THE PHYSICS BEHIND THE SODACONSTRUCTOR. by Jeckyll

Transcription

1 THE PHYSICS BEHIND THE SODACONSTRUCTOR b Jeckll

2 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll /3 CONTENTS. INTRODUCTION 5. UNITS OF MEASUREMENT 7 3. DETERMINATION OF THE PHYSICAL CONSTANTS ADOPTED BY THE APPLET 3. THE GRAVITY Epemen 5 Epemen 6 Epemen 3 6 Epemen 4 7 Epemen 5 7 Resuls Analss 8 3. THE STIFFNESS (STATIC METHOD) The Sac Epemens 3 Epemen 6 4 Epemen 7 4 Epemen 8 5 Resuls Analss THE STIFFNESS (DYNAMIC METHOD) 7 Epemen 9 8 Epemen 9 Epemen 9 Epemen 3 Epemen 3 3 Resuls Analss THE FRICTION 3 Epemen 4 35 Epemen 5 36 Epemen 6 37 Epemen 7 38 Epemen 8 39 Epemen 9 39 Epemen 4 Epemen 4 Epemen 4

3 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 3/3 Epemen 3 43 Resuls Analss COLLISIONS 45 The Coecen o Elasc Resuon n he Ohogonal Impac 47 The Coecen o Elasc Resuon n he Tangenal Impac FINAL RESULTS 5 4. CINEMATIC SYSTEM OF COORDINATES THE POSITION VECTOR THE VELOCITY THE ACCELERATION TOPOLOGY OF A MODEL FORCES ANALYSIS 6 6. THE GRAVITY FORCES 6 6. THE DAMPING FORCES THE ELASTIC FORCES 6 7. SPRINGS AND MUSCLES THE SPRINGS THE MUSCLES RESULTANT OF THE ELASTIC FORCES ON A FREE MASS 7 8. EQUATIONS OF THE MOTION OF A MODEL 7 9. NUMERICAL RESOLUTION OF THE EQUATIONS OF MOTION 74. THE STATIC AND QUASI-STATIC SIMPLIFIED PROBLEM 8. SYSTEM OF EQUATIONS 8. REGULAR POLYGONS 8.3 THE PANDORA S TOYBOX PHENOMENON 86.4 TENSION SPRINGS / PRE-STRESSED PARTS 89 The Dec Poblem 9 The Invese Poblem 93.5 LINEAR MOTORS 94

4 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 4/3. CONCLUSIONS AND ACKNOWLEDGEMENTS 98 APPENDIX A: THE PENDULUM 99 APPENDIX B: DAMPED FREE VIBRATIONS (SINGLE DEGREE OF FREEDOM) 3 The Undamped Fee Vbaons APPENDIX C: OTHER NUMERICAL TECHNIQUES 4 The Eule s Mehod: Non-Lnea Deenal Equaon o Fs Ode 4 The Eule s Mehod: Ssem o Non-Lnea Deenal Equaons o Fs Ode 7 The Eule s Mehod: Non-Lnea Deenal Equaons o Second Ode 8 The Eule s Mehod: Ssems o Deenal Equaons o Second Ode The Eule s Mehod: Fnal Consdeaons The Runge-Kua s Algohm

5 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 5/3. INTRODUCTION The sodaconsuco apple s smpl a smulao o he mechancal laws o phscs. In he ollowng chapes o hs acle, we wll nvesgae hese phscal laws and he wa he have been mplemened n he apple. Hee I wsh o epess jus a ew pelmna consdeaons. Fs, I wsh o sa ha n hs wok ou wll no nd anhng eall ognal n he scenc meanng o he em. Evehng epoed n hs pape s a well-known noon o phscs and mahemacs. Theeoe, hs acle can be caalogued n he caego o scenc wok, bu ahe n he caego o ddaccal publcaon. Thanks o hs pape, anone wh a mnmal knowledge o he man mahemacal ules o analss can dscove wh hs own hands how he sodaconsuco apple woks. A second hng ha I wsh o sa hee s paculal addessed o anone who because o hs oung age o pacula eld o sud doesn oall undesand he mahemacs n hs wok. Fo ou I epea anohe me ha hee s nohng specal pesened n hs wok. All hese hngs, scencall speakng, ae val. I ou wsh o see somehng eall ognal, hen look o models n he sodazoo. Thee ou wll nd genune ceav. Le me use a smple analog ha I hnk bee eplans wha I m ng o sa. The sodaconsuco apple s somehng lke a muscal nsumen. I sn pacula mpoan who made a pano; s moe mpoan who plas he pano! I m no sue o hs, bu I would be mone ha Moza and Beehoven knew nohng abou he mechansm behnd he own panos. So, I wsh o sess one hng: please, don sop plang he pano. Fnall, I wsh o povde hee a sho descpon o he conens o hs wok. Ths pape s oganzed n chapes, uhe sepaaed, whee necessa, no subsecons. In he second chape pacula uns o measuemen ae suded. In he hd chape vual epemens ae ealzed n ode o nvesgae he phscal consans adoped b he apple. In hs chape he unconales o he apple s cusos o gav, con and sness ae epemenall deemned. All hese epemens have been adequael descbed and he coespondng lnks ae ncluded n he e o he eplanaons. In he ouh chape, he man cnemac quanes o he descpon o he moon o ee masses ae noduced and dscussed. In he h chape, a smple denon o a model s noduced. The sh chape gves a dealed

6 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 6/3 analss o he oces mplemened n he apple. The sevenh chape povdes a dealed mahemacal eplanaon o he behavo o spngs and muscles. In hs chape he unconales o he cusos o he muscle conol panel ae compleel claed. In he eghh chape, he equaons o moon o a genec model ae omulaed, whle n he nnh chape, we nd a smple numecal pocedue o he negaon. In he enh chape, a smpled mehod o he sud o models s noduced. Ths chape sudes polgons, enson spngs, and lnea moos. Fnall, he elevenh chape conans he concluson and acknowledgemens. The pape s also accompaned b hee appendes, n whch specc mahemacal agumens ae suded n-deph. Novembe Jeckll

7 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 7/3. UNITS OF MEASUREMENT In he eal wold, especall n scenc and echncal communcaons, all phscal quanes ae epessed n uns o measuemen dened b he Inenaonal Ssem (S.I.). We all know ha he un o measuemen o lengh s he mee (m), he un o measuemen o me s he second (s), he un o measuemen o mass s he klogam (kg), and so on o he ohe undamenal quanes (empeaue, nens o cuen, ec.). In he soda unvese, howeve, we can use hese uns o measuemen. To llusae wh, le s eploe hs seemngl smple queson: how long, o eample, s he old chamng danwalke? I have a 7-nch mono, and I measue a lengh o 7.9 cm. Is hs he gh answe? Ceanl no. Anohe peson wh a deen sze mono would nd a deen measue. Fuhemoe, s even possble ha wo deen people, each wh he same sze mono, would nd wo deen measues because he monos ae o deen bands. I s clea wh sn adequae o use he mee (and s submulples) as he un o measuemen o lengh n he sodaconsuco: he measuemen would be slghl deen o each peson. Obvousl, n he soda unvese s essenal o choose anohe un o measuemen o lengh. Snce he amoun o pels n he sodaconsuco wndow s he same o all uses ( 657 pl 48 pl : see Dmensons o he Sodaconsuco Wndow), ndependen o he dmensons o he mono sceen, n hs pape we wll use he pel as he un o measuemen o lengh. The smbol ha we wll adop wll be: pl. The man poblem wh hs new un o measuemen s hs: how s possble o measue somehng n pels whou counng all he pels one b one? Ths s no a val queson, because o coun man pels on he sceen s ecucangl panul. Theeoe, I ve used he ollowng echnque: I know ha a ed mass (I eall don lke hs denomnaon, I would have peeed ed pon) has he ollowng dmensons: 6 pl 6 pl. So, b aangng man ed masses lke a chessboad, s possble o measue he dsance beween wo deen pons on he sceen b smpl counng he ed masses beween he wo pons and mulplng he amoun b 6. Obvousl, s possble ha he lengh we wan o measue s no a mulple o 6; n hs case we wll aange he ed masses lke a chessboad unl we ae less han 6 pels om he endpon, and we wll coun he emanng pels, 5 a he mos. Somemes, howeve, he lengh ha we wan o measue s so gea ha he above ed masses mehod would be equall panul o ou ees and paence. In hs case s possble

8 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 8/3 o appl a med mehod, such as usng ed masses o measue dsances geae han 6 pels. One o m avoe was o ge equal dsances beween ed pons s b consucng spngs a, 45 and 9, angles whch ae eas o pecsel ceae n he sodaconsuco. Usng hese lnes o make peec squaes, s eas o eplcae a ed lengh man mes, n ode o measue usng lage ncemens. I ve used hs echnque oen, as ou wll see n eamples eeed o lae (ou wll nd one eample n Dmensons o he Sodaconsuco Wndow). Anohe phscal quan nvolved n hese sudes s me. Agan, s mpossble o use he usual un o measuemen (seconds s). In pncple should be possble o use he second (s) as a measuemen o me, because advanced smulaon pogams lke ou beloved Sodaconsuco should snchonze he calculus wh he nenal mcopocesso clock. Fo hs eason, he duaon o he smulaon should be unom n all deen compues. Howeve, all compues ae deen om each ohe. Compues oda ae assembled wh man deen pas: hee ae he mcopocessos, he man boads, he acceleaed gaphc devces, and man ohe such devles. (Ae hee an o ou oung people old enough o emembe he smplc o he Commodoe 64? I be ou don.) The eal peomance o a compue depends geal on all s deen componens. Theeoe, s ve dcul o nd wo deen compues wh dencal peomances. Fo hese easons, I suspec ha he same smulaon could have deen duaons on deen compues. I ma seem ha I am beng unnecessal pecse. Neveheless, as ou wll see n he ne chape, I wll need mamum pecson o ge elable esuls. Jus as wh pels, he numbe o ames necessa o complee a smulaon s eacl he same o each compue. So, o avod an ambgu, we wll use he ame as ou un o measuemen o me. The smbol ha we wll adop wll be:. Now he poblem s: how can we coun he amoun o ames ha akes o a pacula model o complee a smulaon? The supsng answe s: usng an odna chonomee. Ths seems lke a conadcon, bu sn. The man poblem s ha n he sodaconsuco apple hee sn a ame coune, so he onl hng ha we can use s a chonomee. Usng he new sodaace meal apple, each o us can nd how man ames hee ae n a second o ou own compue s peomance. Those o us wh a compue o hgh peomance wll ge man ames n a second. Those o us wh a compue o less powe wll no ge as man ames n

9 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 9/3 a second. Thanks o he snchonc beween he smulao s compuaon and he nenal clock o he compue, he deence beween wo deen compues wll pobabl neve be ve hgh. Sll, I hnk hee wll alwas be a deence. In an case, each o us nds o hs own compue how man ames hee ae n a second, solves he poblem. Ae hs has been deemned, he lengh n ames o a pacula smulaon can be calculaed b mulplng he numbe o seconds b he numbe o ames pe second. The esul n ames should be he same o eveone, compleel ndependen o he compue s peomance. The las queson s: how can I nd how man ames hee ae n a second o m compue s wok? Ths can be deemned usng an model n he sodaace meal. In ode o make hs as accuae as possble, I made a ve slow model (Slow walke). In m compue coveed he whole oue n 966 sec (6 6 : an een) o a oal o 736. So, he me conveson ao ( c ) o m compue s: 736 c. 8 (.) 966 sec Ths ao s ve mpoan because I use n he vual epeences descbed n he ne chape. To evew, we wll be usng he ollowng uns o measuemen: Table.: The new uns o measuemen Phscal quan Un o measuemen Lengh Tme pl (pel) (ame) Veloc pl Acceleaon pl Wha abou mass? We know ha he ohe undamenal phscal quan nvolved n he dnamc poblem s mass. Well, n he 8 h chape I wll show ou mahemacall ha sn

10 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll /3 necessa o assume an value o he masses. Ths s because all masses n he sodaconsuco apple ae he same.

11 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll /3 3. DETERMINATION OF THE PHYSICAL CONSTANTS ADOPTED BY THE APPLET As we all know, hee ae man consans pesen n he phscal laws ha goven ou wold. One o he mos mpoan consans, one ha oen pesecues sudens n school, s he eah s gavaonal acceleaon consan g. In he pom o he Eah s suace, he value o hs consan s abou g 9.8 m sec. Thanks o he knowledge o hs consan, we can sa man hngs. Fs, we can sa ha he veloc o a ee allng bod close o he Eah s suace nceases 9.8 m sec eve second o s allng (hs s ue onl unl a con begns o have an eec;.e. jus n he s ew seconds o s allng). Anohe hng ha hs consan allows us o sa s ha he wegh W (epessed n Newon N) o a genec mass m (epessed n kg) s obaned b he ule: W m g (3.) These appaenl smple hngs allow us o see how he knowledge o he consan g s, whou a doub, ve mpoan o a lo o phscal and engneeng applcaons. Obvousl, n addon o he consan g, hee ae a lo o ohe consans ha ae equall mpoan. All hese consans have specc values commonl known b he scenc commun. The queson s: how wee hese consans ognall ound? The answe s ve smple: b means o epemenal ess. How does all hs elae o he sodaconsuco? To answe hs queson we mus s undesand eacl wha he sodaconsuco s. The sodaconsuco s smpl a smulao o phscal laws. In he sodaconsuco hee ae a numbe o mechancal laws ha, lke n eal, canno be volaed. Theeoe, n pncple, should be possble o deemne he phscal consans o he laws used n he sodaconsuco b means o vual epemens, much lke he epemenal ess ha allow us o undesand naue n he eal wold. O couse, I wee an epe n compue languages I would smpl nd he sodaconsuco s phscal consans b lookng o hem n he soda algohm. Unounael, I am no an epe. So he bes hng ha I can do s ansom msel no a The Newonan laws o dnamcs, Hooke s law o he spngs, he Newonan law o he lud s con, he laws o he quas-elasc mpac beween masses and walls.

12 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll /3 vual Galleo (Ialan scens o he seveneenh cenu) n ode o nvesgae he soda unvese. I ve ceaed a sodaconsuco Laboao whee I ve eecued man vual epemens. Thanks o hese epemens I was able o nd how he well-known cusos o gav (g), con () and sness (k) acuall wok (see Fg. 3.). Fgue 3.: The phscal sodaconsuco consans In he ollowng pas o hs chape I wll eplan eacl wha I ve deemned abou each one o he above consans. I ve also nvesgaed abou he coecens o dnamcal esuons o he walls and gound. 3. THE GRAVITY In he eal unvese we could hnk o a smple epemenal es o deemne he gavaonal acceleaon consan g. We could le a lle objec lke a sone all om a ed hegh h. Meanwhle we could measue he me akes he sone o all b means o a chonomee. Knowng ha he allng sone moves wh a unoml acceleaed moon, we could use he ollowng omula: h g (3.) o ge he acceleaon g: h g (3.3)

13 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 3/3 In ac, hs mehod o deemnng he consan g sn ve accuae, because a con wll have an eec. A moe accuae mehod o deemnng he consan g s he pendulum mehod. I s possble o nd he consan g b smpl measung he peod o a complee oscllaon o a long pendulum. Wha can we do n he soda unvese? Eacl he same hngs! We could ge he acceleaon consan 3 g b usng he epemen o a mass allng. Founael, n he soda unvese we have he opon o un o he a con compleel, so we don need o wo abou he poblem descbed above. Sll, hs mehod s a b poblemac. Fs, I don have an accuae chonomee; I jus measue me wh m analog clock, whch means I onl have he accuac o one second (an een compaed o he accuac equed n hs so o epemen). Second, s ve dcul o sa he chonomee eacl when he mass begns allng. Thd, s also ve dcul o sop he chonomee eacl when he mass hs he gound. As a esul, I would ge a measue wh an noleable eo. Wha could I do o hs poblem? I could conduc hs epemen a numbe o mes, so ha I could educe he magn o eo. Bu hs mehod s oo long and edous, even o m paence. Howeve, hee s a moe convenen and accuae soluon: o use he pendulum mehod. Usng as long a pendulum as possble and measung s peod, we can dscove he gavaonal acceleaon consan g. In Append A, I wll eplan n a dealed manne he mahemacal heo behnd he pendulum mehod. Theeoe, n he ollowng descpon, I wll esc msel o eplanng onl he man elaon ha we wll use. As dscussed n Append A, when he mamum angula ecuson α ma (epessed n adan ad) o a pendulum s such ha s possble o use he ollowng appomaon: sn α ma α ma (3.4) The me ha he pendulum needs o each s mamum ecuson sang om he dencal poson. 3 Thee s a pon I wan o make clea abou he wod consan n he soda unvese. As we well know s possble o change he gav b movng he above menoned gav cuso. Theeoe, n pncple, he gav sn consan. Neveheless, we choose a pacula level o gav, ou models ae movng wh a value o gav ha doesn change ove me; n hs case he gav s consan.

14 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 4/3 (jus when he value o α ma s ve small; see Fg. 3.) he peod T o an oscllaon depends onl on he lengh l o he pendulum and he gav consan g, b means o he omula: T l π (3.5) g α l α ma α ma Fgue 3.: Angula ecuson o a pendulum B measung he peod T and knowng he lengh o he pendulum s possble o ge he consan g b means o he ollowng nvese omula: l g 4π (3.6) T Some mgh ague ha hs unnecessal complcaed, because o measue he peod T I wll have he same dcules as descbed o he allng mass mehod. Ths sn ue! I con s dopped o zeo, he pendulum s oscllaon could las oeve. Theeoe, wll be possble o measue no jus one complee oscllaon bu man oscllaons. In hs wa he nevable eos menoned above wll be dsbued n man oscllaons, educng he eec on he nal compuaon. Ths s eacl wha I ve done. Fng a pacula value o he gav b

15 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 5/3 movng he well-known cuso 4, I measued he elapsed me o man oscllaons. Then, I dvded he oveall me b he numbe o complee oscllaons, gvng me he peod T o a sngle oscllaon wh he equed accuac. Beoe I sa eplanng he vual epeences abou gav, I wll sa jus one hng abou he cusos shown n Fgue 3.. Each one o hese cusos can assume 8 deen posons. Each o hese 8 posons des om he pevous o he ollowng poson b one pel o dsplacemen. We have poson when he cuso s a he boom (n hs poson he phscal quan assocaed wh he cuso has he value o zeo), and we have poson 7 when he cuso s a he op (n hs poson he phscal quan assocaed wh he cuso has he mamum value). In he ollowng secon we wll look o he ules o vaaon o he consans g,, and k n espec o he cuso poson. So, namng hese posons p, p and p, we wll look o he ollowng hee laws: g k g k ( p g ) ( p ) ( p ) k wh p, p, p,,, K,7 g k (3.7) Epemen Usng Epemen was possble o nd he value o he gavaonal acceleaon consan g (n pl ) when he gav cuso poson s. The lengh o he p g pendulum n hs case s l 374 pl. I measued he me ook o complee oscllaons. On m compue hs me was 93 sec ( 4 53 ), so he peod o a sngle oscllaon was: 93 T. 93 sec B usng he me conveson ao (.) he peod T becomes: 4 I ve also dopped he con o zeo and aken a he mamum value he gd o he spng ha connecs he ee mass a he ed pon. In he pendulum heo he connecon beween he mass and he ed pon should be peecl gd.

16 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 6/3 T.93 sec sec Applng (3.6) I nall ound: pl ( ) g Fo now we wll gnoe he queson o sgncan dgs. We wll dscuss hs a he end o he chape. Epemen Usng Epemen was possble o nd he value o he gavaonal acceleaon consan g (n pl n hs case s l 374 pl. ) when he cuso poson s 9. The lengh o he pendulum I measued he me ook o complee oscllaons. On m compue hs me was sec ( 3 ), so he peod o a sngle oscllaon was: T. sec B usng he me conveson ao (.) he peod T becomes: T. sec sec Applng (3.6) I nall ound: pl ( 9) g p g Epemen 3 Usng Epemen 3 was possble o nd he value o he gavaonal acceleaon consan g (n pl ) when he cuso poson s 4. The lengh o he pendulum p g n hs case s l 375 pl. I measued he me ook o 5 complee oscllaons. On m compue hs me was sec ( 3 4 ), so he peod o a sngle oscllaon was:

17 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 7/3 T. 47 sec 5 B usng he me conveson ao (.) he peod T becomes: T.47 sec sec Applng (3.6) I nall ound: pl ( 4) g Epemen 4 Usng Epemen 4 was possble o nd he value o he gavaonal acceleaon consan g (n pl ) when he cuso poson s 5. The lengh o he pendulum p g n hs case s l 375 pl. I measued he me ook o 5 complee oscllaons. On m compue hs me was 76 sec ( 56 ), so he peod o a sngle oscllaon was: 76 T. 7 sec 5 B usng he me conveson ao (.) he peod T becomes: T.7 sec.8 3. sec Applng (3.6) I nall ound: pl ( 5) g Epemen 5 Usng Epemen 5 was possble o ge he value o he gavaonal acceleaon consan g (n pl n hs case s l 376 pl. ) when he cuso poson s 6. The lengh o he pendulum I measued he me ook o complee oscllaons. On m compue hs me was 96 sec ( 3 6 ), so he peod o a sngle oscllaon was: p g

18 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 8/3 96 T. 98 sec B usng he me conveson ao (.) he peod T becomes: T.98sec sec Applng (3.6) I nall ound: pl ( 6) g Resuls Analss Dsplang he above esuls n a gaph n whch he hozonal as epesens he cuso poson p g he cuve o Fgue 3.3: and he vecal as epesens he gavaonal acceleaon consan g, we ge pl g Fgue 3.3: gav end p g Lookng he cuve o Fgue 3.3 we can mmedael see ha he elaon beween g and p g s no lnea. Snce he cuve looks moe smla o a paabola, we can o calculae he ao beween g and p g. The ollowng able shows he esul o hs calculaon o all he above epeences:

19 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 9/3 Table 3.: Rao g p g o he above epeences p g pl g g p g Snce he ao g p g s paccall consan o all 5 o he above vual epeences, we can am ha beween g and p g hee s a quadac popoonal: g pl g g P p g (3.8) ( p ) In he omula above I ve noduced he gav paamee g p whch s a consan paamee mplemened n he sodaconsuco apple. I I wee eemel pecse I would nd he gav paamee g p b means o a quadac nepolaon o he above daa, bu emembeng ha all o hs s jus pla, I wll esc msel o he calculaon o he medum value o g. Fom Table 3. we can nd he ollowng medum value o g : p p pl g p.3569 (3.9) pg Fgue 3.4 dsplas he numecal daa o Table 3. and he connuous cuve made om (3.8) and (3.9). As can cleal be seen, n he ange o ( p [, 6] ) hee s a peec ovelappng. g p g ha was used n he epemens

20 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll /3 pl g Fgue 3.4: Ovelappng beween numecal daa and heoecal cuve p g 3. THE STIFFNESS (STATIC METHOD) The man componens o he sodaconsuco ae spngs and muscles. These componens have elasc popees ha ae dened b a phscal quan called sness. To undesand wha he elasc popees o spngs eacl ae and how hese popees ae epesened b sness, we could make some obsevaons o spngs n he eal wold. We all know wha eal spngs ae and wha chaacescs he possess. We know, o eample, ha a spng changes lengh onl when oce s appled o s ends, and ha when hs oce sops he spng euns o s ognal lengh. We also know ha he oce necessa o pull a spng nceases wh he spng s eenson. Fnall, we know ha wo spngs o deen senghs subjeced o he same oces have deen eensons. These obsevaons show he basc conceps o elasc and sness. We wll sa ha a bod s an elasc bod s deomaons vansh when he causes ae emoved. We wll sa ha a bod s a lnea elasc bod s deomaons ae decl popoonal o he oces appled o he bod. Fnall we wll dene sness as how well an elasc bod can manan s ognal shape (o lengh we ae speccall speakng o a spng) when s subjeced o eenal oces. Fo spngs, all hese chaacescs can be mahemacall dened n a ve smple wa.

21 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll /3 l l F Fgue 3.5: Eenson o a spng subjeced o acon b a oce F Fgue 3.5 shows a spng beoe and ae he acon o a acon oce F. Callng l he lengh o he spng a es and l he lengh o he spng ae eended, we wll dene he quan o he spng s eenson: l l l (3.) B denon, he eenson o a spng wll be posve s nal lengh l s geae han he nal lengh l nal lengh. l. The eenson o a spng wll be negave s nal lengh l s less han he I he spng s a lnea elasc spng (lke he spngs n he sodaconsuco apple), hen he spng s eenson l wll be decl popoonal o he oce s nens F b means o he elaon: F k l (3.) Ths law s known as Hooke s law 5. The consan k ha appeas n (3.) s he sness o he spng and s a phscal quan ha s alwas posve. Is value s epesenave o he spng s sengh. Usng (3.), we can sa: 5 Robe Hooke was an Englsh scens o seveneenh cenu, conempoaneous o Isaac Newon. I seems ha he wo scenss ween paculal ond o each ohe. I s onc ha n ou beloved sodaconsuco apple he laws lve ogehe hamonousl.

22 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll /3 F l (3.) k I s eas o undesand ha unde he same oce, spngs wh hgh values o sness k wll have eensons smalle han hose o spngs wh low sness. Fom (3.) we can also see ha he oce s posve, s eec on he spng wll be posve (eenson), whle he oce s negave, s eec on he spng wll be negave (compesson). The sness k ha appeas n (3.) and (3.) s he same phscal quan ha can be changed wh he cuso k (see Fg. 3.) n he sodaconsuco apple. Thee s one moe hng I mus emphasze abou he sodaconsuco s vual spngs: all spngs, egadless o he nal lengh, have he same sness 6. In he cuen veson o he apple s mpossble o have spngs o ndependen sness n he same smulaon. B means o sac vual epemens, he epesson (3.) wll allow us o deemne he value o he sness k dependng on he poson o he coespondng cuso. Obvousl, n ode o use (3.) o nd he sness k, we wll need a sac oce F. Bu how can we ceae a sac oce? We can smpl use he oce o gav. We know ha a mass m n a consan gavaonal eld has a wegh gven b he epesson (3.). Theeoe, n ode o ceae a consan oce n ou epemens, we wll use he vual wegh o ee masses n he sodaconsuco apple. I mgh be mpoan a hs pon o sa somehng abou he sodaconsuco s ee masses. B means o ve smple vual epemens s possble o pove ha all ee masses ae equal 7, so we wll have jus one value o mass m o all ee masses. A he momen, we wll leave hs value unknown. In he Chape 8 we wll see wh s possble o avod havng o assume an specc value o m. 6.e. all spngs, egadless o he nal lengh, wll be subjec o he same eenson unde he same oce. 7 I wll leave o ou he pleasue o devsng some smple epemens.

23 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 3/3 The Sac Epemens Jus as we dd wh he gav g, ou objecve wll be o nd how he spng s sness k changes accodng o he poson o he coespondng cuso;.e. ou objecve wll be o dscove he law: ( pk k k ) (3.3) In ode o nd hs law we wll appl he wegh o a ee mass o spngs o deen lenghs 8, usng deen ed values o sness. Theeoe, he consan oce F n he elaon (3.) o us wll be he wegh W o a ee mass gven b (3.). Then we wll have: F W m g k l (3.4) so ha: k g m (3.5) l In (3.5), we know he value o g hanks o (3.8) and (3.9), and we wll nd he eenson l b measung. As I ve sad beoe, a he momen we wll leave he value o m unknown, so wll be helpul o dene he ollowng phscal quan: k k g (3.6) m l whch s ndependen o he value o m. We wll call k specc sness. Thanks o he noducon o specc sness we can we (3.5) as: k k m (3.7) 8 I ve used spngs o deen lenghs o pove whou an doub ha he sness o a spng s absoluel ndependen o s nal lengh.

24 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 4/3 Epemen 6 In Epemen 6 he eensons o spngs o deen lenghs have been measued wh he sness cuso poson equal o p k. The oce F has been vaed b changng he poson o he gav cuso om p 3 o b ncemens o. Fo each seng g o gav, he eenson o he spngs has been measued and he esuls epoed n he ollowng Table 3.: p g Table 3.: Spng s eenson o deen values o gav when p k p g pl g l [ pl] g l Usng he above esuls, we nd he aveage value o he specc sness o he cuso poson p k : k ( ) (3.8) Epemen 7 In Epemen 7 he eensons o spngs o deen lenghs have been measued wh he sness cuso poson equal o p k 5. The oce F has been vaed b changng he

25 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 5/3 poson o he gav cuso om p 3 o b ncemens o. Fo each seng g o gav, he eenson o he spngs has been measued and he esuls epoed n he ollowng Table 3.3: p g Table 3.3: Spng s eenson o deen values o gav when p k 5 pl p g g l [ pl] g l Usng he above esuls, we nd he aveage value o he specc sness o he cuso poson p k 5 : k ( 5) (3.9) Epemen 8 In Epemen 8 he eensons o spngs o deen lenghs have been measued wh he sness cuso poson equal o p k 3. The oce F has been vaed b changng he poson o he gav cuso om p 3 o b ncemens o. Fo each seng g p g o gav, he eenson o he spngs has been measued and he esuls epoed n he ollowng Table 3.4:

26 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 6/3 Table 3.4: Spng s eenson o deen values o gav when p k 3 p g pl g l [ pl] g l Usng he above esuls, we nd he aveage value o he specc sness o he cuso poson p k 3 : k ( 3) (3.) Resuls Analss In hs case we have he values o he specc sness o onl hee posons on he cuso p k, so wll be mpossble o ge as good a cuve as we dd wh he gav epemens. Theeoe we wll lm ouselves o a smple analss o he above numecal daa. Based on he esuls o ou gav epemens, s easonable o guess ha he elaonshp beween he specc sness and p k s once agan quadac. In ode o ve hs hpohess we wll calculae he ao (3.8), (3.9) and (3.). ( p k ) p k k o each o he hee esuls epoed n

27 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 7/3 Table 3.5: Rao g p k o he above epemens p k k k p k Because he esuls o he ao k ( p k ) p k ae appomael consan, we can assume wh ve lle doub he ollowng ule o specc sness: k ( p k ) k p p k (3.) n whch has been noduced he paamee as wh he gav paamee g p k p ha we wll call he sness paamee. Jus, he sness paamee mplemened n he sodaconsuco apple. Is medum value s (see Table 3.5): k p s a consan paamee k p.4477 (3.) Theeoe he ule o he sness k, hanks o (3.7) and (3.), wll be: k ( p k ) k ( p k ) m k p m p k (3.3) 3.3 THE STIFFNESS (DYNAMIC METHOD) Thee s anohe and moe accuae mehod o deemnng he sness o a spng. Ths mehod s ve smla o he dnamc mehod ha we used o he gav epemens. B smpl measung he peod o he oscllaon o a spng conneced o one mass, s possble o calculae he spng s sness. In he ollowng secon I wll esc msel o eplan jus

28 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 8/3 he man elaon ha we wll use. A moe dealed eplanaon o hs mehod can be ound n Append B. k m Fgue 3.6: Ssem spng-mass Callng k, m and T especvel he sness o a spng, he value o he mass conneced wh he spng and he peod o a complee oscllaon o he ssem spng-mass (see Fg. 3.6), s possble o show ha he ollowng elaon s vald: 4π k m (3.4) T Usng hs elaon, s possble o ge he sness o a spng b measung he peod T. Takng no accoun (3.7), om (3.4) ollows mmedael he epesson o he specc sness k : k 4π (3.5) T Obvousl, lke n he gav deemnaon, wh con equal o zeo he oscllaons o he spng wll las oeve. Theeoe, o ge a moe accuae eadng, we wll measue he duaon o moe han one oscllaon. Epemen 9 In Epemen 9, was possble o nd he value o he specc sness k (n ) when he cuso poson was p k 6. The duaon o complee oscllaons has been

29 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 9/3 measued. In m compue hs me was 4 sec ( ), so he peod o a sngle oscllaon was: 4 T. 4 sec B usng he me conveson ao (.) he peod T becomes: T.4 sec sec Applng (3.5) I nall ound: ( 6).5945 k Epemen In Epemen, was possble o nd he value o he specc sness k (n ) when he cuso poson was 8. The duaon o 5 complee oscllaons p k has been measued. In m compue hs me was 6 sec ( 4 ), so he peod o a sngle oscllaon was: 6 T. 7 5 sec B usng he me conveson ao (.) he peod T becomes: T.7 sec.8 8. sec Applng (3.5) I nall ound: () k Epemen In Epemen, was possble o nd he value o he specc sness k (n ) when he cuso poson was. The duaon o 5 complee oscllaons has been measued. In m compue hs me was 8 sec ( 8 ), so he peod o a sngle oscllaon was: p k

30 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 3/3 8 T. 85 sec 5 B usng he me conveson ao (.) he peod T becomes: T.85 sec sec Applng (3.5) I nall ound: ( ) k Epemen In Epemen, was possble o nd he value o he specc sness k (n ) when he cuso poson was. The duaon o 5 complee oscllaons has been measued. In m compue hs me was 7 sec ( 47 ), so he peod o a sngle oscllaon was: 7 T. 7 sec 5 B usng he me conveson ao (.) he peod T becomes: T.7sec sec Applng (3.5) I nall ound: ( ) k p k Epemen 3 In Epemen 3, was possble o nd he value o he specc sness k (n ) when he cuso poson was 4. The duaon o 5 complee oscllaons has been measued. In m compue hs me was 9 sec ( 3 ) so he peod o a sngle oscllaon was: 9 T. 6 sec 5 p k

31 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 3/3 B usng he me conveson ao (.) he peod T becomes: T.6sec sec Applng (3.5) I nall ound: ( 4) k Resuls Analss Dsplang he above esuls n a dagam n whch he hozonal as epesens he cuso poson n Fgue 3.7: pk and he vecal as epesens he specc sness k, we ge he cuve k p k Fgue 3.7: specc sness end Fs o all, s possble o see how he cuve n Fgue 3.7 conms ha hee sn a dec popoonal beween k and p k. Theeoe, also akng no accoun he esul obaned n he pevous secon 3., we wll o calculae he ao beween k and p k o he above esuls. I hee wee no eos n he epemens o calculaons, hen he value o he ao k p k can be much deen han he value obaned n (3.). The esuls o hs ao o all he above epemens s shown n he ollowng able:

32 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 3/3 Table 3.6: Rao k p k o he above epemens p k k k p k Snce he ao k p k s paccall consan o all ve o he above vual epemens, we have uhe evdence o he vald o he epemenal elaon (3.). Moeove, s possble o see ha all he values o he ao k ae paccall he same o he value (3.) obaned b means o he sac epemens. Snce he dnamc epemen gves moe accuae esuls, o he sness paamee we wll ake he ollowng medum value: p k k p.449 (3.6) The quas-peec concdence beween he values o k p obaned b means o wo deen mehods o epemens s poo o he vald o hs secon o vual epemens. 3.4 THE FRICTION The con dscussed hee s he con ha a bod mees whle n movemen n a lud. Theeoe, hs knd o con s elaed o he vscos o he medum n whch he bod s movng. Ths con has essenall wo chaacescs: he s s ha he oce o he con s alwas popoonal o he bod s veloc; he second s ha he oce o he con s such ha s eec s alwas n opposon o he movemen. In ode o mahemacall dene hs knd o con we wll noduce a new phscal quan ha n

33 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 33/3 he ollowng pas o hs pape wll be called he dampng consan 9 and ha wll be ndcaed wh he lee. I s hs quan ha we equenl change n ou models b adjusng he second cuso o Fgue 3.. Snce veloc sn a scala quan, bu ahe a vecoal quan, when denng veloc s nsucen o spec jus s nens (o eample epessed n m sec ). I s also necessa o spec s decon. Theeoe, veloc, lke all ohe vecoal quanes, can be epesened b vecos (see Fgue 3.8 a). Foce, lke veloc and acceleaon, s a vecoal quan, so can also be epesened b vecos. Hee we ae eeng o oces om a bod s moon n a vscous lud. So, akng no accoun wha was sad eale abou hese oces, we can epesen hem b means o a veco whch has: nens equal o he veloc s nens mulpled b he dampng consan, and he oppose decon as he veloc (see Fgue 3.8 b). v v F v a) b) Fgue 3.8: Relaon beween veloc and con oce. Obvousl, he oce o con wll end o educe he moon o a bod, so ha anhng causes moon (o eample somehng lke a muscle), he bod evenuall wll sop. The man wa o deemne he dampng consan s o sud he ee oscllaon o a spngmass ssem (lke n he dnamc deemnaon o he sness) n an envonmen o vscous 9 In sodalanguage, hs paamee s usuall smpl called con. Neveheless, hee, n ode o avod an conuson beween he deen knds o cons ha ae encouneed n he eal wold, I have named hs paamee moe appopael he dampng consan. I s possble o epesen a veco b means o an aow. The aow s lengh wll epesen n he appopae scale he veco s nens; he oenaon o he aow wll dene he veco s decon.

34 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 34/3 con: he damped ee vbaon. A dealed eplanaon o hs mehod can be ound n Append B, so o now, we wll jus dscuss he man elaons ha wll be used. Leavng a spng-mass ssem lke he one epesened n Fgue 3.6 ee o oscllae n pesence o vscous con, and callng () he mass dsplacemen n espec o he que poson a he genec nsan, s possble o epo n a Caesan dagam he evoluon o he ssem. B epong he mass dsplacemen n he vecal as and he me n he hozonal as, as we wll see n Append B, wll eld a gaphc smla o he ollowng Fgue 3.9. () n n+ν Fgue 3.9: Eec o vscous dampng on a ee vbaon. Consdeng wo posve peaks beween a numbe o ν complee ccles o oscllaons and callng and he especve values s possble o calculae he ollowng n n+ ν logahmc paamee : n δ ln (3.7) n+ν Usng hs, s possble o ge he dampng consan value: We ae consdeng he naual logahm;.e. he nvese uncon o he eponenal uncon e.

35 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 35/3 k m (3.8) νπ + δ In (3.8) we have especvel ndcaed b k and m he spng s sness and he mass value. Theeoe, akng no accoun he poson (3.7) abou he specc sness o a spng, we can ewe he elaon (3.8) as: k m (3.9) νπ + δ Jus as wh sness, s possble hee o noduce a new paamee ha we wll call specc dampng consan whch doesn need he ee mass value m: m k νπ + δ (3.3) Thanks o he second omula o (3.3) we wll be able o nd he sodaconsuco s law o vaaon wh he con cuso s poson p o he specc dampng. Epemen 4 In Epemen 4, was possble o nd he value o he specc dampng (epessed n ) when he cuso posons o con and sness wee p and 6 p k especvel. The value o he specc dampng should be ndependen o he sness, bu n ode o ve hs ndependence, we wll epea eve epemen (.e. o a ed value o he cuso poson p ) wh wo deen values o he specc sness.

36 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 36/3 The peak values o he s and 73 d ccles o oscllaons wee measued, and he ollowng values wee obaned: n n + ν 73 ν 7 n 9 pl n+ ν pl Fom (3.7) ollows: 9 δ ln Moeove, usng he elaons (3.) and (3.6) n ode o ge he value o he specc sness, ollows: ( 6) k Fnall, akng no accoun he pevous values, om he nd omula (3.3) ollows: () π Epemen 5 In Epemen 5, was possble o nd he value o he specc dampng (epessed n ) when he cuso posons o con and sness wee p and 8 p k especvel. The peak values o he s and s ccles o oscllaons wee measued, and he ollowng values wee obaned: n n + ν ν n 9 pl n+ ν 73 pl Fom (3.7) ollows:

37 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 37/3 9 δ ln Fo he specc sness we have he value: () k Fnall, akng n accoun he pevous values, om (3.3) ollows: () π The esuls o he las wo epemens esablsh ha, neglecng he nevable small eos, he specc dampng s ndependen o he specc sness. The medum value o he specc dampng when p wll be: () Epemen 6 In Epemen 6, was possble o nd he value o he specc dampng (epessed n ) when he cuso posons o con and sness wee p and 6 p k especvel. The peak values o he s and 5 h ccles o oscllaons wee measued, and he ollowng values wee obaned: n n + ν 5 ν 4 n 3 pl n+ ν 5 6 pl Fom (3.7) ollows: 3 δ ln Fo he specc sness we have he value:

38 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 38/3 ( 6).5954 k Fnall, akng n accoun he pevous values, om (3.3) ollows: ( ) π Epemen 7 In Epemen 7, was possble o nd he value o he specc dampng (epessed n ) when he cuso posons o con and sness wee p and 8 p k especvel. The peak values o he s and 33 d ccles o oscllaons wee measued, and he ollowng values wee obaned: n n + ν 33 ν 3 n 3 pl n+ ν 33 6 pl Fom (3.7) ollows: 3 δ ln Fo he specc sness we have he value: () k Fnall, akng n accoun he pevous values, om (3.3) ollows: ( ) π The medum value o he specc dampng when p wll be:

39 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 39/3 ( ) Epemen 8 In Epemen 8, was possble o nd he value o he specc dampng (epessed n ) when he cuso posons o con and sness wee p 3 and 6 p k especvel. The peak values o he s and s ccles o oscllaons wee measued, and he ollowng values wee obaned: n n + ν ν n 36 pl n+ ν 88 pl Fom (3.7) ollows: 36 δ ln Fo he specc sness we have he value: ( 6).5954 k Fnall, akng n accoun he pevous values, om (3.3) ollows: () π Epemen 9 In Epemen 9, was possble o nd he value o he specc dampng (epessed n ) when he cuso posons o con and sness wee p 3 and 8 p k especvel.

40 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 4/3 The peak values o he s and 8 h ccles o oscllaons wee measued, and he ollowng values wee obaned: n n + ν 8 ν 7 n 38 pl n+ ν 8 87 pl Fom (3.7) ollows: 38 δ ln Fo he specc sness we have he value: () k Fnall, akng n accoun he pevous values, om (3.3) ollows: () π The medum value o he specc dampng when p 3 wll be: () Epemen In Epemen, was possble o nd he value o he specc dampng (epessed n ) when he cuso posons o con and sness wee p 4 and 6 p k especvel. The peak values o he s and 8 h ccles o oscllaons wee measued, and he ollowng values wee obaned:

41 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 4/3 n n + ν 8 ν 7 n 98 pl n+ ν 8 46 pl Fom (3.7) ollows: 98 δ ln Fo he specc sness we have he value: ( 6).5954 k Fnall, akng n accoun he pevous values, om (3.3) ollows: ( 4) π Epemen In Epemen, was possble o nd he value o he specc dampng (epessed n ) when he cuso posons o con and sness wee p 4 and 8 p k especvel. The peak values o he s and 3 d ccles o oscllaons wee measued, and he ollowng values wee obaned: n n + ν 3 ν n 33 pl n+ ν 3 49 pl Fom (3.7) ollows: 33 δ ln Fo he specc sness we have he value:

42 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 4/3 () k Fnall, akng n accoun he pevous values, om (3.3) ollows: ( 4) π The medum value o he specc dampng when p 4 wll be: ( ) Epemen In Epemen, was possble o nd he value o he specc dampng (epessed n ) when he cuso posons o con and sness wee p 5 and 6 p k especvel. The peak values o he s and h ccles o oscllaons wee measued, and he ollowng values wee obaned: n n + ν ν n 89 pl n+ ν 5 pl Fom (3.7) ollows: 89 δ ln Fo he specc sness we have he value: ( 6).5954 k Fnall, akng n accoun he pevous values, om (3.3) ollows:

43 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 43/3 () π Epemen 3 In Epemen 3, was possble o nd he value o he specc dampng (epessed n ) when he cuso posons o con and sness wee p 5 and 8 p k especvel. The peak values o he s and 4 h ccles o oscllaons wee measued, and he ollowng values wee obaned: n n + ν 4 ν 3 n 95 pl n+ ν 4 55 pl Fom (3.7) ollows: 95 δ ln Fo he specc sness we have he value: () k Fnall, akng n accoun he pevous values, om (3.3) ollows: () π The medum value o he specc dampng when p 5 wll be: ()

44 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 44/3 Resuls Analss As wh he gav and sness epemens, lookng a he ollowng Table 3.7, whch conans he values obaned n he pevous vual epemens, s possble o esablsh ha he specc dampng s elaed o he cuso poson popoonal. p b means o a quadac Table 3.7: ao beween he specc dampng and he cuso poson p p Jus as wh he pevous phscal quanes, s possble o assume o he specc dampng he ollowng law: ( p ) p p (3.3) n whch he value o he dampng paamee p can be obaned as medum value om he values o he ao p epoed n he above Table 3.7. We wll assume o he dampng paamee he ollowng value: p (3.3)

45 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 45/3 In he ollowng gaph ae epoed boh he epemenal values o he specc dampng and he coespondng nepolaon value obaned om (3.3) and (3.3) Fgue 3.: Epemenal and nepolang value o he specc dampng. p 3.5 COLLISIONS The onl collsons ha es n he sodaconsuco apple ae collsons beween masses and walls. Thee aen collsons beween masses hemselves: hese enes, even hough he appea o have a pacula sze, ae mahemacall jus pons (.e. couples o Caesan coodnaes). Fo hs eason we wll avod speakng n deph abou hs phenomenon. The onl hng ha we need o undesand s he wa n whch collsons beween masses and gd walls occu. Thee ae wo classc knds o collsons: elasc and non-elasc. In elasc collsons, he nens o he veloc o a mass mmedael beoe and ae s mpac on a gd wall s eacl he same. Onl he decon o he veloc changes, dependng on he angle o mpac. In non-elasc collsons, he veloc o he mass mmedael ae he mpac dops o zeo. I s as he wall wee spnkled wh glue: he mass emans aached o he wall. In eal, nehe o hese pes o mpacs ess. I elasc collsons esed n he eal wold, hen, n pncple, would be possble o have a ball ha bounces on he gound oeve. Moeove, n eal, non-elasc collsons alwas esul n a leas a small ecol o he mass. B wall I mean he walls, he gound and he celng.

46 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 46/3 u u o v o v u v Fgue 3.: veloces beoe and ae an mpac o a mass on a gd wall. We wll dene u and v, especvel, as he veloces o a mass mmedael beoe and ae s collson wh a gd wall. In eeence o he wall s suace, we wll ndcae wh vo he ohogonal componens, and wh u and v and he angenal componens (see Fgue 3.). I s possble o see n Fgue 3. ha he ohogonal componens o veloces have he oppose decon beoe and ae he collson, whle he angenal componens o veloces have he same decon beoe and ae he collson. Wha abou he values o veloc? Snce, as menoned eale, collsons n he eal wold wll neve be peecl elasc o nonelasc, n ode o dscuss mahemacall eacl wha happens dung he mpac o a mass on a gd wall, we wll noduce he coecens o elasc esuon hese coecens, s possble o we: c u o and c. Thanks o o v v o c c o u u o (3.33) Takng no accoun ha n a ealsc collson < o c, c <, we can mmedael undesand ha he value o veloc ae he mpac s alwas less han he value o veloc beoe he mpac. Obvousl, we would have a ue elasc mpac he coecens and wee c c o equal o, and we would have a ue non-elasc mpac he coecens c and equal o. c o wee

47 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 47/3 The queson now s: wha happens n he sodaconsuco apple? I eas o ve ha n collsons beween masses and walls he masses alwas lose pa o he knec eneg 3, so n he ne pa o hs secon we wll nvesgae he coecens o elasc esuon o walls n he sodaconsuco apple. The Coecen o Elasc Resuon n he Ohogonal Impac Thanks o he vual epemen on ohogonal mpac, s possble o see ha, n absence o con 4, he mamum hegh o he ee allng mass ae each bounce s alwas less han he pevous. Callng u and v especvel he veloc o a mass mmedael beoe and ae s mpac on he gound, n accod wh (3.33), he coecen o elasc esuon o he ohogonal mpac wll be: v c o (3.34) u In ode o nd he veloc o he mass mmedael beoe he mpac on he gound we wll use he well-known laws o unoml acceleaed moon. Usng he hegh he mass begns allng, on he gound he veloc wll be: h om whch u gh (3.35) Wha abou he veloc v? We alead know ha he veloc v wll be less han he veloc u, so, nevabl, he hegh ha he mass wll each ae he s ebound mus be less h han he hgh h. I s possble o calculae he veloc v b smpl measung he hegh h eached b he mass ae he s ebound: v gh (3.36) 3.e. he phscal quan elaed o he veloc o masses. 4 The dampng n he pevous secon.

48 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 48/3 Takng no accoun (3.34), (3.35) and (3.36) we nall have: gh h c o (3.37) gh h Measung he heghs h and h n he pevous vual epemen we nd h 4 pl and h 38 pl. Applng he (3.37) we ge: c o Wha does hs mean? I means ha n ohogonal mpacs wh he walls, ee masses alwas lose abou 5% o he veloc! The Coecen o Elasc Resuon n he Tangenal Impac In ode o nd he value o he coecen o elasc esuon n angenal mpacs, a vual epemen has been devsed eamnng a pacula angenal mpac. In ode o ad he eplanaon o he epemen I have also ealzed he schemac Fgue 3.. In hs epemen, a mass wh an nal hozonal veloc s ee o all n absence o con. The mass s moon, n s paabolc allng, can be dened n wo componens: he vecal moon and he hozonal moon. The vecal moon wll be a unoml acceleaed moon, whle he hozonal moon wll smpl be a unom moon 5. h h d d Fgue 3.: Scheme o he vual epeence 5.e. he hozonal componen o veloc s consan.

49 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 49/3 Callng h he hegh om whch he mass sas s paabolc allng, he me wll ake o he mass o each he gound s: h (3.38) g Callng d he hozonal dsplacemen o he mass n hs me, he hozonal componen o he veloc wll be: d g u d (3.39) h Ths s he angenal veloc o he mass beoe he mpac on he gound. In ode o calculae he coecen o elasc esuon n he angenal mpac now we need he hozonal componen o he veloc ae he mpac. I s possble o calculae hs veloc b consdeng ha ae he mpac he moon o he mass can agan be dened n he wo pas: he unoml acceleaed vecal moon and he unom hozonal moon. Theeoe, callng h me necessa o each he gound agan s: he mamum hegh eached b he mass ae he s ebound, he h (3.4) g Callng d he hozonal dsplacemen o he mass dung hs me, he hozonal componen o he veloc ae he mpac wll be: d d g v (3.4) h

50 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 5/3 In accod wh he s omula o (3.33), and akng no accoun he veloces (3.39) and (3.4), he coecen o elasc esuon o he angenal mpac wll be: v d h c (3.4) u d h Fom he vual epemen menoned eale, s possble o measue he ollowng dsances: d d h h 377 pl 57 pl 4 pl 38 pl Fom (3.4) ollows: c Wha does hs mean? Ths means ha n angenal mpacs wh he walls, ee masses alwas lose abou 9% o he veloc! Ths ac s eemel enlghenng (a leas o me). Indeed, s he low value o hs coecen ha allows ou models o walk. In he apple, hee sn con on he gound n he eal phscal meanng o he wod. I he value o hs coecen wee geae, ou models would walk wh much dcul. I hs coecen wee equal o, ou models would no be able o walk a all. 3.6 FINAL RESULTS A he end o hs long chape I wll smpl summaze n a concse wa he man esuls obaned b means o he pevous vual epemens. We have dened wh p, p and p he poson o he cusos o Fgue 3.. These paamees can assume values om o 7. g k

51 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 5/3 The law o vaaon o he acceleaon consan g wh he cuso s poson p g s: g pl g g p p g (3.43) ( p ) n whch he value o he gav paamee g p epemenall obaned s: 4 pl g p (3.44) The law o vaaon o he spng s sness k wh he cuso s poson p k s: k k ( p ) k ( p ) k ( p ) k k p k p m k (3.45) n whch he value o he sness paamee k p epemenall obaned s: 5 k p 4.49 (3.46) The law o vaaon o he dampng wh he cuso s poson p s: ( p ) ( p ) ( p ) p p m (3.47) n whch he value o he dampng paamee p epemenall obaned s:

52 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 5/3 5 p (3.48) The componens o he veloc beoe and ae an mpac beween masses and walls ae elaed b means o he equaon: v v o c c o u u o (3.49) n whch he values o he coecens o elasc esuon epemenall obaned ae: c c o..75 (3.5) All o he above consans ae ceanl aeced b nevable eos n he measuemens, so I would nuvel el on jus he s wo sgncan dgs. I would have been moe hoough o calculae he eos usng he well-known sascal echnques, bu snce hs s all jus pla, I decded no o.

53 THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll 53/3 4. CINEMATIC Ths chape wll dscuss he man cnemac quanes 6 ha we wll encoune n he ollowng chape o hs pape. These cnemac quanes ae he poson veco, he veloc, and he acceleaon. All ae elaed o he masses on he sceen SYSTEM OF COORDINATES The s hng we need s a ssem o coodnaes. We wll use a pa 8 o ohogonal Caesan s aes wh he ogn ed n he lowe le cone o he sodaconsuco wndow. The hozonal as wll be he -as and s decon wll be om he le o he gh. The vecal as wll be he -as and s decon wll be om he boom o he op (see Fgue 4.). Thanks o hs ssem o Caesan s aes wll be possble o dene he poson o an ee mass on he sceen wh a pa o Caesan coodnaes. O m Fgue 4.: Ssem o eeence adoped n he es o he pape. 6 The quanes elaed o he moon o masses. 7 Fom now on we wll alk eclusvel abou elemens o he sodaconsuco apple. 8 Onl wo ae needed because, as we know, he sodaconsuco s onl a wo-dmensonal applcaon o he man laws o mechancs.