Center of Mass and Linear

Transcription

1 PH 22-3A Fall 2009 Cente of Mass and Linea Moentu Lectue 5 Chapte 8 (Halliday/Resnick/Walke, Fundaentals of Physics 8 th edition)

2 Chapte 9 Cente of Mass and Linea Moentu In this chapte we will intoduce the following new concepts: -Cente of ass (co) fo a syste of paticles -The velocity and acceleation of the cente of ass -Linea oentu fo a single paticle and a syste of paticles We will deive the equation of otion fo the cente of ass, and discuss the pinciple of consevation of linea oentu Finally we will use the consevation of linea oentu to study collisions in one and two diensions and deive the equation of otion fo ockets 2

3 The Cente of Mass: Conside a syste of two paticles of asses and 2 2 at positions x and x, espectively. We define the position of the cente of ass (co) as follows: x co = x + x We can genealize the above definition i i fo a syste of n paticles as follows: x x+ x + x x x+ x + x x n n n n n co = = = x i i n M M i = Hee M is the total ass of all the paticles M = n We can futhe genealize the definition fo the cente of ass of a syste of paticles in thee diensional space. We assue that the the i-th paticle ( ass i ) has position vecto i n co = i i M i = 3

4 The position vecto fo the cente of ass is given by the equation: The position vecto can be witten as: ˆ ˆ ˆ co = xcoi + yco j+ zcok The coponents of co ae given by the equations: n n n xco = x i i yco = i yi zco = z i i M M M i= i= i= co = M n i= i i The cente of ass has been defined using the equations given above so that t it has the following popety: The cente of ass of a syste of paticles oves as though all the syste's ass wee concetated thee, and that the vecto su of all the extenal foces wee applied thee The above stateent will be poved late. An exaple is given in the figue. A baseball bllbat is flipped dinto the ai and oves unde the influence of the gavitation foce. The cente of ass is indicated by the black dot. It follows ows a paabolic path as discussed in Chapte 4 (pojectile otion) All the othe points of the bat follow oe coplicated paths 4

5 The Cente of Mass fo Solid Bodies Solid bodies can be consideed as systes with continuous distibution of atte The sus that ae used fo the calculation of the cente of ass of systes with discete distibution of ass becoe integals: xco = xd yco yd zco zd M = M = M The integals above ae athe coplicated. A siple special case is that of d M unifo objects in which the ass density ρ = is constant and equal to dv V xco = xdv yco ydv zco zdv V = V = V In objects with syety eleents (syety point, syety line, syety plane) it is not necessay to eveluate the integals. Thecenteofassliesonthesyety on the syety eleent. Fo exaple the co of a unifo sphee coincides with the sphee cente In a unifo ectanglula object the co lies at the intesection of the diagonals C. C 5

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7 F F 2 2 F 2 O x z 3 y Newton's Second Law fo a Syste of Paticles Conside a syste of paticles of asses,,..., and position vectos, 2, 3,..., n, espectively. The position vecto of the cente of ass is given by: n 2 3, n Mco = n n We take the tie deivative of both sides d d d d d M co = n n dt dt dt dt dt Mvco = v + 2v2 + 3v nvn Hee vco is the velocity of the co and v is the velocity of the i-th paticle. We take the tie deivative once oe i d d d d d M vco = v+ 2 v2 + 3 v n vn dt dt dt dt dt Maco = a + 2a2 + 3a nan Hee aco is the acceleation of the co and a is the acceleation of the i-th paticle i 7

8 F F 2 2 F 2 O x z Ma a a a a co = n n We apply Newton's second law fo the i-th paticle: 3 a i i = Fi Hee Fi is the net foce on the i-th paticle y Maco = F+ F2 + F Fn The foce Fi can be decoposed into two coponents: applied and intenal app int Fi = Fi + Fi The above equation takes the fo: app int app int app int app int Maco = ( F + F ) + ( F2 + F2 ) + ( F3 + F3 ) ( Fn + Fn ) Ma = F app + F app + F app F app + F int + F int + F int F int ( 2 3 ) ( 2 3 ) co n n The su in the fist paenthesis on the RHS of the equation above is just F net The su in the second paethesis on the RHS vanishes by vitue of Newton's thid law. The equation of otion fo the cente of ass becoes: In tes of coponents we have: Fnet, x = Maco, x Fnet, y = Maco, y Fnet, z = Ma co, z Ma co = F net 8

9 Ma co = F net F F F = Ma net, x co, x = Ma net, y co, y = Ma net, z co, z The equations above show that the cente of ass of a syste of paticles oves as though all the syste's ass wee concetated thee, and that the vecto su of all the extenal foces wee applied thee. A daatic exaple is given in the figue. In a fiewoks display a ocket is launched and oves unde the influence of gavity on a paabolic path (pojectile otion). At a cetain point the ocket explodes into fagents. If the explosion had not occued, the ocket would have continued to ove on the paabolic tajectoy (dashed line). The foces of the explosion, even though lage, ae all intenal and as such cancel out. The only extenal foce is that of gavity and this eains the sae befoe and afte the explosion. This eans that the cente of ass of the fagents folows the sae paabolic tajectoy that the ocket would have followed had it not exploded 9

10 p v = v Linea Moentu p Linea oentu p of a paticle of ass and velocity v is defined as: p= v The SI unit fo lineal oentu is the kg./s Below we will pove the following stateent: The tie ate of change of the linea oentu of a paticle is equal to the agnitude of net foce acting on the paticle and has the diection of the foce dp In equation fo: Fnet = We will pove this equation using dt Newton's second law dp d dv p = v = ( v) = = a = Fnet dt dt dt This equation is stating that the linea oentu of a paticle can be changed only by an extenal foce. If the net extenal foce is zeo, the linea oentu cannot change F net = dp dt 0

11 z p p 3 2 p 2 x O 3 y The Linea Moentu of a Syste of Paticles In this section we will extedend the definition of linea oentu to a syste of paticles. The i-th paticle has ass i, velocity v i, and linea oentu p i We define the linea oentu of a syste of n paticles as follows: P = p + p2 + p pn = v + 2v2 + 3v nvn = Mv co The linea oentu of a syste of paticles is equal to the poduct of the total ass M of the syste and the velocity v co of the cente of ass dp d The tie ate of change of P is: = ( Mvco ) = Maco = Fnet dt dt The linea oentu P of a syste of paticles il can be changed only by a net extenal foce F. If the net extenal foce F is zeo P cannot change net P = p + p + p + + p = Mv n co net dp Fnet dt =

12 Exaple. Motion of the Cente of Mass 2

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14 Collision and Ipulse We have seen in the pevious discussion that the oentu of an object can change if thee is a non-zeo extenal lfoce acting on the object. Such foces exist duing the collision of two objects. These foces act fo a bief tie inteval, they ae lage, and they ae esponsible fo the changes in the linea oentu of the colliding objects. Conside the collision of a baseball with a baseball bat The collision stats at tie ti when the ball touches the bat and ends at t f when the two objects sepaate The ball is acted upon by a foce F ( t) duing the collision The agnitude Ft ( ) of the foce is plotted vesus tin fig.a The foce is non-zeo only fo the tie inteval t i < t < t f dp Ft ( ) = Hee pis the linea oentu of the ball dt d tf tf p= Ftdt () dp= Ftdt () t i t i 4

15 dp = F () tdt dp = p p =Δ p = change in oentu t t t f f f t t t i i i t f F () tdt is known as the ipulse J of the collision t i t f J = F( t) dt The agnitude of J is equal to the aea t i unde the F vesus t plot of fig.a Δ p= J f In any situations we do not know how the foce changes with tie but we know the aveage agnitude F ave of the collision foce. The agnitude of the ipulse is given by: J = F Δt whee Δ t = t t ave f i Geoetically this eans that the the aea unde the F t l t (fi ) i l t th d th F vesus ave t plot (fig.b) Δ p = J vesus plot (fig.a) is equal to the aea unde the J = FaveΔt 5 i

16 Collisions. Ipulse and Moentu 6

17 The Ipulse-Moentu Theoe 7

18 F ave Seies of Collisions Conside a taget which collides with a steady stea of identical paticles of ass and velocity v along the x-axis A nube n of the paticles collides with the taget duing a tie inteval Δ t. Each paticle undegoes a change Δp in oentu due to the collision with the taget. Duing each collision a oentu change Δp is ipated on the taget. The Ipulse on the taget duing the tie inteval Δt is: J = nδp The aveage foce on the taget is: J n Δ p n Fave = = = Δv Hee Δv is the change in the velocity Δt Δt Δt of each paticle along the x-axis due to the collision with the taget Δ Δ Fave = Δv Hee is the ate at which ass collides with the taget Δt Δt If the paticles stop afte the collision then Δ v = 0 v = v If the paticles bounce backwads then Δ v = v v = 2v 8

19 z p p 3 2 p 2 x O 3 y Consevation of Linea Moentu Conside a syste of paticles fo which F net = 0 dp F net 0 P Constant dt = = = If no net extenal foce acts on a syste of paticles the total llinea oentu P cannot change total linea oentu total linea oentu at soe initial tie t = i at soe late tie t f The consevation of linea oentu is an ipotan pinciple in physics. It also povides a poweful ule we can use to solve pobles in echanics such as collisions. Nt Note : In systes in which h F net = 0 we can always apply consevation of linea oentu even when the intenal foces ae vey lage as in the case of colliding objects Note 2: We will encounte pobles (e.g. inelastic collisions) in which the enegy is not conseved but the linea oentu is 9

20 The Pinciple of Consevation of Linea Moentu 20

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22 Moentu and Kinetic Enegy in Collisions Conside two colliding objects with asses and 2, initial velocities v and v and final velocities v and v, i 2i f 2 f espectively If the syste is isolated i.e. the net foce F net = 0 linea oentu is conseved The conevation of linea oentu is tue egadless of the the collision type This is a poweful ule that allows us to deteine the esults of a collision without knowing the details. Collisions ae divided into two boad classes: elastic and inelastic. A collision is elastic if thee is no loss of kinetic enegy i.e. K = K A collision is inelastic if kinetic enegy is lost duing the collision due to convesion it into othe fos of enegy. In this case we have: K < K A special case of inelastic collisions ae known as copletely inlelastic. In these collisions the two colliding objects stick togethe and they ove as a single body. In these collisions the loss of kinetic enegy 22 is axiu f i i f

23 One Diensional Inelastic Collisions In these collisions the linea oentiu of the colliding objects is conseved p i + p2i = pf + p2 f v + v = v + v i 2i f 2 f One Diensional Copletely Inelastic Collisions In these collisions the two colliding objects stick togethe and ove as a single body. In the figue to the leftwe show a special case in which v 2i = 0. v i = V + V 2 V = v i + 2 The velocity of the cente of ass in this collision P p i + p2i v i is vco = = = In the pictue to the left we show soe feeze-faesfaes of a totally inelastic collition 23

24 One-Diensional Elastic Collisons Conside two colliding objects with asses and 2, initial velocities v and v and final velocities v and v, espectively i 2i f 2 f Both linea oentu and kinetic enegy ae conseved. Linea oentu consevation: v + v = v + v (eqs.) i 2i f 2 f v i v v 2 f v i 2 2 f Kinetic enegy consevation: + = + (eqs.2) We have tw o equations and two unknowns, v and v 2 2 f i 2i f f f f If we solve equations and 2 fo v and v we get the following solutions: 2 v = v + v 2 v = v + v 2 2 f 2 + i i

25 The substitute v = 0inthe two solutions fo v and v ( v = ) Special Case of elastic Collisions-Stationay Taget 0 2i f f 2i 2 v = v + v v = f i 2i f i v = v + v v = 2 2 f i 2i 2 f i Below we exaine seveal special cases fo which we know the outcoe of the collision fo expeience. Equal asses = = 2 v = v = v = 2 f i i v = v = v = v 2 f i i i The two colliding objects have exchanged velocities 0 v v v i v 2i = 0 v f = 0 v 2f x x 25

26 v i 2 v 2i = 0 x v f 2 v 2f 2. A assive taget 2 v = v = v v f i i i x v2 f = v i = v i 2 v i Body (sall ass) bounces back along the incoing path with its speed pactically unchanged. Body 2 (lage ass) oves fowad with a vey sall speed because

27 v 2 i 2. A assive pojectil e 2 v 2i = 0 x v f v 2 2f v = v = v v x 2 2 f i i i v = v = v 2v f i i i Body (lage ass) keeps on going scacely slowed by the collision. Body 2 (sall ass) chages ahead at twice the speed of body 27

28 Elastic Collision 28

29 Inelastic Collision 29

30 Collisions in Two Diensions In this section we will eove the estiction that the colliding objects ove along one axis. Instead we assue that the two bodies that paticipate in the collision ove in the xy -plane. Thei asses ae and 2 The linea oentu of the syte is conseved: p + p = p + p i 2i f 2 f If the syste is elastic the kinetic enegy is also conseved: K + K = K + K 2 i 2 i f 2 f We assue that is stationay and that afte the collision paticle and paticle 2 ove at angles θ and θ with the initial diection of otion of axis: i = f θ+ v 2 2 f θ2 2 In this case the consevation of oentu and kinetic enegy take the fo: x v v cos cos (eqs.) y axis: 0= v sin θ + v sin θ (eqs.2) f 2 2 f v i = v 2 f + v 2 2 f (eqs.3) We have thee equations and seven vaiables: Two asses:, 2 thee speeds: v i, vf, v2 f and two angles: θ, θ2. If we know the values of fou of these paaetes we can calculate the eaining thee 30

31 Poble 72. Two 2.0 kg bodies, A and B collide. The velocities befoe the collision ae ˆ ˆ v (5 30 ) /s and ( 0 ˆ 5.0 ˆ A = i + j vb = i + j) /s. Afte the collision, v = ( 5.0iˆ + 20 ˆj) /s. What ae (a) the final velocity of B and (b) the change ' A in the total kinetic enegy (including sign)? (a) Consevation of linea oentu iplies v + v = v' + v'. A A B B A A B B Since A = B = = 2.0 kg, the asses divide out and we obtain v (5i ˆ 30ˆj)/s ( 0ˆi 5j)/s ˆ ( 5ˆi 20ˆ B = va + vb v A = j)/s = (0 ˆi + 5 ˆj) /s. (b) The final and initial kinetic enegies ae c h J K v 2 v f = ' A+ ' B = ( 2. 0) ( 5) = Ki = va + vb = ( 2. 0) c ( 0) + 5 h = 3. 0 J The change kinetic enegy is then ΔK = J (that is, 500 J of the initial kinetic enegy is lost). 3

32 Systes with Vaying Mass: The Rocket A ocket of ass M and speed v ejects ass backwads dm at a constant ate. The ejected ateial is expelled at a dt constant speed vel elative to the ocket. T hus the ocket loses ass and acceleates fowad. We will use the consevation of linea oentu to deteine the speed v of the ocket In figues (a) and (b) we show the ocket at ties t and t+ dt. If we assue that thee ae no extenal foces acting on the ocket, linea oentu is conseved ( ) ( )( ) p( t) = p t + dt Mv = UdM + M + dm v + dv (eqs.) Hee dm is a negative nube because the ocket's ass deceases with tie t U is the velocity of fthe ejected gases with espect to the inetial efeence fae in which we easue the ocket's speed v. We use the tansfoation equation fo velocities (Chapte 4) to expess U in tes of v el which is easued with epsect to the ocket. U = v+ dv vel We substitute U in equation and we get: 32 Mdv = dmvel

33 Using the consevation of linea oentu we deived the equation of otion fo the ocket Mdv = dmv el (eqs.2) We assue that ateial is ejected fo the ocet's nozzle at a constant ate dm = R (eqs.3) Hee R is a constant positive nube dt dv dm We devide both sides of eqs.(2) by dt M = vel = Rvel dt dt Ma = Rvel (Fist ocket equation) Hee a is the ocket's acceleation We use equation 2todeteinetheocket's the ocket speedasfunction of tie t f f dm dm dv = vel We integate both sides dv vel M = M f [ ] [ ] M M i f i el M el el i M f M f i v v = v ln M = v ln M =v ln M i vf vi = vel ln (Second ocket equation) v i M i /M f M 33 f v v i M M M i O v f

34 Poble 78. A 6090 kg space pobe oving nose-fist towad Jupite at 05 /s elative to the Sun fies its ocket engine, ejecting 80.0 kg of exhaust at a speed of 253 /s elative to the space pobe. What is the final velocity of the pobe? M i 6090 kg vf = vi + vel ln = 05 /s + (253 /s) ln = 08 /s. M f 600 kg 34