Linear Momentum. Equation 1
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1 Lnear Momentum OBJECTIVE Obsere collsons between two carts, testng or the conseraton o momentum. Measure energy changes durng derent types o collsons. Classy collsons as elastc, nelastc, or completely nelastc. INTRODUCTION The collson o two carts on a track can be descrbed n terms o momentum conseraton and, n some cases, energy conseraton. I there s no net external orce experenced by the system o two carts, then we expect the total momentum o the system to be consered. Ths s true regardless o the orce actng between the carts. In contrast, energy s only consered when certan types o orces are exerted between the carts. Collsons are classed as elastc (knetc energy s consered), nelastc (knetc energy s lost) or completely nelastc (the objects stck together ater collson). Sometmes collsons are descrbed as super-elastc, knetc energy s ganed. In ths experment you can obsere most o these types o collsons and test or the conseraton o momentum and energy n each case. APPARATUS computers Verner computer nterace Logger Pro two Verner Moton Detectors dynamcs cart track two low-rcton dynamcs carts wth magnetc and Velcro bumpers THEORY The lnear momentum o a partcle s gen by: p = m Equaton Where, p [kg m/s] s the lnear momentum, m [kg] s the mass o the partcle and [m/s] s the elocty o the partcle. Note that as the elocty s a ector quantty (magntude and drecton) so, too, s the momentum. It can be shown that there are no external orces actng on a system, then the total momentum o the system s held constant. To begn ths explanaton, t s necessary to state the equaton orm o Newton's second law. Lnear Momentum - Page
2 F = m Equaton a Where, F [N] s an appled external orce that causes mass m [kg] to hae acceleraton a [m/s ]. Acceleraton s dened as the change n the elocty oer the change n the tme. Thus, makng ths substtuton n Equaton and takng adantage o Equaton we get the relaton: m p F = = t t Equaton 3 Ths shows how an appled external orce leads to the change n momentum o a system. Howeer, n the absence o an appled external orce (F = 0) Equaton 3 becomes: p 0= or p=0 or p - p=0 or p = p t Equaton 4 Ths states that the total momentum o a system remans constant (s consered) n the absence o external orces. Addtonally, Newton's rst law can be appled to momentum conseraton. It ollows rom Equaton 4 that: m = m or = Equaton 5 Ths equaton states that an object wll reman n a state o constant moton n the absence o an external orce. When two objects nteract, each o ther respecte momentums s consered n addton to the momentum o the two-body system. Thus, conseraton tells us that the total momentum beore a collson should be equal to the total momentum ater a collson. Lnear Momentum - Page
3 Ths concept o momentum conseraton or the two-body system can be expressed by makng use o Equatons and 4. ( m ( p + m + p p = p or ) =( or ) =( p m Equaton 6 + p ) + m ) Where and reer to the alue assocated wth objects # and # respectely. The elocty wll be determned by the change n poston o the objects as a uncton o tme ( = Δx/Δt). Remember that elocty s a ector quantty and has both magntude and drecton. Thus, beore a collson the elocty o one object may be poste (let to rght) but ater the collson t may be negate (rght to let). Let us consder the knetc energy o the system. Knetc energy s dened by the relaton: KE = Equaton 7 m A perectly elastc collson s dened as one n whch there s no loss o knetc energy n the collson. An nelastc collson s one n whch part o the knetc energy s changed to some other orm o energy n the collson. There s also a perectly nelastc collson where two objects stck together. Thus, one can test lnear ar track collsons or energy conseraton n addton to momentum conseraton. Smlar to momentum conseraton, knetc energy conseraton can be expressed n terms o the ntal and nal masses and eloctes o the colldng objects. ( m + m ) =( m + m ) Equaton 8 Fnally, just as Newton's rst and second laws o moton were appled to momentum conseraton, so too can Newton's thrd law o moton be appled to momentum conseraton. Lnear Momentum - Page 3
4 Newton's thrd law o moton states that or eery acton, there s an equal and opposte reacton: = - F Equaton 9 Equaton 9 ndcates that the orce on object # due to object # s equal n magntude to, but opposte n drecton (the negate sgn), to the orce on object # due to object #. Thereore, based on momentum conseraton, the change n momentum or one object wll be equal n magntude and opposte to the change n the momentum or the second object. By opposte, or ths laboratory and lnear momentum n general, we are reerrng to the reersal n drecton o the two objects. F EXPERIMENTAL PROCEDURE. Set up the track so that t s horzontal. Test ths by releasng a cart on the track rom rest. The cart should not moe.. Practce creatng gentle collsons by placng the blue cart at rest n the mddle o the track, and release the red cart so t rolls toward the blue cart, magnetc bumper toward magnetc bumper. The carts should smoothly repel one another wthout physcally touchng. 3. Place a Moton Detector so ts ace s about 0cm rom the end o the track, allowng or the 0.4 m mnmum dstance between detector and cart. Connect the Moton Detectors to the DIG/SONIC and DIG/SONIC channels o the nterace. 4. Open the le 8 Momentum Energy Coll rom the _Physcs wth Verner older. Enlarge the s. t graph to ull screen...ths s the only one you wll use! 5. Clck to begn takng data. Repeat the collson you practced aboe and use the poston graphs to ery that the Moton Detectors can track each cart properly throughout the entre range o moton. You may need to adjust the poston o one or both o the Moton Detectors. The graph/data should resemble (does not hae to be EXACT) the sample data below: Lnear Momentum - Page 4
5 6. Place the two carts at rest n the mddle o the track, wth ther Velcro bumpers toward one another and n contact. Keep your hands clear o the carts and clck. Select both sensors and clck. Ths procedure wll establsh the same coordnate system or both Moton Detectors. Very that the zerong was successul by clckng and allowng the stll-lnked carts to roll slowly across the track. The graphs or each Moton Detector should be nearly the same. I not, repeat the zerong process. Part I: Magnetc Bumpers 7. Reposton the carts so the magnetc bumpers are acng one another. Clck to begn takng data and repeat the collson you practced n Step. Make sure you keep your hands out o the way o the Moton Detectors ater you push the cart. Push hard enough to hae as near a constant elocty (horzontal lne) beore the collson as well as ater the collson...otherwse ths s not elocty but acceleraton/deceleraton (Force NOT Momentum). 8. From the elocty graphs you can determne an aerage elocty beore and ater the collson or each cart. To measure the aerage elocty durng a tme nteral, FIRST zoom n on the desred area (the secton o the moton where the cart had constant elocty), then drag the cursor across the nteral (ntal elocty o the blue cart or example). The nteral s the horzontal secton (constant elocty) o the graph. There should be FOUR o these sectons showng on the graph. Clck the Statstcs button to brng up a statstcs box rom whch you can read the mean alue o the elocty along that nteral. Drag that statstcs box out o your way (DO NOT delete t) and repeat ths procedure or the three remanng eloctes o nterest (red cart's ntal elocty, blue cart's nal elocty, and red cart's nal elocty); keepng each o the statstcs boxes on the screen. You wll enter the magntude (Absolute Value) o these our mean alues n the data table. Prnt a copy o ths graph, wth the FOUR statstcs boxes, or ncluson n your laboratory report. 9. Repeat Step 8 as a second run wth the magnetc bumpers; agan recordng the mean eloctes n the data table. There s NO NEED to prnt a copy o the graph or ths second tral. Just be sure to extract the statstcs box normaton or your data table. Lnear Momentum - Page 5
6 Part II: Velcro Bumpers 0. Change the collson by turnng the carts so the Velcro bumpers ace one another. The carts should stck together ater collson. Practce makng the new collson, agan startng wth the blue cart at rest.. Clck to begn takng data and repeat the new collson. Usng the procedure n Step 8, measure and record the magntude (Absolute Value) o the cart mean eloctes n your data table. Prnt a copy o ths graph, wth the FOUR statstcs boxes, or ncluson n your laboratory report.. Repeat the preous step as a second run wth the Velcro bumpers. There s NO NEED to prnt a copy o the graph or ths second tral. Just be sure to extract the statstcs box normaton or your data table. Part III: Velcro to Magnetc Bumpers 3. Face the Velcro bumper on one cart to the magnetc bumper on the other. The carts wll not stck, but they wll not smoothly bounce apart ether. Practce ths collson, agan startng wth the blue cart at rest. 4. Clck to begn data collecton and repeat the new collson. Usng the procedure n Step 8, measure and record the magntude (Absolute Value) o the cart mean eloctes n your data table. Prnt a copy o ths graph, wth the FOUR statstcs boxes, or ncluson n your laboratory report. 5. Repeat the preous step as a second run wth the Velcro to magnetc bumpers. There s NO NEED to prnt a copy o the graph or ths second tral. Just be sure to extract the statstcs box normaton or your data table. COVER PAGE REPORT ITEMS (To be submtted and stapled n the order ndcated below) (-5 ponts ths s not done properly) Lab Ttle Each lab group member s rst and last name prnted clearly Color Group Date DATA (worth up to 0 ponts) Data tables aalable as a downloadable Excel le Lnear Momentum - Page 6
7 DATA ANALYSIS (worth up to 5 ponts) The our requred sample calculatons, to be shown n your laboratory report, are hghlghted n yellow on the downloadable Excel data table spreadsheet. o These are to be done or a sngle tral only. For Part I, Part II & Part III. I the total momentum or a system s the same beore and ater the collson, we say that momentum s consered. I momentum were consered, what would be the rato o the total momentum ater the collson to the total momentum beore the collson? Be sure to explan your answer not merely ndcate a rato. For your sx runs, nspect the momentum ratos. Een momentum s consered or a gen collson, the measured alues may not be exactly the same beore and ater due to measurement uncertanty. The rato should be close to one ( ), howeer. Is momentum consered n your collsons?. I the total knetc energy or a system s the same beore and ater the collson, we say that knetc energy s consered. I knetc were consered, what would be the deal rato o the total knetc energy ater the collson to the total knetc energy beore the collson? Be sure to explan your answer not merely ndcate a rato. For your sx runs, nspect the knetc energy ratos. Is knetc energy consered n the magnetc bumper collsons? How about the Velcro collsons? Is knetc energy consumed n the thrd type o collson studed? Classy each o the three collson types as elastc, nelastc, or completely nelastc. GRAPHS (worth up to 0 ponts) Part I: Magnetc Bumpers (Tral # only) Part II: Velcro Bumpers (Tral # only) Part III: Velcro to Magnetc Bumpers (Tral # only) GRAPH ANALYSIS (worth up to 0 ponts) For Part I, Part II & Part III (Tral # Graph Only) -- Place analyss on each o the respecte graph prntouts themseles -- Careully explan each regon o the graph (beore, durng, and ater the collson) and what t ndcates about the moton o the cart(s) durng that nteral. Be sure to also explan why the regon selected or the statstcs was approprate. o Color-codng these graph sectons may proe helpul. Lnear Momentum - Page 7
8 CONCLUSION (worth up to 0 ponts) See the Physcs Laboratory Report Expectatons document or detaled normaton related to each o the our questons ndcated below.. What was the lab desgned to show?. What were your results? 3. How do the results support (or not support) what the lab was supposed to show? 4. What are some reasons that the results were not perect? QUESTIONS (worth up to 5 ponts) DO NOT orget to nclude the answers to any questons that were asked wthn the expermental procedure Lnear Momentum - Page 8
ONE-DIMENSIONAL COLLISIONS
Purpose Theory ONE-DIMENSIONAL COLLISIONS a. To very the law o conservaton o lnear momentum n one-dmensonal collsons. b. To study conservaton o energy and lnear momentum n both elastc and nelastc onedmensonal
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