C3 Inteactions tansfe momentum C4 - Paticles and Systems Geneal Physics 1
Solutions to HW Download PobViewe 1.4 www.physics.pomona.edu/sixideas/sicp.html Passwod fo daily HW poblems kllxjks8 Geneal Physics 2
Displacement " = 2 # $ "x' $ & ) & & "y ) = & %& "z() %& x 2 y 2 z 2 1 ' ) ) # () $ & & %& x 1 y 1 z 1 ' ) ) () = $ x 2 # x 1 ' & ) & y 2 # y 1 ) %& z 2 # z 1 () (x 2, y 2, z 2 ) (x 1, y 1, z 1 ) Geneal Physics 3
Velocity Change in position in a small time inteval Column vecto v d = dt & v = $ $ v $ % v x y #!!!" 1 = dt & dx# & dx $! = $ $ dy! $ dy $ % dz! " $ % dz dt# dt!! dt!" z Geneal Physics 4
Momentum Quantity of motion Vecto Defined as p " mv in tems of mass and velocity Units ae kg m/s Geneal Physics 5
Momentum - skill building poblem What is the momentum of an 8-kg bowling ball olling at 2 m/s? Geneal Physics 6
Think-Pai-Shae What is the magnitude of the momentum of a 28-g spaow flying with a speed of 8.4 m/s? Geneal Physics 7
Impulse Amount of momentum that an inteaction (A) tansfes to eithe paticle duing a shot inteval [ ] A dp Geneal Physics 8
Change in momentum Multiple inteactions contibute thei own impulses esulting in a change in momentum d p = d p [ ] A + d p [ ] B + d p [ ] C +... Geneal Physics 9
Impulse - skill building poblem What is the impulse on an 8-kg ball olling at 2 m/s when it bumps into a pillow and stops? Geneal Physics 10
Impulse - skill building poblem How much impulse stops a 50-kg caton sliding at 4 m/s when it meets a ough suface? Geneal Physics 11
Impulse Questions In tems of impulse and momentum, why do aibags in cas educe the chances of injuy in accidents? When jumping fom a significant height, why is it advantageous to land with you knees bent? Geneal Physics 12
Financial Analogy Paticle s momentum is a peson s net financial woth Inteaction is a financial inteaction Inceases one s net woth at the expense of the othe s Inpulse is a check that is witten o eceived Momentum in geneal is like money Geneal Physics 13
Foce Rate at which momentum flows F A = d p dt [ ] A Geneal Physics 14
Impulse Change in momentum Vecto quantity The magnitude of the impulse is equal to the aea unde the foce-time cuve The impulse can also be found by using the time aveaged foce [ dp] F dt =! A A This would give the same impulse as the timevaying foce does Geneal Physics 15
Think-Pai-Shae An estimated foce-time cuve fo a baseball stuck by a bat is shown. Fom this cuve, detemine the impulse deliveed to the ball. Geneal Physics 16
Mass and Weight In physics, mass and weight ae completely distinct concepts F = mg g Whee mass is a scala, g is the gavitational field vecto F g is the weight acting on an object Geneal Physics 17
Momentum flow and motion - 1 Multitap Geneal Physics 18
Momentum flow and motion - 2 Resevoi Geneal Physics 19
Multitap and Resevoi - Execise C3X.7 Imagine a pool ball that is initially olling due noth on a pool table. You opponent blows constantly diectly eastwad on the ball. Model the ball as a paticle moving initially nothwad but expeiences a constant eastwad foce. Sketch momentum aows fo the ball at successive times using the multitap and esevoi models. Geneal Physics 20
Solution Geneal Physics 21
Consevation of Momentum A system s total momentum is the vecto sum of the momenta of its constituent paticles p " p + p + p +...+ p tot 1 2 3 N The law of consevation of momentum: The total momentum of an isolated system is conseved; that is, it does not change in time. Geneal Physics 22
Consevation of Momentum p ix The total momentum of a system is constant. Fo collisions, use p p befoe 1i + = p 2i p = afte p 1 f In component fom, the total momenta in each diection ae independently conseved = p fx p iy = p fy + p 2 f p iz = p fz Geneal Physics 23
Think-Pai-Shae A 95-kg halfback moving at 4.1 m/s on an appaent beakaway fo a touchdown is tackled fom behind. When he was tackled by an 85-kg coneback unning at 5.5 m/s in the same diection, what was thei mutual speed immediately afte the tackle? Geneal Physics 24
The Cente of Mass Thee is a special point in a system o object, called the cente of mass, that moves as if all of the mass of the system is concentated at that point The system will move as if an extenal foce wee applied to a single paticle of mass M located at the cente of mass M is the total mass of the system Geneal Physics 25
Motion of the Cente of Mass A pojectile is fied into the ai and suddenly explodes With no explosion, the pojectile would follow the dotted line Afte the explosion, the cente of mass of the fagments still follows the dotted line, the same paabolic path the pojectile would have followed with no explosion Geneal Physics 26
Cente of Mass CM is the system s cente of mass CM " 1 M m 1 1 + m 2 2 +...+ m N N ( ) 1, 2,..., N ae the positions of the system s N paticles m 1,m 2,...,m N ae the masses is the system s mass M " m 1 + m 2 +...+ m N Geneal Physics 27
Cente of mass - column vecto notation " 1 CM M m + m +...+ m 1 1 2 2 N ( ) N " $ $ # $ x CM y CM z CM % ' ' &' ( = 1 " * M m $ 1$ * ) # $ x 1 y 1 z 1 % " x 2 % " x N % + ' $ ' $ '- ' + m 2 $ y 2 ' +...+ m N $ y N ' &' # $ z 2 &' # $ z N &' -, Geneal Physics 28
Cente of Mass, Example Both masses ae on the x-axis The cente of mass is on the x-axis The cente of mass is close to the paticle with the lage mass Geneal Physics 29
Cente of Mass Dick Fosbuy intoduced a new high jump technique - the Fosbuy Flop - at the 1968 Olympics. Raised wold ecod 30 cm! Geneal Physics 30
Cente of mass example Inset example on p. 67 to intoduce activity Geneal Physics 31
Cente of mass activity - 1 You will need Mete stick Fulcum Mass hanges masses Geneal Physics 32
Cente of mass activity - 2 Use column vecto notation fo two examples Same mass Find location of and compae to expeimental esult Diffeent masses Geneal Physics 33
System s Momentum A system s cente of mass esponds to its extenal inteactions exactly as a point paticle would espond to those inteactions p tot = M v CM Geneal Physics 34
Inteacting with the eath-1 Does the eath s momentum change when inteacting with you? It must if momentum is conseved! But if I jump, the eath doesn t move! Use a efeence fame that includes you AND the eath that is fixed to the CM of you and the eath Feely floating fame Geneal Physics 35
Inteacting with the eath - 2 v f % Assume Peson s mass ~ 50 kg Peson jumping ~ 3 m/s vetically Peson s impulse upwad = 150 kg m/s Eath s impulse is the same, 150 kg m/s Calculate velocity ( ) mag( p ) eath kg m s mag v 150 $ f = = # 25" 10! 24 m 6" 10 kg eath 24 m s Impossible to detect Geneal Physics 36