2-D Momentum Conservation

Transcription

1 -D Mmentum Cnseratin Saleback Cllege Physics Department Purpse: T cnirm that linear mmentum is cnsere in tw-imensinal cllisins. T shw that kinetic energy is nearly cnsere in tw-imensinal near-elastic cllisins. Equipment: cllisin ramp steel balls abut cm in iameter (with apprximately the same mass) plumb line meter stick sheets paper r each lab partner, abut 70 cm square pieces carbn paper Sketch: In ur present experiment, the balls unerg a glancing cllisin (i.e. center masses misaligne) at the en a ramp an all nt the lr, where their impact psitins are marke with carbn paper as shwn in the igure belw. Ball Ball Plumb bb paper Figure. Carbn paper

2 Thery: CONSERVATION OF MOMENTUM In any cllisin, with n net external rces acting n the system, linear mmentum is cnsere. A typical case is shwn in the igure belw, where the incming ball,, unerges a glancing cllisin with a target ball, which is initially at rest. Ater the cllisin, bth balls me at an angle t the riginal path ball. Nte that = 0 an therere is nt shwn in igure belw. P ball P 0 P 0 ball P ball P P P y x Figure. P P Nte that the ectr sum the inal mmenta is equal t the initial mmentum. In the example rawn abe, the cllisin ccurs in a plane althugh the meth cul be use t analyze a similar three-imensinal cllisin. Applying cnseratin linear mmentum t the situatin escribe abe, using the shwn crinate system, yiels: Px 0 Px Px = m x = m x mx () Py 0 Py Py = m y = m y m y where P x an P y represent the crrespning x an y cmpnents mmentum. The subscripts an represent initial (i.e. bere the cllisin) an inal (i.e. ater the cllisin) respectiely, an ball is the incming ball while ball is the target. () As bth the balls all at the same rate an strike the paper at the same time, the hrizntal istance they me (their range) will be prprtinal t the elcity the balls in the plane the lr. Multiplying this istance by the mass each ball will yiel a number prprtinal t the mmentum. I the masses are equal ( m = m = m ) then equatins () & () becme x = x x (3) y = y y (4)

3 Since there is n acceleratin in the hrizntal plane x = t an x =, where = istance. Sling r gies = (5) t Substituting equatin (5) int equatins (3) & (4) x = t t x t x y y y = t t t Realizing the balls all take the same time, t, t all yiels x = x x (6) y = y y r = (8) (7) CONSERVATION OF KINETIC ENERGY I the cllisin is perectly elastic, then the kinetic energy, K, the system is als cnsere. K = K K (9) r m (0) = m m I the masses are equal ( m = m = m ) equatin (0) becmes = () Using the same reasning as r mmentum cnseratin abe, equatin () becmes = () Prceure: (a) Initial Set-Up. The cllisin ramp shul be securely astene t the ege the lab table. Ajust the target ball (ball ) supprt s that the centers bth balls lie in a hrizntal plane when they cllie ( a -D cllisin is esire, nt a 3-D cllisin).. Align the plumb line s that it hangs irectly belw the pint cllisin. Tape the paper (n carbn paper yet) t the lr s that it is psitine as shwn in igure. An mark the psitin the plumb bb n the paper. 3. Place the target ball (ball ) n its supprt. Place the secn ball n the ramp at the 0-cm mark an release it. Nte apprximately where the tw balls strike the paper. I either ball

4 misses the paper, change the angular ajustment the target supprt until bth strike in a satisactry psitin. 4. Reme the target ball (ball ) rm its supprt an allw the incient ball (ball ) t leae the ege the ramp withut a cllisin (still release it rm the 0-cm mark!); it shul als strike the paper. I it es nt, either me the paper (an make a new plumb line mark) r release ball rm a lwer pint n the ramp. Make sure that the incient ball (ball ) es nt strike the target ball supprt screw as it leaes the ramp. (b) Taking Data. Place a piece carbn paper ace wn in the apprximate psitin that the ball struck the paper in part 4 abe. Try t ajust the psitin the carbn paper s that secnary bunces the ball will nt leae marks. Release the ball rm the 0-cm mark n the ramp 5 times, checking between trials t see that clear marks are being let uner the carbn paper.. Place the target ball (ball ) n it s supprt an release ball rm the 0 cm mark, bsering the apprximate psitin where the tw balls strike the paper. Place a sheet carbn paper at the psitin each strike, again being careul t place the carbn paper s that secnary bunces will nt leae marks. 3. Recr the strike psitin the tw balls n the paper r 5 cllisins. I the strike pints n the paper are reasnably well clustere (say n mre than a ew cm sprea) reme the paper (replacing it with a new piece) an repeat steps 3 prceure (b) until each lab partner has a cmplete ata sheet. Be sure t mark the psitin the plumb line (yur rigin) n each ata sheet an raw a set axes with ne axis parallel t the cllisin ramp! 4. Change the angular psitin the target ball, ball, an repeat the entire prceure (b). I it is necessary t release ball rm a ierent pint n the track, when changing the angular psitin r ball, be sure t release ball rm this same pint r all ata runs ne uring this secn hal the experiment. 5. Bere leaing the lab, btain the instructr s signature n yur ata sheets. Each lab partner shul hae ierent cmplete ata sheets with 5 sets ata n each. Analysis: (a) On each ata sheet make a careul cnstructin the ectr sum using the parallelgram meth, like that shwn belw. plumb mark P P P P

5 Yu will hae t estimate the mile each cluster, ignring any pints that are biusly in errr. Secnary bunce marks can usually be istinguishe because they are nticeably lighter than true ata pints. In yur cnclusin gie the percent ierence between the magnitue the initial mmentum ectr P an the ectr sum the inal mmenta ( P P 0 ), use the aerage P an ( P ) in the enminatr the percent ierence equatin. As percent ierence 0 P is imensinless, yu can wrk in arbitrary mmentum units, such as the istances measure n the ata page. In yur cnclusin yu shul als gie the angular ierence between P an 0. Repeat the abe calculatins r yur secn ata sheet. P P (b) Measure the magnitue each isplacement ectr,, n the ata pages an use equatin () = () where the s are the istances measure irectly n the ata sheets. In yur cnclusin, gie the percent ierence between the let-han an right-han sies eqn () [where the enminatr is the aerage the let an right sies eqn ()]. Elementary Errr Prpagatin: (Nte: This es nt ulill yur errr prpagatin requirements!) Estimate the uncertainty in, δ, taking int accunt the iiculty in estimating the mile each cluster. Cmpare this uncertainty t the ierence between an. Repeat r yur secn ata sheet. It is nt necessary t estimate the ranm errr r uncertainty r the angle measurements. Yu ecie i yur experiment is successul base upn whether δ is greater than, less than r equal t ( ).