cos B By B v' 0 v' sn ' v' sn ' 1 2 v 2 1 2 v' 2 1 2 2 Bv' B Bx conservaton of y-oentu: B before x cos Two equatons, can be solved for two unknowns If collson s elastc, we get a thrd equaton (conservaton of echancal energy) Three equatons, can be solved for three unknowns p p B before y p p after x after y v n 2-D collsons v B ( v vb )
Exaple: 2D collson Ball ovng at 4 /s strkes ball B (of equal ass) at rest. fter the collson, ball travels forward at an angle of +45º, and ball B travels forward at -45º. What are the fnal speeds of the two balls? conservaton of y-oentu 0 v' sn 45 v' sn( 45) ( 4 ) v' cos 45 v' cos( 45) s 0 v' sn 45 v' B sn(45) v' v' B conservaton of x-oentu 4 ( 1 2 )v' ( 1 2 )v' B 4 2 v' v' B B B v' v' B 2 2 v v B 2.83 s
Center of ass ()
Center of ass () How to descrbe otons lke these? The general oton of an object can be consdered as the su of the translatonal oton of a certan pont, plus rotatonal oton about that pont. That pont s called the center of ass pont
Center of ass () Ths allows us to fnd a translatonal oton; however, t doesn t tell us anythng about rotaton about the. Pure translatonal oton Translatonal plus rotatonal oton
How to fnd the center of ass? pont depends only on the ass dstrbuton of an object.
Center of ass: Defnton Poston vector of the : r 1 r 2 1 2 r r 1 2 3 ( x, y, z ) n 1 r 1 total ass of the syste Coponent for: n 1 x x y 1 1 n 1 y r 3 3 z 1 n 1 z
Center of ass (2 partcles, 1D) x=0 1 2 x 1 x 2 x-axs x 1 where n 1 x Poston of the : 1x1 2x2 x 1 2 Velocty of the : cceleraton of the : 1v 1 2 v 2v 1a1 2 a 2a
Exaple: Center of ass (2 partcles) What s the center of ass of 2 pont asses ( =1 kg and B =3 kg), at two dfferent ponts: =(0,0) and B=(2,4)? By defnton: x y x Bx ( B ) y B y ( B ) x y =1 kg and B =3 kg (1 0) (3 2) 1 3 (1 0) (3 4) 1 3 1.5 3 B B Or n a vector for: =(0,0) B=(2,4) r 1.5î 3ĵ
oton Consder velocty : 1v 1 2v2 v 1 2 v X v v v ( v) 2 0 v v/2 X v v (0) 2 v / 2 For unequal asses v v/3 X 2 v v 2(0) 2 v / 3
Two equal-ass partcles ( and B) are located at soe dstance fro each other. Partcle s held statonary whle B s oved away at speed v. What happens to the center of ass of the two-partcle syste? ConcepTest 1 oton of ) t does not ove B) t oves away fro wth speed v C) t oves toward wth speed v 1D) t oves away fro wth speed 2v E) t oves toward wth speed 2v Let s say that s at the orgn (x = 0) and B s at soe poston x. Then the center of ass s at x/2 because and B have the sae ass. If v = x/t tells us how fast the poston of B s changng, then the poston of the center of ass 1ust be changng lke (x/2)/t, whch s sply v.? v/2 v v 2X (0) ( v) 2 1 v / 2 1v 1 2 v 2v
of a sold object Let s fnd of an extended body: Before, for any partcles we had r n 1 r 1 r Now, let s dvde ass nto saller sectons r 1 r 1 r l 0 1 r r rd x 1 xd y 1 yd z 1 zd
of sold syetrcal objects The easest trck s to use syetry If we break a syetry, the wll be shfted Old Old
of Sold Objects (nce trck) How to deal wth objects lke ths? Dvde t nto syetrcal objects 2 1 2 1 of the orgnal object
ConcepTest 2 Center of ass The dsk shown below n (1) clearly has ts center of ass at the center. Suppose a saller dsk s cut out as shown n (2). Where s the center of ass of (2) as copared to (1)? ) hgher B) lower C) at the sae place D) there s no defnable n ths case (1) (2) X
Before we used Newton s 2 nd law for ponts (whch had asses but not szes) Now, Newton s 2 nd law for a syste of partcles (or extended bodes)
Newton s 2 nd law for a syste of partcles n 1 Poston vector of the : r r 1 d 2 r Snce a, then acceleraton s : a 2 d t n a a 1 Where F forces actng on r r 2 1 r 3 r However nt ernal a F F n F 1 1 2 external ext n 1 a 1 n F 1 F nt ernal F ext net partcle forces cancel each other ( N.3rd law) The of a syste (total ass ) oves lke a sngle partcle of ass acted upon by the sae net external force.
F pont descrbes translatonal oton of a syste F It doesn t atter where you appled an ext. force, transl. oton of the syste wll be the sae F F
Newton s 2 nd law for a syste of partcles (II) a F net ext In the absence of external forces, the oton of the center of ass of a syste of partcles (or an extended object) s unchanged.
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Revew: Ballstc Pendulu devce used to easure the speed of a bullet. v o F T + F T v 1 F T F T h g g Collson: Swngng: ech. Energy s not conserved (wood s crushed). Ln. oentu s conserved (ext. forces cancel each other) ech. Energy s conserved. Ln. oentu s not conserved (ext. forces don t cancel each other)
ConcepTest 1 uranu nucleus (at rest) undergoes fsson and splts nto two fragents, one heavy and the other lght. Whch fragent has the greater oentu? Nuclear Fsson I ) the heavy one B) the lght one C) both have the sae oentu D) possble to say The ntal oentu of the uranu was zero, so the fnal total oentu of the two fragents ust also be zero. Thus the ndvdual oenta are equal n agntude and opposte n drecton. 1 2