Chapter 7 Impulse and Momentum

Transcription

1 Chapter 7 Ipulse and Mentu Gals r Chapter 7 T study pulse and entu. T understand cnseratn entu. T study entu changes durng cllsns. T understand center ass and hw rces act n the c... T apply entu t rcket prpulsn. When the bat strkes the ball, the agntude the rce exerted n the ball rses t a axu alue and then returns t zer DEINITION O IMPULSE The pulse a rce s the prduct the aerage rce and the te nteral durng whch the rce acts: J t The cllsn te between a bat and a ball s ery shrt, ten less than a llsecnd, but the rce can be qute large. The te nteral durng whch the rce acts s t, and the agntude the aerage rce s. Ipulse s a ectr quantty and has the sae drectn as the aerage rce. SI unt: newtn secnds (N s) Mentu transer (cllsn) tescales Cllsns typcally nle nteractns that happen quckly. rce and Ipulse sae area Durng ths bre te, the rces nled can be qute large t t t bg, sall t t J t ntal nal The balls are n cntact r a ery shrt te. V t t t t t sall, bg

2 DEINITION O LINEAR MOMENTUM p The lnear entu an bject s the prduct the bject s ass tes ts elcty: SI unt: klgra eter/secnd (kg /s) Lnear Mentu Mentu a partcle s dened as the prduct ts ass and elcty p Mentu cpnents p x = x and p y = y Apples t tw-densnal tn as well Mentu (agntude) s related t knetc energy K p Relatn between Ipulse and Mentu (Newtn nd) a a t t t IMPULSE-MOMENTUM THEOREM When a net rce acts n an bject, the pulse ths rce s equal t the change n the entu the bject pulse t J p t nal entu ntal entu Ipulse s a ectr quantty; SI Unt: N s r kg / s pulse = change n entu! Ipulse and aerage rce We can use the ntn pulse t dene aerage rce, whch s a useul cncept. Dene aerage rce such that (een s nt cnstant), pulse s gen by a Exaple: A Well Ht Ball A baseball ( = 0.4 kg) has an ntal elcty = - 38 /s as t appraches a bat. The bat apples an aerage rce that s uch larger than the weght the ball, and the ball departs r the bat wth a nal elcty = 58 /s. (a) Deterne the pulse appled t the ball by the bat. (b) Assung that the te cntact s t = s, nd the aerage rce exerted n the ball by the bat. J a (t t ) a t r: a p t

3 Exaple : A Ranstr Ran ces dwn wth a elcty -5 /s and hts the r a car. The ass ran per secnd that strkes the r the car s kg/s. Assung that ran ces t rest upn strkng the car, nd the aerage rce exerted by the ran n the r. t t Neglectng the weght the randrps, the net rce n a randrp s sply the rce n the randrp due t the r. t kg s5 s 0.90 N (rce n the randrp) n-the-r = (Newtn s thrd law) Cnceptual Exaple: Halstnes ersus Randrps Instead ran, suppse hal s allng. Unlke ran, hal usually bunces the r the car. I hal ell nstead ran, wuld the rce be saller than, equal t, r greater than that calculated preusly? J t t r a randrp, the change n elcty s r (dwnward) t zer. r a halstne, the change s r (dwnward) t (upward). Thus halstnes hae a larger t Ipulse appled t aut cllsns The st prtant actr s the cllsn te r the te t takes the persn t ce t a rest Ths wll reduce the chance dyng n a car crash Ways t ncrease the te Seat belts Ar bags a p ( ) t t The ar bag ncreases the te the cllsn and absrbs se the energy r the bdy Cnseratn Lnear Mentu WORK-ENERGY THEOREM CONSERVATION O ENERGY Apply the pulse-entu there t the dar cllsn IMPULSE-MOMENTUM between tw bjects.. THEOREM??? Internal rces rces that bjects wthn the syste exert n each ther. External rces rces exerted n bjects by agents external t the syste. e.g. Weght=W Cnseratn Lnear Mentu t OBJECT W t OBJECT W t Cnseratn Lnear Mentu W t + W t W W t The nternal rces cancel ut. P Cnsder syste: bth bjects nled P

4 Prncple Cnseratn Lnear Mentu W W t P P su aerageexternalrcest I the su the external rces s zer, then P P P P 0 P P CONSERVATION O LINEAR MOMENTUM The ttal lnear entu an slated syste s cnstant (cnsered). An slated syste s ne r whch the su the aerage external rces actng n the syste s zer. Dentn Ttal Mentu r a Syste Partcles r a syste partcles the ttal entu P s the ectr su the nddual partcle enta: P p A p B p... cpnents ttal entu P (p A C p B p P p x P p y...) A,x A,y P The entu each bject wll change The ttal entu the syste reans cnstant there are n external rces C A A net B B C C pb,x... p... t B.y N N p... Exaple: Asseblng a reght Tran A reght tran s beng assebled n a swtchng yard. Car has a ass = kg and es at a elcty 0 = /s. Car, wth a ass = kg and a elcty 0 = +.3 /s, ertakes car and cuples t t. Neglectng rctn, nd the cn elcty the cars ater they bece cupled. Exaple: Recl a rle A arksan hlds a 3.00 kg rle lsely, allwng t t recl reely when red, and res a bullet ass 5.00 g hrzntally wth a speed B = 300 /s. What s the recl speed the rle? = PT P R P B Bere PT 0 Ater P R P B 0 0 B B R R B R B / s R Cncept Test: Expldng Prjectle A del rcket traels as a prjectle n a parablc path ater ts rst stage burns ut. At the tp ts trajectry, where ts elcty pnts hrzntally t the rght, a sall explsn separates t nt tw sectns wth equal asses. One sectn alls straght dwn, wth n hrzntal tn. What s the drectn the ther part just ater the explsn? A. Up and t the let B. Straght up C. Up and t the rght bere Exaple: Mentu Cnseratn A bx wth ass = 6.0 kg sldes wth speed = 4.0 /s acrss a rctnless lr n the pste drectn an x axs. It suddenly expldes nt tw peces. One pece, wth ass =.0 kg, es n the pste x-drectn wth speed = 8.0 /s. What s the elcty the secnd pece, wth ass = 4.0 kg? ater

5 Exaple: Cnseratn Lnear Mentu - Ice Skaters Startng r rest, tw skaters push aganst each ther n ce where rctn s neglgble. One s a 54-kg wan and ne s a 88-kg an. The wan es away wth a speed +.5 /s. nd the recl elcty the an. P P 54kg.5 s.5 88kg s 0 Cncept Test: Cnseratn Mentu A by stands at ne end a latng rat that s statnary relate t the shre. He then walks t the ppste end, twards the shre. Des the rat e (assue n rctn)?. N, t wll nt e at all. Yes, t wll e away r the shre 3. Yes, t wll e twards the shre Nte: Snce entu s cnsered n the by-rat syste and nether was ng at rst, the rat ust e n the drectn ppste t the by s. In Cllssns Ttal Mentu s Cnsered In cllsns, we assue that external rces ether su t zer, r are sall enugh t be gnred. Hence, entu s cnsered n all cllsns. A cllsn ay be the result physcal cntact between tw bjects Cntact ay als arse r the electrstatc nteractns the electrns n the surace ats the bdes Matheatcally (r tw bjects): Mentu s cnsered r the syste bjects The syste ncludes all the bjects nteractng wth each ther Assues nly nternal rces are actng durng the cllsn Can be generalzed t any nuber bjects Types Cllsns Mentu s cnsered n any cllsn Elastc cllsns bth entu and knetc energy are cnsered Inelastc cllsns Knetc energy s nt cnsered Se the knetc energy s cnerted nt ther types energy such as heat, sund, wrk t peranently der an bject Perectly nelastc cllsns ccur when the bjects stck tgether Nt all the KE s necessarly lst Mst cllsns all between elastc and perectly nelastc cllsns Elastc Cllsns Elastc eans that knetc energy s cnsered as well as entu. Ths ges us re cnstrants We can sle re cplcated prbles!! Bllards (-D cllsn) The clldng bjects hae separate tns ater the cllsn as well as bere. rst: spler -D prble Intal nal Inelastc and Elastc Cllsns A cpletely nelastc cllsn An elastc cllsn

6 Applyng the Prncple Cnseratn Mentu. Decde whch bjects are ncluded n the syste.. Relate t the syste, denty the nternal and external rces. 3. Very that the syste s slated. 4. Set the nal entu the syste equal t ts ntal entu. Reeber that entu s a ectr. Prble Slng r One-Densnal Cllsns Set up a crdnate axs and dene the elctes wth respect t ths axs It s cnenent t ake yur axs cncde wth ne the ntal elctes In yur sketch, draw all the elcty ectrs wth labels ncludng all the gen nratn Draw bere and ater sketches Label each bject nclude the drectn elcty keep track subscrpts Prble Slng r One-Densnal Cllsns Sketch r perectly nelastc clllsn The bjects stck tgether Include all the elcty drectns The ater cllsn cbnes the asses Wrte the expressns r the entu each bject bere and ater the cllsn Reeber t nclude the apprprate sgns Wrte an expressn r the ttal entu bere and ater the cllsn --- entu the syste s cnsered I the cllsn s nelastc, sle the entu equatn r the unknwn --- Reeber, KE s nt cnsered I the cllsn s elastc, yu can use the KE equatn t sle r tw unknwns Perectly Inelastc Cllsns Suppse, r exaple, =0. Cnseratn entu beces ( ) 0 ( ) E.g., 000 kg, (000kg)(50 s) 0 (500kg) 4 50 kg s 0 s kg 500 kg :, Perectly Inelastc Cllsns What aunt KE lst durng cllsn? KE bere (000 kg)(50 s) KEater ( ) (500 kg)(0 s) KE lst J lst n heat and sund 6.50 J J Exaple: A Ballstc Pendulu The ass the blck wd s.50-kg and the ass the bullet s kg. The blck swngs t a axu heght abe the ntal pstn. nd the ntal speed the bullet. Apply cnseratn entu t the cllsn:

7 Applyng cnseratn energy t the swngng tn: gh gh gh g h s Elastc Cllsns Bth entu and knetc energy are cnsered Typcally hae tw unknwns Sle the equatns sultaneusly s s kg.50 kg kg Incng and utgng elctes are ery ass dependant ass ntally at rest Tw-densnal Cllsns r a general cllsn tw bjects n threedensnal space, the cnseratn entu prncple Elastc Cllsn () () ples that the ttal entu the syste n each drectn s cnsered Use subscrpts r dentyng the bject, ntal and nal, and cpnents and x y x y x y x y A Cllsn n Tw Densns Use entu cnseratn t deterne the agntude and drectn the nal elcty ball ater the cllsn. x x x x y y y y

8 Center Mass The center ass s a pnt that represents the aerage lcatn r the ttal ass a syste. x c x x Center Mass Crdnates The crdnates the center ass are x y CM CM x y Tw asses n x-axs x x = M s the ttal ass CM the syste x Velcty Center Mass x c c x x x t In an slated syste, the ttal lnear entu des nt change, therere the elcty the center ass des nt change. p c c M x t c p p x t Exaple: Center Mass Mtn c = cnst. In slated syste BEORE c 0 Mtn the Center Mass The syste wll e as an external rce were appled t a sngle partcle ass M lcated at the center ass c ATER 88 kg.5 s 54 kg.5 s 88 kg 54 kg