Scattering of a ball by a bat in the limit of small deformation

Transcription

1 INVESTIGACIÓN Revista Mexicana de Física AGOSTO 01 Scattering o a all y a at in the limit o small deormation A. Cao Deartamento de Física Teórica, Instituto de Ciernética, Matemática y Física, Calle E, No. 309, Vedado, La Haana, Cua. L. Soler Instituto de Investigaciones del Transorte, MITRANS, La Haana, Cua. C. González Deartamento de Ultrasónica, Instituto, de Ciernética, Matemática y Física, Calle E, No. 309, Vedado, La Haana, Cua. Reciido el 18 de octure de 011; acetado el de mayo de 01 The rolem o the mechanical evolution o a shock etween a cylindrically symmetric oject and a sherical one is solved in the strict rigid small deormations aroximation or aritrary values o the initial conditions. The riction during the imact is assumed to satisy the standard rules. Firstly, when it is assumed that the only source o energy dissiation is riction, the rolem is ully solved y determining the conditions at the searation oint etween the two odies. A relation determining whether the contact oints o the two odies slides etween them or ecome at rest to e ure rotation state at the end o the imact, is determined or this case o the urely rictional energy dissiation. In second lace, the solution is generalized to include losses in addition to the rictional ones. It ollows that, whatever the mechanism o the additional orm o dissiation is, assumed that it did not aects the usual orms o the laws o riction, the comlementary losses only can change the ending value o the imulse I done y the normal orce o the at on the all at searation. Then, the dynamical evolution o all the mechanical quantities with the value o I during the shock rocess remains invariale. Thus, under the adoter assumtions o strict rigidity and validity o the standard rules or riction, the solution o the rolem is also exactly ound, whenever the total amount o dissiated energy is considered as known y examle, measuring the ending mechanical energy o the system. The analysis allows to determine the values o the tangential and normal coeicients o restitution or the class o shocks examined. Finally, the results are alied to the descrition o exerimental measures o the slow motion scattering o a aseall y a at. The evaluations satisactorily reroduce the measured curves or the outut center o mass and angular velocities o the all as unctions o the scattering angle and the imact arameter, resectively. Keywords: Mechanical imact; shock; scattering; hysics o aseall. PACS: ; ; Nk; F; 45.0.dg; 45.0.D- 1. Introduction Classical Mechanic is an ancient ield o Physics [1,7]. An innumerale amount o rolems had een already solved, which y now constitute a main dataase or technological alications. In articular, the scattering rolem in the ramework o Particle, Nuclear and Atomic Physics has een the suject o intense investigation along centuries o research [8-10]. On another hand, as stated in Re. 11, at dierence with the situation in microscoic Physics, the scattering etween macroscoic odies, had not een a similarly attended area o study. However, in relatively recent times, and as motivated y the relevance within the aseall o the shocking rocess etween the at and the all in the aseall sort, a research activity on the theme had een stimulated [11-14]. An extended study o the hysical rocess in imact mechanics can e ound in Re. 13. Reerences 11 and 14 resented detailed studies o the shock rolem o a at and a all directed to investigate the otimal atting conigurations and the scattering results o the imact rocess at low velocities. The solution o the shock rocess given in those works were ound under the restrictions o: 1 an assumed two dimensional character o the shock, and the use o articular models or the imulse orces aearing during the imact. The work resented in Re. 1 is devoted to investigate the inluences o the lack o rigidity o the at and the all, and the dynamical evolution o the all in the air on the ating rocess results. In these cited works reerences to a numer o additional studies stimulated y the relevance o mechanical rocesses in sorts can e ound. In the resent study we investigate the rolem o the shock etween a rigid and sherical oject to e called as the all and an also rigid ody to e named as the at showing a cylindrical symmetry axis. The general motivation o the work is to clariy u to what extent the assumtion o rigidity o the odies, when taken in conjunction with the satisaction o the usual laws or static and dynamic riction could allow or a ull solution o the rolem. When considered or ossile ractical alications, it can e noted that the strict rigidity aroximation adoted determines that the results might e o use to study real shock events or which the all and the at are not areciale deormed. This is the case o imacts occurring at small velocities in aseall,

2 354 A. CABO, L. SOLER, AND C. GONZÁLEZ or slow direct normal relative velocities at the contact oints etween the at and the all, with ossile higher values o the relative tangential velocity. Such a study is done in the last section o the work. In the case o illiard alls the rigidity aroximation seems to e oeyed or most o the shocks occurring in normal laying. The initial conditions or the shock in this work also generalize the ones emloyed in Re. 11 and 14, y including aritrary starting data or the center o mass and angular velocities. In addition the values o the normal and tangential coeicients o restitution are redicted. The results are also alied to descrie the exerimental measures o the slow motion scattering o a all y a at resented in Re. 11. For ookkeeing uroses, the simler exlicit solution o the conservative and riction less imact is also resented in an aendix. The discussion starts y considering the case in which sliding rictional orces in the contact oints develos in the assumed strictly rigid at and all. As mentioned aove, the situation generalizes the one studied in Res. 11 and 14, y removing their two main assumtions: 1 The shock will considered as ully three dimensional with aritrary initial conditions, and the at orm is only restricted to e a solid o rotation showing a cylindrical symmetry axis, No model aout the nature o the normal imact orce will e adoted. Only the standard connection etween the sliding riction and the normal orce will e assumed. That is, the modulus o the sliding riction orce will e equal to the riction coeicient µ times the magnitude o the instantaneous normal orce at the contact oint. As usual, the riction orce over one o the odies will e directed in the oosite sense to the tangent comonent or the relative velocity o the contact oint on that oject with resect to the contact oint on the other ody. In addition, a criterium is ound or deciding aout whether the ending state o the shock corresonds to sliding tangent suraces, or to a ure rotation state. The ure rotation state will e called that one in which the tangent suraces end the shocking rocess y showing null tangential relative velocity. Analytic and integral exressions o the ending values o the center o mass and angular velocities are given, in terms o the solution o a simle dierential equation or the two comonents o the relative velocities etween the tangent oints. It is an interesting outcome that the imulse done y the normal orce o the at on the all can e emloyed in lace o the time in descriing the dynamical evolution during the shock. This roerty was noted in the work [14]. The solution or the ending velocities o the all and the at resents two kinds o ehavior: In one case, the riction orce is unale to reduce the tangential comonent o the relative velocity to zero during the small time interval o the shock. In this otion, the two odies end the imact with a remaining sliding etween their contact suraces. In the alternative ending state, the rictional orce ecomes ale in reducing to zero the relative velocity o the two contact oints etween the odies at an intermediate instant o the shocking interval. In general, at this time instant t r in which the sliding etween suraces ends, the normal orce is not yet vanishing. This means that a inite ortion o the initial mechanical energy o the system is yet stored in the orm o elastic deormation energy. Thereore, ater the attaining o the ure rotation state, the system, which is yet within the shocking rocess, evolves in an alternative way than in the revious sliding eriod. In this second interval, the evolution ecomes conservative, and the equations are similar ut not identical to the ones corresonding to the riction less conservative rolem solved in the Aendix A. Their dierence rests in that within this eriod, a static kind o rictional orce can contriute to the conservative mutual imulses etween the odies. As it was remarked eore, in order to make the solution alicale to the realistic cases in which there exist areciale energy losses due heat, deormations, sound, etc., in addition to the rictional one, the solution o the rolem allowing only rictional dissiation is here generalized to include alternative energy losses. The generalization was suggested and heled y two actors: a The ossiility o roerly identiying the amount o stored elastic energy in the system at any moment when the energy losses are urely due to riction; The helul technical act that due to the assumed extreme rigidity aroximation, the exact evolution o the system, no matter the nature o the energy losses, only deends on one single variale: the net imulse It transmitted u to a given instant t y the normal orce exerted y the at on the all [14]. It should e underlined, that the ound solution or this general case o dissiation, can e considered as an exact one, when one considers as a known quantity the dissiated mechanical energy at the end o the shock y examle y measuring the total energy at the end o the imacts in reeated exeriments. Since all the comonents or the relative normal and tangential velocities etween the contact oints are redicted or this general solution, it ollows that the values o imortant henomenological quantities as the tangential and normal coeicients o restitution are also redicted, assumed that the total energy loss is known. In the case in that the riction is the only source o dissiation, the energy loss is also ollowing and is not needed to e measured. The generalized analysis is then alied to the descrition o the exeriments on the scattering o a all y a at reorted in Re. 11. Ater henomenologically ixing a single exerimentally measured quantity: the value o the center o mass velocity o the all at zero values o the imact arameter and its initial angular velocity, the calculated results urnish a satisactory descrition o the reorted measurements. In articular, the curves or the ending velocity o the all as unctions o the scattering angle coincided with the measured ones within the range o the disersion o the data, or each o the three values o the initial angular velocity o the all emloyed in the exeriences. Rev. Mex. Fis

3 SCATTERING OF A BALL BY A BAT IN THE LIMIT OF SMALL DEFORMATION 355 The exosition o the work roceeds as ollows. In Sec., the starting equations or the shock rolem are resented and the notation and asic deinitions are given. Aterwards, the solution o the imact rolem or the case in which the ending state is assumed to corresond to sliding contact suraces is exosed. The Sec. 3 continues y resenting the solution or the situation in which the contact oint o the all and the at inish the shock rocess in the ure rotation state. Section 4 exoses how the solution or the case in which the losses are only due to riction can e readily generalized or the more general situation including additional tyes o energy dissiation. Finally, in Sec. 5 the results o revious sections are alied to solve the scattering rolem o the all y the at in the articular coniguration considered in the exeriments reorted in Re. 11. The results or the descrition o the measurement reorted in that reerence are resented. Finally, in Sec. 5 a summary o the results is given.. Only rictional dissiative imact: the sliding inal state This section will exose some asic considerations and deinitions which will e o use along the resentation. The Fig. 1, illustrates the shock rocess etween the all and the at in the recise instant at which they ecome in contact. The adoted laoratory system o reerence to e named as the La system in what ollows, will e situated on the center o mass o the at and having its z x 3 coordinate axis eing collinear with the cylindrical symmetry axis o the at. The three unitary vectors o the La reerence rame axes will e {i, j, k}. In what ollows, old letters will indicate vectors. The unit vector k along the z axis, will oint in the direction o the arrel o the at, and thus, the vectors i, j will e contained in a transversal section o the at as illustrated in Fig. 1, and they are chosen to orm a direct triad with k. The vector r c deicted in Fig. 1, deines the osition o the contact oint o the two odies in the aove deined reerence rame. Note that the adotion o the La rame does not restrict the generality o the discussion. The vector r cm gives the osition o the center o mass o the all in the La coordinate system. The set o three unit vectors {t 1, t, t 3 } are deined as ollows: t 3 is normal to the common tangent surace o oth odies at the contact oint, and is directed as ointing outside the volume o the at. Further, t can e deined as a tangential unit vector eing contained in a common lane with the unit vector k and having a ositive scalar roduct with it. Finally, t 1 is deined as eing orthogonal to t and t 3 y also orming with them a direct triad. Let us seciy now ew hysical assumtions that will e adoted or the solution o the rolem. Firstly, as it was already stated, it will e considered that the all and the at are ideally rigid odies. That is, the elastic orces are suosed as eing so strong, that the sacial orms o oth ojects remain almost the same during the whole shocking time inter- FIGURE 1. The igure illustrate the deined systems o reerence to e systematically emloyed along the work. The all and the at are shown in the moment at which the imact starts. val δt ch. Naturally, this lase will e suosed as eing extremely short. These assumtions will e relected in the discussion to ollow, in which the whole geometry o the arrangement will e suosed to e invariale during the time interval δt ch. The only quantities that will allowed to change are the linear and angular velocities o the two odies. Let us write the general equations o motions or the evolution o the center o mass linear momenta P, P and the angular momenta L, L o the all and the at, resectively, during the shocking interval δt ch. They can e written as d dt P t = Ft, 1 d dt P t = Ft, d dt L t = r c r cm Ft, 3 d dt L t = r c Ft, 4 in which Ft is the very raidly varying imulsive orce which is exerted y the at on the all. The deinitions o the momenta are P t = m v cm = m d dt r cmt, P t = m v cm = m d dt r cmt, L t = Î w, L t = Î w t + r cm t m. vcm t, 5 in which m and m are the masses o the ate and the all, resectively. The angular momenta o the at is deined with resect to the La reerence rame, and or the simlicity o Rev. Mex. Fis

4 356 A. CABO, L. SOLER, AND C. GONZÁLEZ the urther discussion the angular momenta o the all was deined with resect to its center o mass. Note that the angular imulse in Eq. 3 is consistent with this deinition. The inertia tensor o the all Î is the identity matrix and the one associated to the at Î has cylindrical symmetry in the La reerence system. The exlicit orms o the inertia tensor are m dv cm t = µ vt v t t 3 di, 8 I dw t = µdir c r cm vt 9 v t, Î = I Î = I I I 3, = I t I t I 3. 6 Note, that the vector r cm t is the osition o the center o mass o the at, which during the shocking interval δt ch remains to e very close to the origin o the La reerence rame deicted in Fig. 1, i the odies are suiciently rigid or the given initial relative velocities and angular momenta. That is, in the here adoted strict rigidity aroximation, in which the two odies are assumed e comletely invariale in orm and osition during the imact, it will assumed that r cm t = 0. In a irst instance, we will include energy dissiation only through the resence o sliding riction during the imact. The inclusion o additional sources o energy losses will e incororated in last sections. As stated in the introduction the usual laws o riction will e assumed. That is, we will consider that the modulus o the sliding riction vector will e the sliding riction coeicient µ, times the modulus o the normal orce etween the odies. Further, the direction o the riction orce over one o the two odies contact oint, will e oosite to the relative velocity o this oint with resect to the contact oint o the other ody. Let us note that, ecause the normal orce grows starting rom zero at the eginning o the shocking eriod, in general the irst stage o the shock rocess should corresond to a situation in which the contact oints o the all and the at slide etween them at the eginning o the rocess. Only in the articular situation in which the tangent velocity is already vanishing at the eginning o the imact, this eriod will not exist. In such a case the solution is directly given y the one resented in the next section. Then, consider the equations o motion as written or this initial rocess. For an instant t eing inside the very small time interval δt ch during which the odies are in contact, the equations can e written in the orm m dv cm t = µ vt v t + t 3 di, 7 d w t=µdiî 1 r c vt v t 1 I t dir c t 3, 10 where a new magnitude aearing is the dierential imulse di = N dt done y the normal orce N exerted y the all on the at. It allows to deine the net imulse done y this orce u to a given time t y It = t 0 N dt. 11 Another quantity aearing is the tangential comonent o the relative velocity v t etween the contact oint o the all and the corresonding contact oint o the at or which v t means its modulus. As deined aove, µ is the sliding riction coeicient and the inverse o the inertia moment tensor o the at Î 1 can e exlicitly written as ollows 1 Î 1 I 0 0 = 1 0 I I 3 1 I 0 0 = 1 0 I I + = 1 1 I δ + 1 k k, I 3 I I 3 1 I i = 1, 0, 0, j = 0, 1, 0 and k = 0, 0, 1, 1 where k k means the diadic tensor k k k i k j. Let us deine now a simliied notation or the tangential comonent v t t o the ull relative velocity v t etween the contact oints o the all the at as ollows v t t vt = v 1t t 1 + v t t, v vt = v 1 + v Then, v t can e written as ollows v t = v t + v t t 3 t 3 = v 1 tt 1 + v tt + v t 3 t 3 = v cm t v cm t + w t r c r cm w t r c, 14 which allows to write or the variation o v t in a time interval dt in the considering sliding interval the exression dv t = dv cm t dv cm t + dw t r c r cm d w t r c, Rev. Mex. Fis

5 SCATTERING OF A BALL BY A BAT IN THE LIMIT OF SMALL DEFORMATION 357 where, as descried eore, the geometry o the system has een assumed as invariant due to the erect rigidity o the all and the at. This assumtion can e satisied, in articular, i all the velocities are chosen to e scaled to suiciently small values roducing small enough deormations. Emloying the equations o motion allows to exress the aove variation in terms o the dierential imulse o the normal orces in the ollowing linear orm 1 dv t = µ + 1 vt 1 m m vt di t 3 di m m µ dir c r cm r c r cm vt I vt + µdiî 1 r c vt vt 1 I t di r c t Ater rojecting the aove relation over t 1 and t, the ollowing set o two dierential equations or the variation o the relative tangential velocity comonents as unctions o the imulse o the normal orce o the at on the all can e otained v 1 I dv 1 I = s 1 v1 I + v I di, v I dv I = s v1 I + v I di + s 0 di, 16 in which the arameters aearing deend on the system s roerties as ollows 1 s 1 = µ r c r cm m m I + r c t 1 r c k, 17 I t I 3 I t 1 s = µ r c r cm m m I + r c r c t, 18 I t I t s 0 = r c t 3 r c t. 19 At this oint it can e underlined that the time had disaeared orm the equations o motion or the tangent velocities. That is, the time deendence is all emodied in the time deendence o net imulse o the normal orce o the at on the all It deined y 11. This unctional relation etween the imulse I and the time t will e assumed in what ollows or studying the evolution o the system during the shock interval in terms the imulse I in lace o the time t. A very imortant consequence o the unique deendence o the evolution on the net imulse I, is the act the resence o any orm o dissiation in addition to riction will not aect the result o the evolution o all the roerties u to a given state with deinite value o I. Thereore, the only eect o the resence o such sulementary orms o dissiation will e to determine a dierent instants in which searation end o the imact will occur. This roerty will e emloyed in last sections to extend the solution to include non dissiation eects. The solution o the aove set o equations or the relative velocities, ater determining their initial values allows to determine the evolution with the imulse I o all these velocities. As it will e veriied in the articular examles solved in next sections, the general ehavior o the solutions is such that oth comonents evolve with I in a continuous way u to a critical value o the normal imulse, at which oth comonents simultaneously tend to vanish. This oint corresond to the attainment o the ure rotation state. However, whether this critical situation would e aroached or not deends on the dynamical equations o motion: they in act should determine whether or not ositive values o the normal orce done y the at on the all can exist u to the arriving to the ure rotation state. Veriying the aove remarks, the set o equations o motion 7-10 during the imact rocess can e written in terms o the most aroriate evolution arameter I as ollows m dv cm I di m dv cm I di I dw I di d w I di = = µ vi vi + t 3 µ vi vi t 3, 0, 1 = µ r c r cm vi vi, = µ Î 1 r c vi vi 1 I t r c t 3. 3 These equation clearly evidence that the evaluation o all the velocities is only determined y the value o the imulse o the normal orce I eing exerted y the at on the all. This imulse is deined y Eq. 11. Since the solution o the dierential Eqs. 16 determines the tangential velocity as a unction o I, the Eqs. 0-3 can e integrated to ind out all the velocities in terms o I as ollows v cm I = v cm m I r I + It 3, 4 v cm I = v cm m I r I It 3, 5 w I = w 0 1 I r c r cm I r I, 6 w I = w 0 + Î 1 r c I r I 1 I t Ir c t 3, 7 where the imulse done y the tangential rictional orce is deined as a unction o I y the ollowing integral I I r I = µ 0 di vi vi. 8 Rev. Mex. Fis

6 358 A. CABO, L. SOLER, AND C. GONZÁLEZ For a coming reerence to them, let us deine now the increments in the velocities with resect to their initial values v cm, in v in cm, win and w in when the imulse rises to a deinite value It y: v cm I = v cm I v cm 0, v cm I = v cm I v cm 0, w I = w I w 0, w I = w I w 0, 9 v cm 0 = v in cm, v cm 0 = v in cm, w 0 = w in, w 0 = w in. 30 The aove ormulae indicate that the shock rolem in this assumed initial sliding rocess, will ecome solved, ater inding a condition determining the inal value o the tangent velocity comonent at the moment o searation o the odies. Let us consider this oint in what ollows. In the assumed in this section case, in which the shock is inalized when the contact oints are yet sliding etween them, the osed equations remain eing valid along all the shock interval. In this situation, the aroriate condition or ixing the ending value o the imulse I is that total dissiative work W done due to riction or non elastic rocesses u to the value o the imulse I, should e equal to the decrease o the total kinetic energy E kin along the evolution u to the same value o I. The hysical reason or this condition is that when the normal orce vanishes, which deines the searation o the at and the all, the conservation o energy imlies that all the non already dissiated art o the mechanical energy should aear in the orm o the translational and rotational kinetic energies o the at and the all. Thereore, since as it has een concluded, the evolution as a unction o I is comletely indeendent o the nature o the dissiation, it ollows that the solution o the rolem in this eriod is only deending o the raction o the initial kinetic energy o the two odies which ecomes dissiated in the shocking rocess, a quantity which only will determine the value o the imulse transmitted at searation I out. This condition leads to the equation or I out W I out = E kin I out. 31 The art o the total dissiative work W which is done y the riction u to the value o the time t, or which the imulse has the value It can e calculated to e t W r It = 0 = µ µn v dt It 0 vi di. 3 In what ollows, we will also consider also the existence o additional sources o dissiation, as the one associated to the non comlete elastic ehavior o the all and the at. Then, the total dissiative work W I will e written as W I = W r I + W add I, 33 where W add I reresents the amount o mechanical energy dissiated in the system u to the instant in which the normal imulse takes the value I due to mechanisms additional to the rictional one. It can e understood that the deendence on I o W add I will deend on the concrete orms o the dissiation rocess acting in the at at the all. However, it is a remarkale roerty that, assumed that i the system ecomes ale to arrive to the ure rotation state, the velocities at this oint result to e in the here considered erect rigidity situation and only at this articular instant comletely indeendent o the existence o non rictional kinds o dissiation. The articular cases o the exerimental results to e considered in next sections elong to the situation in which ure rotation is estalished. The exlicit orm o the condition 31 determining the searation oint under sliding regime in case that it eectively occurs is comleted ater deining the ormula or the increase in the total kinetic energy as a unction o I. It ollows ater exressing the values o the inal lineal and angular velocities in terms o their initial values lus their increment. Making use o the Eqs. 9 this quantity has the exression E kin I = m + m + I + 1 v cm in v cm I + v cm I v in cm v cmi + v cm I w in w I + w I w in I. w I + w I I w I, 34 in which all the increments in the linear and angular velocities are given y the ormulae urnished y the integrated equations o motion 4-7 and the solution o the dierential equations or the tangent velocity 16. Thereore, the ull solution o the rolem in the assumed situation in which along all the shock rocess the contact oints o the all and at slide etween them, can e otained ater inding the value o I out making equal to zero the unction SI out = W I out E kin I out. 35 It should e remarked that or the case in which all the dissiation is determined y riction, the rolem is comletely solved ecause the evaluation o the work o riction as a unction o I out is comletely deined y 3 in terms o the ound solutions o the tangential velocities as unctions o the imulse I. The case o additional sources o losses, Rev. Mex. Fis

7 SCATTERING OF A BALL BY A BAT IN THE LIMIT OF SMALL DEFORMATION 359 or to e analogously solved needs or a deinition o the seciic mechanism leading to dissiation. A model or taking in consideration such a mechanism will e constructed in the coming Sec. 4 in order to aly the results to the descrition o realistic exerimental measures given in Re. 11. This comletes the inding o the solution or the evolution o the imact rocess in this irstly assumed situation. It can e helul to recall that all the shocking events in which the contact oints slide at the eginning o the imact, start evolving as guided y this just considered sliding case. In the case that no solution exists or the aove equation, the situation should corresond to a rolem in which the work which is done y the riction along the whole initial sliding eriod is not ale to ecome equal to the decrease in the kinetic energy o the at and the all. Let us consider the inding o the solution in this second regime. 3. Only rictional dissiative imact: the ure rotation inal state As ollowed rom the revious section, we will now consider the situation in which the maximal work that can e done y the riction u to the oint in which the sliding etween the contact oints vanish, is not ale to dissiate the mechanical energy down to the value needed to coincide with the total kinetic energy at this same sliding state. In other words, at the instant t r or the corresonding imulse o the at on the all I r at which sliding stos, and the ure rotation state is attained, a ortion o the total initial energy should e yet stored in the orm o deormation energy. In addition, the resence o deormation energies orces is reresented in the considered rolem y a non vanishing normal orce. Thus, the shock rocess is divided in two arts, each one eing governed y dierent dynamical equations. The irst one was discussed in the ast section, in which sliding occurs and stos at the imulse value I r at which the slice etween the contact oints o the all and at ends. At this value o the imulse the increments in the velocities are given y the ormulae 4-7, in which the tangential sliding velocity vanishes. These linear and angular velocity increments have the exlicit exressions v cm I r = v cm m I r I r + I r t 3, 36 v cm I r = v cm m I r I r I r t 3, 37 w I r = w 0 1 I r c r cm I r I r, 38 w I r = w 0 + Î 1 r c I r I r 1 I t I r r c t 3, 39 These quantities ully determine the linear and angular velocities o the all and the at in the next intermediate ure rotation state in which a ortion o the total mechanical energy is yet stored in the orm o elastic deormation energy. The value o the total kinetic energy at this oint E r can e calculated through the ormula E r = m v cmi r + m v cmi r + I w I r + 1 w I r I w I r. 40 As remarked eore, the existence o deormation energy at the transmitted imulse I r is reresented y the existence o a yet not vanishing normal orce, which should tend to zero in the new stage o evolution. In this second eriod o the imact rocess, the stored elastic energy transorms in a contriution to the kinetic energy o the all and the at at the real ending state o the shock, in which the normal orce tends to vanish. Deormation losses can occurs also in this rocess. Thereore, ater the instant at which the imulse is I r, the system will e governed y a similar, ut not identical, set o equations valid or the conservative shock studied in aendix A. The dierence is related with the act that during this last interval, a static rictional orce can e dynamically required to remain acting. This ossiility was excluded in the case o the asence o riction o the conservative case, ut here it can occurs due to the ossile existence o a static riction. Then, the increments o the center o mass velocities and angular velocities u to the value o the imulse I at any moment t within this inal ure rotation eriod can e written in the orm m v r cm I = I t 3 + I r, 41 m v r cm I = I t 3 I r, 4 I w r = r c r I r, 43 Î w r I = r c t 3 I r c I r, 44 I r im = Ir 1 t 1 + I r t. 45 The arameters I, I r 1 and I r are the imulses o the normal and rictional orces roduced y the at on the all starting rom the time t r u to the instant t, and are deined y the integrals I t = t t I r 1 t = t I r t = t r N t dt, 46 t r r t dt, 47 t r r t dt. 48 Rev. Mex. Fis

8 360 A. CABO, L. SOLER, AND C. GONZÁLEZ A comlete set equations or determining the evolution o all the quantities can now e deined, ater assuming that we are eectively in the ure rotation regime occurring eore the ending o the shock. The additional conditions or comletely ixing the solution o the equations o motion are asically the vanishing o the two tangential comonents o the relative velocities o the contact oints at any moment during this ending rocess. This condition or the relative velocity at the tangential oint can e written as ollows 0 = v r cm I t i v r cm I t i + w r I r c r cm t i w r I r c t i, 49 i = 1,. In these two equations, all the increments in the velocities can e sustituted in terms o the just deined three values o the imulses. They in turns can e solved or the two values o the rictional imulses in terms o the unique values o the normal orce imulse as ollows I r i t = I S ij v j, 50 j=1, S ij = t i t 1 t 1 t j D 1 D 1 = 1 m + 1 m + r c r + t i t t t j D, 51 I + r c k r c, 5 I t I 3 I t D = 1 m + 1 m + r c r I + r c I t 1 I t r c t. 53 Thereore, all the velocity increments in can e exressed as linear unctions o the imulse done y the normal orce o the at on the all I. Note that this imulse is deined as the one transmitted ater the instant in which the system arrives to the ure rotation state u to any moment t. In exlicit orm the values o the velocities as unctions o I are deined y Eqs in the orm v r cm I = v r cm 0 + I m v r cm I = v r cm 0 I m t 3 + w r I = w r 0 t 3 +, j=1,, j=1, + I r c r I w r I = w r 0 I Î 1 r c t 3 +r c S ij v j t i, 54 S ij v j t i, 55, j=1,, j=1, S ij v j t i, 56 S ij v j t i. 57 In terms o these velocities, the total amount o kinetic energy E I at a ixed value o the imulse I is deined y the ormula E I = E r + m + I w I r w r v cm I r v cm r I + v cm r I + m I + w r I + 1 w I r Î w r v cm I r v r cm I + v r cm I I + w r I Î w r I. 58 This comletes the solution or the evolution equations or the second rocess in which ure rotation occurs. In order to ully deine the solution it only rests to determine the searation oint. The condition or determining it is resented in the next susection or the case in which dissiation is only roduced y riction. However, it needs or a clear deinition aout the concrete mechanisms o energy dissiation in the system. However, as it was already remarked, one helul outcome is that, under the assumed rigidity assumtions, no matter the orm o the existing dissiation mechanisms, their dierences only can alter a single arameter o the rolem: the articular value o the total imulse done y the normal orce o the at on the all at the just end o the whole imact. This strong roerty will e used in next section to deine a general solution o the rolem in which additional sources o dissiation exist A criterium or determining the shock case rom the initial conditions when dissiation is only rictional Let us consider in this susection the condition allowing to determine in advance which kind o state will show the all and the at at the end o the shock event, y only knowing the initial conditions. This comletes the determination o the solution or the case in which dissiation is only imle- Rev. Mex. Fis

9 SCATTERING OF A BALL BY A BAT IN THE LIMIT OF SMALL DEFORMATION 361 mented y the riction and the odies attain the ure rotation state. It can e noticed that under the assumtion o existence o additional non rictional sources o dissiation, such a criterium is exected to deend on the seciic mechanism to e considered. However, assuming that we know the total amount o energy dissiated in the shock, the solution o the mechanical rolem will e also ound in the next section. In this case o the sole resence o rictional losses, the criterium is directly given y the sign o the quantity C v in cm, v in cm, win =E in v in cm, v in, w in cm, win, w in +W r I r E r, 59 where W 0 and E r are imlicit unctions o the initial velocities v cm, in v in cm, win, w in, which were otained in the course o the revious discussion. I the sign o C is ositive, then the total energy ater the at and the all arrives to the ure rotation state E in v cm, in v in cm, win, w in + W r I r is larger than the kinetic energy o the two odies in this same state E r. Thereore, at this moment the system has energy stored in the orm elastic deormations, and a non vanishing normal orce should remain existing. That is, the shock is not yet ended and the second kind o the solution should e considered. In another hand, i C is negative, it indicates that the total energy o the system ater attaining ure rotation results to e smaller than the kinetic energy in the same ure rotation state. This means either, that energy is not eing conserved in the rocess, or that the system could not in act attains the ure rotation state, and the shock ended with sliding contact oints. In this case, the kind o solution to emloy should e the one discussed in Sec.. The case C = 0 indicates that the shock ends recisely at the moment in which the system arrives to the ure rotation state. 4. General dissiative imact solution In this short section we descrie how the solution ound in revious sections lend the asis or a more general solution in which losses in addition to the rictional ones are resent in the rocess. Let us only assume that the new kind o losses do not aects the rules governing the relations etween the normal imact orce and the tangential rictional one. Then, it can e oserved that along the raid mechanical evolution during the imact rocess, all the equations o motions or the two odies are exactly the same at any instant at which the odies are not yet searated. That is, the mechanical state o the all and the at at any moment within the shock time interval, is ully determined y the total amount o linear momentum which had een transmitted u to this moment y the normal orce o the all on the at. Thereore, under the adoter assumtions o strict rigidity and validity o the standard rules or riction, all the mechanical quantities o the rolem coincide with the ones in the case o the sole resence o rictional dissiation, at any given instant eore the searation etween the odies. Then, let us consider the exact searation instant. It is clear that the condition o searation will e that the at this recise moment, the already known rictional dissiation energy, lus the additional energy lost y heat, sound, lastic deormations,... should e equal to the dierence etween the initial value o the total mechanical energy at the eginning o the shock and the inal value o this same quantity at the searation instant. Thereore, i we consider the amount o mechanical energy dissiated during the shock certainly a measurale quantity as a known value, then the ull solution o the imact rolem is otained. That is, all the mechanical roerties are determined at all times. Since all the tangential and normal to the contact lane velocities can e calculated, it also ollows that results redicts the tangential and normal coeicients o restitution or the class o shocks studied. This analysis will e emloyed in the next section to descrie exerimental data o the slow motion imacts o a all and a at. Beore, in the next susection, let us resent the exlicit searation conditions The condition or determining the searation oint Let us discuss now the exlicit condition to e imosed or determining the moment in which the all and the at start to searate. It can e constructed in terms o the energy alance in the system. The searation oint can e deined as that one at which the initial total mechanical energy minus all the mechanical energy lost u to this oint, just ecomes equal to the total kinetic energy o the all and the at. In exlicit terms, it can e written in the orm E in + W r I r + W add I r + I out = E I out 60 where the total dissiative work o the riction W r is given y the general exression 3 ater sustituting I = I r W r I r = µ I r 0 v di, 61 and the losses due to other sources o dissiation in addition to riction u the total value o the imulse done y the normal orce at the searation oint I r + I out, is the term W add I r + I out. The right hand side o the equation is the total kinetic energy o the all and the at at the searation instant. That is, the total kinetic energy at the oint in which ure rotation is estalished lus the increment due to the changes in all the velocities during the ure rotation evolution u to the searation oint. This kinetic energy is deined y Eq. 58 as evaluated at the imulse done y the normal orces at the searation oint. As it was remarked at the start o this section, without seciying the nature o the additional sources o dissiation Rev. Mex. Fis

10 36 A. CABO, L. SOLER, AND C. GONZÁLEZ in addition to the rictional one, is not ossile to deine the searation oint assumed that we only know the initial data. However, i we consider as a known quantity the total mechanical energy ater the shock or alternatively, the amount o energy lost in the imact, the rolem can e considered as solved in this section. Let us now consider again the simler case in the only energy dissiating source is the riction. Then the additional losses term W add I r + I vanishes. In this case relation 60 ecomes a simle quadratic equation or I out ater sustituting the exressions or the velocity increments, which all are homogeneous linear unctions o I out. In Eq. 60 all the quantities E in, W r 0, E r, v cm I r, v cm t r, w I r and w I r are already known rom the solutions just otained in the revious stage o the shock. Henceorth, the comlete analytic solution o the shock rolem ollows in the considered case in which ure rotation is attained and riction is the only source o energy losses. The alication o the analysis to descrie exeriments in which additional sources o losses add to riction will e discussed in next section. 5. Descrition o exerimental measures In this section we will consider the alication o the results resented eore to the descrition o the measures relative to the scattering o a all y a at in Re. 11. The exeriment consisted in droing a ree alling all at certain height which determines a vertical velocity o 4.0 m/sec at the instant in which it shocks with an horizontally oriented static at. Then, a high video recording o the ree all o the all allowed the authors to determine all the kinematic arameters eore and ater the imact. Exeriments were done, in which the all was itched a numer o times with ixed values o the angular velocities along the symmetry axis o the at. Exeriences or three values o the angular velocities w o = 79, 0, 7 rad/sec were done. The at has a arrel diameter o 6.67 cm, a length o 84 cm and a mass o kg. The center o mass o the at is situated at 6.5 cm o its arrel end. The all or which measures were done landed on the at at distances along the at axis ranging etween cm rom the arrel end. Then, we will descrie the shocks y assuming that the all imacted the at at an axial distance o 15 cm rom the arrel end. The arameters o the at and the all considered in that work determine the ollowing values or the magnitudes deined in the revious sections m = kg, m = kg, r = r c r cm = m r = m I = / m kg m, I t = kg m, I 3 = kg m, µ = The set o unit vectors sitting at the tangential oint o the odies were chosen in the ollowing way t 1 = cosθ j sinθ i, t = k, t 3 = sinθ j + cosθ i, r c = k sinθ j + cosθ i, 63 which relect the act that the arrel o the at has een ixed as a cylinder o radius cm and that the center o mass o the at has a minimum distance o m rom the lane eing transversal to the at axis and contains the contact oint. The angle θ is the one ormed etween a radius traced rom the axis o the at to the contact oint. For the exerimental arrangement, the initial velocities o the all and the at just an instant eore the imact ecome v in cm = 4, 0, 0, v in cm = 0, 0, 0, w in = 0, 0, w o, w in = 0, 0, The solution o the shock rolem or nearly vanishing imact arameter In starting, let us exemliy the solution o the imact rolem or the situation in which the vertically alling all makes contacts with the horizontally oriented at at a very small value o the imact arameter. That is, when the vertical line o alling asses very close to the at symmetry axis. For concreteness let us suose that the imact occurs at the small value o the angle θ = π/400. This coniguration will serve two uroses o the resentation. In irst lace it will illustrate the alication o the ormal solutions ound in revious sections to a concrete shock rocess. In second hand this articular solution or scattering at zero imact arameter case will serve or henomenologically constructing a descrition o the exerimental data resented in Re. 11. Firstly, consider the solution o the Eqs. 16 or the evolution or the tangential velocities in the irstly occurring sliding eriod. The evaluation o the arameters s 1, s and s o deined in relations leads to the exlicit orm o the equations dv 1 I di dv I di v 1 I = 13.98, 65 v 1 I + v I v I = , 66 v 1 I + v I Rev. Mex. Fis

11 SCATTERING OF A BALL BY A BAT IN THE LIMIT OF SMALL DEFORMATION 363 v 1 0 = meter sec, 67 v 0 = The emloyed initial values o the comonents o the tangent velocities v 1 0 and v 0 along the vectors t 1 and t, resectively were determined y rojecting the relative velocity at the just eginning o the imact, which is deined y Eq. 14, on each o these vectors. Note that the starting value o the tangent velocity is small due to the assumtions o vanishing angular velocity o the all in common with the very small selection o the imact arameter. The solutions o these equations or v 1 I, v I and modulus o the tangent velocity vi = v1 I + v I are deicted in the Figs. -4. Note that the v comonent is very small, although non vanishing, which is consistent with the act that the shock is not strictly two dimensional, ecause the center o mass o the at is out o the lane which is orthogonal to the symmetry axis o the at and asses through the center o mass o the all. The v comonent, although initially vanishing, develos values which grow u to a maximal one or tending to zero again. On another hand the v 1 comonent o velocity start decreasing rom the start to vanish exactly at the same value o the imulse I, or which the v comonent also ecomes equal to zero. Thereore, the system o equations redict that oth comonents simultaneously tend to aroach a vanishing value. This roerty is exhiited y all the solutions o the scattering rolem ound in this work to descrie the exerimental results in Re. 11. The Fig. 4 clearly illustrates the vanishing o the modulus o the tangent velocity. From the Figs. -4 it can e seen that the value o the imulse I r o the normal orce on the all or which the system arrives to ure rotation or this secial scattering coniguration is FIGURE 3. The igure shows the evolution with the imulse I o the comonent v I o the relative velocity etween the contact oints on the at and the all. It was evaluated y solving the corresonding dierential equations. FIGURE 4. The igure shows the evolution with the imulse I o the modulus vi o the tangent relative velocity etween the contact oints in the at and the all. It was also evaluated y solving the corresonding dierential equations. FIGURE. The igure shows the evolution with the imulse I o the comonent v 1I o the tangent relative velocity etween the contact oints on the at and the all. It was evaluated y solving the corresonding dierential equations. I r= kg meter. 69 sec Having ound the evolution o the tangent velocities with the variation o the imulse o the normal orces I, we ecome ale to check whether the shock rocess will end in ure rotation state or in a sliding condition etween the contact oints o the at and the all. For this urose let us evaluate the relation 35 y sustituting the aove deined data or the velocities valid or the exeriment and evaluating W r I r and E r through their resective ormulae 61 and 40. The evolution o the our velocities o the all and the at rom the starting o the shock u to the moment in which the imulse at which ure rotation could e attained, Rev. Mex. Fis

12 364 A. CABO, L. SOLER, AND C. GONZÁLEZ was evaluated rom the ormulae 4-7 ater determining the imulse o the riction rom its deinition 8. The evaluation results in a ositive value or the C unction C v cm, in v in cm, win = E in v in cm, v in, w in cm, win, w in + W r I r E r = Joules, 70 E in v cm, in v in cm, win, w in = 1.16, 71 E r = Joules, 7 W r I r = Joules. 73 Thereore, assumed that the only source o energy losses is the riction dissiation, since C results to e ositive at the instant in which the transmitted imulse is I r or which ure rotation is attained, the total mechanical energy is larger that the kinetic energy E r o the two odies. In addition the values o the dissiated energy y the riction W r I r results to e very small when ure rotation is estalished. Since the kinetic energy E r at ure rotation has a value close to the total initial mechanical energy, it is clear that the ure rotation state is attained just at the very eginning o the imact. This is comatile with the very small imulse I r see 69 transmitted y the normal orce in arriving to vanishing sliding o the contact suraces. Having deined that the scattering situation corresonds to a ure rotation inal state, let us consider now the evolution o the system ater the imulse done y the normal orce continues to e growing under the ure rotation state. The change o all the velocities o the at and the all in this new rocess are determined as simle linear unctions o the imulse I y equations Let us consider irst the case in which only riction is ale to dissiate energy. Then the condition or the searation o the all and the at 60 gives a simle quadratic equation or the determination o the value o I at which the two odies searate. The condition or searation and its exlicit orm are E in + W r I r =E I =1.151+I I, 75 which give or the value o the imulse o the normal orce during the ure rotation interval u to the searation oint, assumed that the only existing energy losses are associated to riction, the result I out,1 = kg meter. 76 sec This value comared with I r = evidences that the ure rotation state was directly estalished at the eginning o the imact. FIGURE 5. The lot o the total kinetic energy o the at and the all E as a unction o the imulse o the normal orce o the at on the all I in the ending ure rotation rocess. The horizontal line deines the conserved value o the total mechanical energy ater the ure rotation state is estalished. The dierence etween the horizontal line and E i gives the amount o energy stored in the orm o elastic deormation at any value o the imulse I. The all deicted on the horizontal indicates the searation oint etween the all and the at when the only source o dissiation is the riction. The similar all laying on the curve o E indicates the searation oint to e deined y the model here constructed or evaluating the non rictional energy losses. The determination o I inishes the solution o the shock rolem in this case, since sustituting this value in the exressions determines all the center o mass and angular velocities o the at and the all at the searation instant Consideration o the losses due to inelastic rocesses Let us consider now the situation in which there exist energy dissiation sources in addition to the riction. For this uroses consider Fig. 5 which illustrates the solution the just discussed solution. The araolic curve shows the deendence o the kinetic energy E as a unction o I. The horizontal line indicates the value o the conserved mechanical energy E in +W r I r in the considered ure rotation eriod. The igure shows how, as the system evolves rom the instant in which ure rotation was estalished, it accumulates energy in elastic orm as signaled y the dierence etween the kinetic energy E and the conserved mechanical energy E in + W r I r, u to a maximum value, which aterwards starts to decrease. This ehavior will e taken into account in what ollows to construct a model or the non elastic dissiation rocesses dierent rom rictional one. The asic urose will e to aly the analysis to the descrition o the measures o scattering o a all y a at given in Re. 11. Assume that we are already in the ure rotation state, as the ormer evaluations in this section had stated. Then, let us Rev. Mex. Fis