PHY191 Experment 5: Elastc and Inelastc Collsons 7/1/011 Page 1 Experment 5 Elastc and Inelastc Collsons Readng: Bauer&Westall: Chapter 7 (and 8, or center o mass deas) as needed Homework 5: turn n the second week o ths experment. Assume random and ndependent errors, and use p v = L / t p = m v K m measured: L = ±.06 t = 1 ±.04 m = ±.0001 (neglgble uncertanty) For each queston below, gve the algebrac ormula, then evaluate the numercal uncertanty n value and n % (e.g. nd δk, then δk/k n %): 1. Calculate the uncertanty n v n terms o the uncertanty n L and n t.. Calculate the uncertanty o p n terms o the uncertanty o v. 3. Calculate the uncertanty o K n terms o the uncertanty o p. 1. Goals 1. Study momentum and energy conservaton n nelastc and elastc collsons. Understand use o Excel n analyzng data 3. Carry out uncertanty calculatons o moderate complexty. Theoretcal Introducton The ollowng experment explores the conservaton o momentum and energy n a closed physcal system (deally: no nteracton o measured objects wth rest o unverse). As you probably know rom the accompanyng theoretcal course, the conservaton o energy and momentum play an mportant role n physcs and ther conservaton s a consequence o undamental symmetres o nature..1 Momentum For a sngle partcle (or a very small physcal object), momentum s dened as the product o the mass o the partcle and ts velocty: p m v (1) Momentum s a vector quantty, makng ts drecton a necessary part o the data. To dene the momentum n our three-dmensonal space completely, one needs to specy ts three components n x, y and z drecton. The momentum o a system o more than one partcle s the vector sum o the ndvdual momenta: p p p m v m () 1 1 1 v
PHY191 Experment 5: Elastc and Inelastc Collsons 7/1/011 Page The nd Newton s law o mechancs can be wrtten n a orm whch states that the rate o the change o the system s momentum wth tme s equal to the sum o the external orces actng on ths system: dp F (3) dt From here we can mmedately see that when the system s closed (whch means that the net external orce actng on the system s zero), the total momentum o the system s conserved (constant).. Energy Another mportant quantty descrbng the evoluton o the system s ts energy. The total energy o a gven system s generally the sum o several derent orms o energy. Knetc energy s the orm assocated wth moton, and or a sngle partcle: mv KE (4) Here v wthout the vector symbol stands or the absolute value o the velocty, vx v y vz In contrast to momentum, knetc energy s NOT a vector; or a system o more than one partcle the total knetc energy s the algebrac sum o the ndvdual knetc energes o each partcle: KE KE 1 KE (5) Another undamental law o physcs s that the total energy o a system s always conserved. However wthn a gven system, one orm o energy may be converted to another (such as potental energy converted to knetc n the Pendulum experment). Thereore, knetc energy alone s oten not conserved..3 Collsons An mportant area o applcaton o the conservaton laws s the study o the collsons o varous physcal bodes. In many cases, t s hard to assess how exactly the colldng bodes nteract wth each other. However, n a closed system, the conservaton laws oten allow one to obtan the normaton about many mportant propertes o the collson wthout gong nto the complcated detals o the collson dynamcs. In ths lab, we wll see n practce how the conservaton o momentum and total energy relate varous parameters (masses, veloctes) o the system ndependently o the nature o the nteracton between the colldng bodes. Assume we have two partcles wth masses m, m and speeds 1 v 1 and v whch collde, wthout any external orce, resultng n speeds o v 1 and v ater the collson ( and stand or ntal and nal). Conservaton o momentum then states that the total momentum beore the collson P s equal to the total momentum ater the collson P : P m1 v1 mv, P m1v1 mv and P P (6)
PHY191 Experment 5: Elastc and Inelastc Collsons 7/1/011 Page 3.4 Elastc and nelastc collsons There are two basc knds o collsons, elastc and nelastc:.4.1 In an elastc collson, two or more bodes come together, collde, and then move apart agan wth no loss n total knetc energy. An example would be two dentcal "superballs", colldng and then reboundng o each other wth the same speeds they had beore the collson. Gven the above example conservaton o knetc energy then mples: 1 1 1 1 m1v1 mv m1v1 mv or KE KE (7).4. In an nelastc collson, the bodes collde and (possbly) come apart agan, but now some knetc energy s lost (converted to some other orm o energy). An example would be the collson between a baseball and a bat. I the bodes collde and stck together, the collson s called completely nelastc. In ths case, all o the knetc energy relatve to the center o mass o the whole system s lost n the collson (converted to other orms). In ths experment you wll be dealng wth a) a completely nelastc collson n whch all knetc energy relatve to the center o mass o the system s lost, but momentum s stll conserved, and b) a nearly elastc collson n whch both momentum and knetc energy are conserved to wthn a ew percent..5 Conservaton laws or macroscopc bodes So ar we were talkng about the system o pont-lke partcles. However, the conservaton o the momentum s also vald or macroscopc objects. Ths s because the moton o any macroscopc object can be decomposed nto the moton o ts center o mass (whch s a pont n space) wth a gven lnear momentum, and a rotaton o the object around ths center o mass. Then, the conservaton o the lnear momentum s agan vald or the moton o deal pont masses located at the center o the mass o each o the objects. However, some o the lnear knetc energy can be transormed nto the rotatonal energy o the objects, whch should be accounted or n a real experment..6 Knetc Energy n Inelastc Collsons. It s possble to calculate the percentage o the knetc energy lost n a completely nelastc collson; you wll nd that ths percentage depends only on the masses o the carts used n the collson, one o the carts starts rom rest. Ater the completely nelastc collson, the carts move together, so that v1 v v3 The ntal KE s gven by: m1v1 m1v KE. But, snce v 0 m1v1 KE (8) The nal KE s gven by: m1 m KE v3 (9)
PHY191 Experment 5: Elastc and Inelastc Collsons 7/1/011 Page 4 From conservaton o momentum: m v m v ( m m v or, snce v 0 1 1 1 ) 3 m1 v1 m1 m ) 3 ( v (10) Snce the collson s nelastc, the ntal KE s not equal to the nal KE. You could use ( KE KE ) equatons (8), (9), and (10) to obtan an expresson or DK (%). Hnt: dene x = KE m 1 /(m 1 +m ) and use t to elmnate v 3. 3 Expermental setup We wll study the momentum and energy conservaton n the ollowng smpled stuaton: a) we wll look on the collson o only objects; b) the moton o these objects wll be lnear and one-dmensonal, so that we can choose the reerence rame n such a way that only x-components o the objects momenta are nonzero; the sgn o these components depends on the drecton o the moton; c) the expermental apparatus can be set up n a way to almost completely elmnate the net external orce on the system. Our objects wll be two carts o derent masses, wth one ntally at rest. The carts move on an ar track, whch ensures that the moton s one-dmensonal and reduces the rcton between the carts and the surace. The veloctes o the carts can be measured wth the help o the photogates, whch are descrbed n more detals below. Beore the begnnng o the measurements, spend at least 15 mnutes to gure out whch external actors can dsturb the moton o the carts on the track, and what you should do to reduce or elmnate these actors. Remember, the successul completon o ths lab strongly depends on your ablty to create an almost closed system. Make a ew practce trals to see you can acheve an unperturbed one-dmensonal collson o the carts. Adjust the level o the ar track and the power o the ar supply necessary. Questons or prelmnary dscusson 3.1 Draw a dagram o all orces actng on each cart when they collde. Whch orces wll nluence the total P and KE most? 3. In our experment, can we acheve a completely elastc collson? a completely nelastc collson? 3.3 In an nelastc collson n a closed system, can some o the total momentum be lost? Some o the knetc energy? 3.4 In an elastc collson n a closed system, can some o the total momentum be lost? Some o the knetc energy be lost? 3.5 Extra Credt: Fnd a ormula or D K (%) as a uncton o x, `as suggested n the end o secton.6. 3.6 I knetc energy s lost, where does t go? Does conservaton o energy apply?
PHY191 Experment 5: Elastc and Inelastc Collsons 7/1/011 Page 5 Measurements 4. Inelastc collsons 4.1 Technques. In the rst part o the lab we make sure that ater the collson the carts stck together and move wth some velocty common to both masses. Thus, we have to measure the velocty o cart 1 beore the collson and the common velocty o the carts 1 and ater the collson. For ths purpose, we use two photogates (see Fgure 1). Each o them allows measurng the tme t takes the cart to go through t. The speed s calculated by dvdng the length o the n on the cart 1 by the measured tme; but to turn speed nto velocty, as requred, you have to pck a + drecton. Fgure 1: Intal state o the carts beore nelastc collson (n on cart can be removed). Warnng: your carts should be balanced, unlke the ones n ths gure! Poston cart close to the gate and set the photogate tmer to "GATE" mode and the memory swtch n ON poston. In ths mode the photogate wll dsplay the rst tme nterval measured. Subsequent measurements wll not be dsplayed (only the rst one s), but the tmes are added n the memory. By pushng the READ swtch you can dsplay the memory contents, whch s the sum o all measurements. Example: the ntal readng or cart 1 (the tme that t took to pass through the gate 1) s 0.300 seconds. Cart 1 colldes wth cart and they go together through the photogate (Fgure ). Suppose t now takes 0.513 seconds. The dsplay wll reman at 0.300, but the memory wll contan.300 + 0.513=0.813 seconds. To nd the second tme, you have to subtract the rst tme rom the contents o the memory. Try ths out by movng the cart through the gate by hand a ew tmes. 4. Uncertanty Estmaton: Frst we test to see whether some o our assumptons are correct (the two tmers gve the same answer, the track s level, rcton s neglgble, drecton doesn t matter). Perorm some trals wth a sngle cart n whch no collson occurs. Do ths at derent speeds, and n derent drectons. Record the results n your notebook n approprate tables. Dd you conserve momentum? Was there a bas? What tme uncertanty would you deduce rom these measurements? What conclusons do you draw? 4.3 Predcton: Wrte n your lab book whch case below wll the change the nal Knetc Energy most. In your report comment on whether your predctons were correct, and not, why.
PHY191 Experment 5: Elastc and Inelastc Collsons 7/1/011 Page 6 4.4 Data Do 3 sets o nelastc collsons consstng o trals each. Vary the masses o the carts by addng the masses (small metal dsks) to them. In these measurements, use the needle and putty bumpers and measure the ntal and nal veloctes or the ollowng sets o masses: Tral 1+: no mass dsks on cart 1, 4 mass dsks on cart ; Tral 3+4: mass dsks on cart 1, mass dsks on cart ; Tral 5+6: mass dsks on cart1, no mass dsks on cart. In each measurement, you need to nd all the ntal and nal masses and veloctes, and use them to calculate the ntal and nal total momentum and knetc energy. Make tables n your lab book to organze your recordngs. You may neglect the uncertanty o all masses. In R:exp6, you wll nd preset datasheets, Inelastc and Elastc. Open them and save nto your secton s older. The spreadsheet has some entres or ractonal uncertantes (δdp(%), or example). These should be dsplayed n % ether by multplyng the relevant racton by 100, or, preerably, by usng the % ormattng button n Excel. When you have completed the spreadsheets, prnt them showng the numbers, then save them. There are many uncertanty calculatons. You should explan n your lab book or report how you calculated them. An alternatve s to re-open your spreadsheet, save t wth a new name, use the ctrl-` key to dsplay ormulas (handy whle checkng ormulas) and prnt agan, showng the ormulas used. It won t be possble to read the ormulas unless you adjust the column wdth, and use page setup to prnt landscape, to t to pages wde x 1 tall. You can use Prnt Prevew beore prntng to check. 4.4 Hnts The correct calculaton o the uncertanty o δd(%) s complcated because the ntal value or p or K s n both the numerator and denomnator. However, or our purposes, t s sucently close to use δd p (%) δd/p and a smlar ormula or K. (Optonal) I you wsh you could add auxlary cells to help you wth uncertanty calculatons. For example, you could also calculate δv/v or δp/p. (Santy checks or uncertantes) Check whether your uncertanty calculatons are makng sense! For ndependent and random uncertantes, you expect ncreasng values o absolute and ractonal uncertantes as your calculaton proceeds (unless your calculaton nvolves ractonal powers < 1). Ths s dscussed n the Reerence Gude secton on Uncertanty Calculatons. 5. Elastc collsons 5.1 Technques. In an almost elastc collson, the man derence rom the prevous part o the lab s that ater the collson the carts move separately. The rubber band bumpers allow carts to collde wth almost no converson o the knetc energy nto the other orms o energy. As beore, cart ntally stays at rest, and beore the collson we have to measure only the velocty o the cart 1 v 1 (Fgure 3). However, ater the collson we have to measure the veloctes o both carts, v 1 and v (Fgure 4). Thus, all n all we have to measure three tmes (, t t ), whle the photogate system can smultaneously measure only two o them. t1 1, We can get out o ths stuaton, ater the measurement o the ntal tme t 1, but beore the collson, we reset the tmer. You have to make several practce trals to quckly remember and reset the contents o the tmer beore the carts collde. Then, we can agan see the contents o the tmer dsplay and the memory to nd t 1 and t.
PHY191 Experment 5: Elastc and Inelastc Collsons 7/1/011 Page 7 Fgure 3: The ntal state o the carts beore elastc collson. Fgure 4: The nal state o the carts ater elastc collson. 5. Predcton: Wrte n your lab book or each case mentoned n secton 5.3 below the predcted drecton and speed ater the collson o the ntal-movng cart: Forward or backward? Faster or slower than the cart ntally at rest? In your report comment on whether your predctons were correct, and not, why. 5.3 Data Measure the n length on the two carts. The experment wll be done wth cart ntally at rest. You wll do 6 trals wth the ollowng choces o m 1 and m : Tral 1+: no mass dsks on cart 1, 4 mass dsks on cart Tral 3+4: mass dsk on cart 1, mass dsks on cart Tral 5+6: 4 mass dsks on cart 1 no mass dsk on cart 5.4 Hnts. Pay attenton to the sgn o the veloctes, whch depends on the drecton o moton o the cart. I one o the carts goes backward wth respect to your chosen + drecton, how wll you make sure the velocty s calculated as negatve? I the percentage change n momentum or knetc energy beore and ater the collson s greater than 10%, repeat the measurement more careully (collde slower/aster, etc.). Snce the datasheet s set up t s easy to see whether momentum/energy s better conserved wth every tral you do. I one o the tmes you measure s too long or the tmer to measure, substtute a large number or the tme n your spreadsheet.
PHY191 Experment 5: Elastc and Inelastc Collsons 7/1/011 Page 8 6 Questons to be dscussed 6.1 Create a graph (one or nelastc and one or elastc collsons) o the relatve change o the total momentum D p (%) ( P P ) / P versus the number (1-6) o the measurement. Ths s just to show graphcally the varous values o D p (%). Show the uncertanty o D p (%) wth the help o error bars. Also, show the theoretcal predcton or D p (%). 6. Accordng to the graphs, was the total momentum conserved n the collsons? Use the two standard devatons rule to justy your answers. Was there any derence n how well momentum was conserved n elastc vs. nelastc collsons? I the momentum was not conserved, dscuss the reason why. What dd you do to try to mprove the momentum conservaton results? 6.3 Dd you correctly predct the moton o the carts n the elastc collson? I not, why? 6.4 For the nelastc collson, plot the knetc energy ractonal change, D K (%) vs. m1 x. Show the error bars or D K (%). What are the slope and ntercept o a m1 m straght lne t to these data? Does a straght lne t these data reasonably? Dd you correctly predct the tral wth the largest ractonal change? Why or why not? 6.5 Extra Credt: Draw your predcted lne on the graph, and dscuss quanttatvely the consstency o your nelastc collson data or D K (%) wth the lne you predcted or the theoretcal predcton. 6.6 Make and analyze a D K (%) graph or the nearly elastc collson (as you analyzed Dp(%) above n steps 6.1-6.3). What was the average loss o the knetc energy n ths part o the experment? Dd you acheve the goal o 10% loss o KE n the nearly elastc collsons? Whch potental systematc errors or setup problems s the loss most senstve to? 6.7 Extra Credt: Use the excel spreadsheets wth your data or ths queston. Save a NEW copy o each sheet beore proceedng, and work on the new copy. a) What happens to your momentum conservaton D p (%) you arbtrarly change the n length n the nelastc collson by a actor o? Explan why ths s so. b) Repeat a) or the ractonal knetc energy loss n the nelastc collson. c) Repeat a) or the elastc collson. Is the result the same as n a)? Why or why not? 6.8 What was the muddest pont o ths lab? 7. Systematc Errors Proessonal scentsts deal wth systematc errors n successve levels o sophstcaton. Level 1:Thnk o possble errors. Ths s one o the hardest parts! That s because dong so means seeng where our assumptons break down. The best way s to go through the measurement step by step and denty where thngs could have been otherwse than we wshed. The result s a laundry lst o possble systematc eects. Some examples o our assumptons: Setup: We assumed that the track was straght and level. Carts were balanced. Tmng: The veloctes were constant whle the carts moved through the gate and untl the collson. Collson: The system o the carts s closed: no momentum (nor energy) enters or leaves the system consstng o the two carts durng the collson. Queston 7.1 Lst ways n whch these assumptons mght have been volated. Level : Sgn o eect. For each volaton o our assumptons, try to understand the sgn o ts eect on our nal result. For example, the carts weren t balanced, then the ar track would
PHY191 Experment 5: Elastc and Inelastc Collsons 7/1/011 Page 9 tend to add momentum o a partcular sgn to the cart. It oten helps to exaggerate (mentally, or by measurng) the eect to see whch way t should aect your results. 7. What sgn would each o your eects have on momentum or energy conservaton? Or could t be o ether sgn? Level 3: Estmatng the sze o the eect. Sometmes we can perorm a sde-experment and actually measure the eect and correct or t. More commonly, we can only provde a bound well t should really have been at worst ths bg. For example, levelng errors ddn t gve the carts enough extra momentum to go the length o the track n 10 seconds. Wth the mass, that bounds the change o momentum; that over the smallest ntal momentum bounds the % error. 7.3 How bg an eect could the ar track eect be on your momentum conservaton data? 7.4 How bg an eect would that have on the energy loss data or nelastc collsons? 7.5 Extra Credt: See what you can oer as a bound or the rest o the eects n your lst.