Lecture 09 Systems of Particles and Conservation of Linear Momentum

Transcription

1 Lecture 09 Systes o Partcles and Conseraton o Lnear oentu 9. Lnear oentu and Its Conseraton 9. Isolated Syste lnear oentu: P F dp dt d( dt d dt a solated syste F ext 0 dp dp F, F dt dt dp dp d F F 0, 0 ( P P dt dt dt P tot const p p p p the law o conseraton o lnear oentu Exaple: Can we really gnore the knetc energy o the Earth? K E E E, EE bb K b bb, E b, b E K K E E b b b E b E 9. Nonsolated Syste (Ipulse I F t P Exaple: How good are the bupers? In a crash test, an autooble o ass 00 kg colldes wth a wall. The ntal and nal eloctes o the autooble are = - /s and =.6 /s. I the collson lasts or 0. s, nd the pulse due to the collson and the aerage orce excerted on the autooble.

2 ( I, N t I F Collsons n One Denson What s a collson? A collson s an solated eent n whch two or ore bodes (the colldng bodes exert relately strong orces on each other or a relately short te. Don t need real touch. Rules: The lnear oentu ust be consered whether t s elastc or nelastc collson. Elastc collson: the total knetc energy ust be consered. Inelastc collson: the total knetc energy o the syste s not the sae beore and ater the collson. Perectly Inelastc Collsons: (lnear oentu s consered (, Elastc Collsons: (both lnear oentu and energy are consered, ( (, ( ( ( ( ( (,, Specal cases: when 0,,

3 . Equal asses:, 0,. A asse target:,,. A asse projectle:,, Saple Exaple: Two etal spheres, suspended by ertcal cords, ntally just touch. Sphere, wth ass = 0 g, s pulled to the let to heght h = 8.0 c, and then released ro rest. Ater swngng down, t undergoes an elastc collson wth sphere, whose ass = 7 g. What s the elocty o sphere just ater the collson? gh, gh ,,, ( (,, (, Saple Exaple: A block o ass =.60 kg ntally ong to the rght wth a speed o 4.00 /s on a rctonless horzontal track colldes wth a sprng attached to a second block o ass =.0 kg ntally ong to the let wth a speed o.0 /s. s 600 N/. (a Fnd the eloctes o the two blocks ater the collson ,, 8 The sprng constant (b Durng the collson, at the nstant block s ong to the rght wth a elocty o +.00 /s, deterne the elocty o block ,. 74 (c Deterne the dstance the sprng s copressed at that nstant. kx

4 (d What s the axu copresson o the sprng durng the collson? when two blocks are ong at the sae speed. Exaple: The Ballstc Pendulu The ballstc pendulu s an apparatus used to easure the speed o a ast-ong projectle, such as a bullet. A bullet o ass s red nto a large block o wood o ass suspended ro soe lght wres. The bullet beds n the block, and the entre syste swngs through a heght h. How can we deterne the speed o the bullet ro a easureent o h? Obtan the speed o and edately ater collson: gh Apply the conseraton o oentu: gh A Saple Proble: A karate expert strkes downward wth hs st (o ass = 0.70 kg, breakng a 0.4 kg board. He then does the sae to a. kg concrete block. The sprng constants k or bendng are 4. the board and N/ or 0 6 N/ or the block. Breakng occurs at a delecton d o 6 or the board and. or the block. (a Just beore the object (board or block breaks, what s the energy stored n t? Board: Block: 4 U kd 4.0 (6 0. J 6 U kd.6 0 (. 0. 6J (b What s the lowest st speed st requred to break the object (board or block? Assue the ollowng: The collsons are copletely nelastc collsons o only the st and the object. Bendng begns just ater the collson. echancal energy s consered ro the begnnng o the bendng untl 4

5 just beore the object breaks. The speed o the st and object s neglgble at that pont. st (, ( U st U ( ( ( U 9. Collsons n Two Densons For conseraton o oentu: Checked by ther coponents p p p p p x px p x p x, p y py p y p y For conseraton o knetc energy: K K K K p p p p p x p y px p y Spled questons: cos cos sn sn Exaple: A proton colldes elastcally wth another proton that s ntally at rest. The ncong proton has an ntal speed o.x0 /s and akes a glancng collson wth the second proton. Ater collson, one proton oes o at an angle o 7 o to the orgnal drecton o oton, and the second delects at an angle o to the sae axs. Fnd the nal speeds o the two protons and the angle. cos7 o cos sn 7 o sn cos cos7 o

6 sn sn 7 o o o ( cos7 ( sn 7 o cos , sn sn 7 o 4 4 Saple Exaple: Two skaters collde and ebrace, n a copletely nelastc collson. Thus, they stck together ater pact, where the orgn s placed at the pont o collson. Alred, whose ass A s 8 kg, s orgnally ong east wth speed A = 6. k/h. Barbara, whose ass B s kg, s orgnally ong north wth speed B = 7.8 k/h. a What s the elocty V o the couple ater they collde? P ˆ 8 6.ˆ 4.6ˆ A A A P B ˆj B B 7.8 ˆj 49 ˆj 4.6ˆ 49 ˆj P ( A B PA PB,.7ˆ.ˆj k / hr 9.6 The Center o ass x x x x x x x n x Three Denson: dscrete ass ponts: r n r 6

7 contnuous ass: x x V xd, y yd, z zd xdv, y ydv, z V zdv V Saple Exaple: Three partcles o asses =. kg, =. kg, and =.4 kg or an equlateral trangle o edge length a = 40 c. Where s the center o ass o ths three-partcle syste? r (0,0, r (40,0, r (70,0 r r r. (0,0. (40,0.4 (70,0 r (...4 (8,8 Saple Exaple: Fgure shows a unor etal plate P o radus R ro whch a dsk o radus R has been staped out (reoed n an assebly lne. Usng the xy coordnate syste shown, locate the center o ass cop o the plate. x 4 0 ( ( R R R c 4 ( x 4 0 ( ( R R R c 4 ( Gratatonal Potental Energy o a Syste Fndng the Center o ass by Integraton Rod: Exaple: Show that the center o ass o a rod o ass and length L les dway between ts ends, assung the rod has a unor ass per unt length. X x L x dx L 0 L Suppose the rod s non-unor wth ts ass densty ares as center o ass. O x. Fnd the O 7

8 xdx L L dx L 0, X L 0 Secrcular Hoop: d ds rd r r cosxˆ r snyˆ L L r rr r cosxˆ r snyˆ rd 0 r r yˆ 9.7 oton o a Syste o Partcles r a a r r r... nrn... a a a... nan F F F... F n n n Saple Exaple: The three partcles are ntally at rest. Each experences an external orce due to bodes outsde the three-partcle syste. The drectons are ndcated, and the agntudes are F = 6.0 N, F = N, and F = 4 N. What s the acceleraton o the center o ass o the syste, and n what drecton does t oe? r (,, r (4,, r (, r r r r c F ( 6,0, F (6,6, F (4,0, F ( 6 8,6 8

9 a F (6 8 ˆ 6 6 ˆj Knetc Energy o a Syste K K, u K u u u Krel The Center-o-ass Reerence Frae I the net external orce on a syste reans zero, the elocty o the center o ass reans constant. F d total 0 F total dt Deorable Systes Pushng on a Sprng: As shown n the rght gure, two blocks are at rest on a rctonless, leel table. Both blocks hae the sae ass, and they are connected by a sprng o neglgble ass. The separaton dstance o the blocks when the sprng s relaxed s L. Durng a te nteral t, a constant orce o agntude F s appled horzontally to the let block, ong t through a dstance x. Durng the te nteral, the rght block oes through a dstance x. At the end o ths te nteral, the orce F s reoed. (a Fnd the resultng speed c o the center o ass o the syste. The external orce s used to ncrease the oentu o the syste. The oentu s related to the nal speed o the syste. F t ( (change o the oentu. The dsplaceent o the center o ass (nal co poston s subtracted by ntal co poston s calculated by the aerage speed o the co.., ag x L x 0 L t x x 0, ag 9

10 x x F ( F x x 9.9 Rocket Propulson P j P, d U ( d ( d rel rel : relate speed between the rocket and the exhaust products d U d U d dd 0 d d U d 0 rel d d 0, d d (a rel d d d rel, R, Rrel a dt dt dt (b d d d rel, d rel, rel ln rel ln Saple Proble: A rocket whose ntal ass s 80 kg consues uel at the rate R =. kg/s. The speed rel o the exhaust gases relate to the rocket engne s 800 /s. a What thrust does the rocket engne prode? T R rel b What s the ntal acceleraton o the rocket? a T s 80 / (c Suppose, nstead, that the rocket s launched ro a spacecrat already n deep space, where we can neglect any gratatonal orce actng on t. The ass o the rocket when ts uel s exhausted s 80 kg. What s ts speed relate to the spacecrat at that te? Assue that the spacecrat s so asse that the launch does not alter ts speed. 0

11 80 0 0, rel ln 800 ln 400 / s 80