IN AN GOGA FAIFAN GILASHI BIYU BARKATAI, ANA SAMUN CIKAKKIYAR SIFFAR ƘWALLO
Kowa zai iya yin madubin na’urar hangen nesa da hannu. Ana kafa faifan gilashi ɗaya, wato “kayan aiki” (tool), a kan tebur, a ɗora na biyu, wato madubin da za a samu, a kansa, a saka ƙasa mai goge (abrasive grit) a tsakaninsu, sannan a tura faifan saman gaba da baya cikin gajerun bugun hannu. Ana zagaya teburin a hankali ana juya madubin a yayin da ake tafiya, ta yadda kowace hanya za ta samu nata rabo. Kowane bugu yana ɗan bambanta da wanda ya gabace shi. Sannan a koma ga ƙasa mai goge mafi laushi, a sake, a sake kuma.
Sakamakon yana da ban mamaki: fuskokin biyu suna kusantar juna da kansu har su zama siffofin ƙwallo masu dacewa da juna sosai, daidai har zuwa ’yan gommai na nanomita. Littattafan jagora suna bayyana hanyar dalla-dalla. Abin da ba su bayar ba, a cewar Emmanuel Schaan, daga SLAC National Accelerator Laboratory da Kavli Institute for Particle Astrophysics and Cosmology na Stanford University, shi ne bayani na zahiri daga tushen ka’idoji (first principles) na dalilin da ya sa kowace fuska ta farko ke ƙarewa a siffar ƙwallo.
Doka ɗaya ta kankarewa
Binciken Schaan ya fara ne daga dokar Preston (Preston’s law), wadda aka wallafa a 1927 don goge gilashin taga (plate glass): a kowane wuri, abu yana kankarewa da gudun da ke tafiya daidai-wa-daida (proportional) da matsin (pressure) da ke wurin da kuma saurin da fuskokin ke zamewa a kan juna. Idan aka ƙara samfurin taɓawa (contact model) mai sauƙi — yayin da fuska ɗaya ta fi fitowa idan aka kwatanta da ɗayar, haka ake ƙara danna ta — komai yana biyo baya daga lissafi (equation) ɗaya.
Sakamako na farko yana fitowa ne daga daidaito (symmetry) kaɗai. Fuskoki biyu suna daina sauyawa ne kawai idan sun yi kama da juna kuma suna kasancewa iri ɗaya bayan kowane motsi na bugun hannun. Bugun da ke zamewa ta hanyoyi biyu suna barin shimfiɗaɗɗun fuskoki (planes) kaɗai; bugun da ke juyawa a kewayen gatura (axes) biyu daban-daban suna barin siffofin ƙwallo kaɗai. Saboda ana tura madubin da aka goge da hannu kuma ana juya shi ta kowace hanya, shimfiɗaɗɗun fuskoki da siffofin ƙwallo ne kaɗai ƙarshe mai yiwuwa. Wannan gaskiya ne ga kowace dokar matsi wadda matsin ke ɓacewa a inda fuskokin suka dace da juna.
Agogo uku
Ko fuska za ta kai ga waɗannan ƙarshe a zahiri ya dogara ne da saurin da lahani (defects) daban-daban ke ɓacewa.
- Kumburin da suka fi bugun hannu ƙanƙanta ana daidaita su har su ɓace. Suna raguwa a tsarin exponential (exponentially) a kan lokacin da ake kira T_grind — ’yan gommai na mintuna zuwa awa ɗaya a aikace.
- Rashin dacewar faifan biyu yana ɓacewa da sauri ninki biyu, saboda dukkan fuskokin biyu suna kankarewa lokaci guda.
- Karkacewar da suka fi bugun hannu girma ba sa raguwa kawai: daidaita su a kan yankin da bugun hannun ke rufewa yana mayar da lissafin zuwa lissafin yaɗuwa (diffusion equation) — irin wanda ke bayyana wani abu da ke bazuwa yana daidaita kansa a tsawon lokaci. Manyan karkacewa suna bazuwa kuma suna shimfiɗewa a hankali. Ta ƙarshe da ke ɓacewa ta mamaye dukkan madubin, kuma ita ce ke kayyade lokacin zama siffar ƙwallo:
T_sphere = (32/π²) × (R_mirror / R_stroke)² × T_grind
Ga bugun hannu na “al’ada” (normal stroke) na gargajiya, wanda cikakkiyar tafiyarsa ta kai kashi ɗaya cikin uku na faɗin (diameter) madubin, wannan kusan ninki 30 na T_grind ne — kusan awanni goma.
Me ya sa shimfiɗaɗɗiyar fuska ke shan kaye
Nauyi (gravity) yana karya daidaiton da ke tsakanin faifan biyu. Duk lokacin da faifan sama ya wuce gefen na ƙasa, matsin yana sake rarrabuwa: ana fi goge faifan sama a tsakiyarsa, na ƙasa kuwa a gefunansa. Na saman yana zurfafa, na ƙasan yana kumbura, don haka shimfiɗaɗɗun fuskoki ba su da tabbas (unstable). Siffofin ƙwallo suna tsira muddin suna samuwa da sauri fiye da yadda nauyi ke karkatar da su, wanda haka yake ga bugun hannu masu isasshen gajarta: a lokacin fuskokin biyu suna ci gaba da zama siffofin ƙwallo masu dacewa da juna waɗanda lanƙwasarsu (curvature) ke ƙaruwa a hankali. Dogayen bugun hannu suna juya wannan tsari, kuma wannan ne ainihin dalilin da ya sa masu yin madubi ke farawa da dogon bugu na “chordal” don haƙa tsakiyar, sannan su koma ga bugun al’ada don su mayar da shi siffar ƙwallo.
Littafin bayanan wani madubi
An tantance girman adadin (orders of magnitude) da littafin bayanan aiki na madubin santimita 25 mallakar mai binciken, wanda aka goge da hannu cikin kusan awanni 50. Da hannaye biyu suna dannawa da kilopascal 1 zuwa 2 kuma suna zamewa da mita 0.1 zuwa 0.3 a kowace daƙiƙa, ƙasa mai goge mafi ƙaƙƙarfa tana cire ’yan micromita a kowane minti. Haƙa ramin milimita 3 a tsakiyar ya kamata ya ɗauki kusan awanni 10: littafin ya nuna awa 11 da minti 50. Mayar da shi siffar ƙwallo ya kamata ya ɗauki kusan 30 × T_grind, wato kuma kusan awanni 10: littafin ya nuna awa 12 da minti 50. Samfurin ma ya bayyana dalilin da ya sa kowane mataki na ƙasa mai goge mai laushi ya ɗauki kusan lokaci ɗaya, daga awa 2 da minti 15 zuwa awa 5 da minti 25: ƙasa mai laushi tana cire abu a hankali, amma kuma tana da layin lalacewa mara zurfi da za ta cire, kuma tasirin biyu suna soke juna.
Duwatsu da takardar goge (sandpaper)
Lissafi iri ɗaya suna shafar abubuwa biyu na yau da kullum. A cikin na’urar juya duwatsu (rock tumbler), duwatsu masu faɗin kusan santimita 1 suna gogayya a kan tafiye-tafiye na kusan milimita 1. Ƙananan kumburi suna ɓacewa cikin makonni, don haka duwatsun suna yin sheƙi; amma ma’aunin (R_rock/R_stroke)² na kusan 100 yana tura lokacin zama siffar ƙwallo zuwa shekaru. Na’urorin juyawa masu zagaye (rotary tumblers), masu dogon gogayya ta zamewa, suna zagaye gefuna fiye da masu girgiza (vibratory). Goge ƙaramin abu mai shimfiɗaɗɗiyar fuska da takardar goge shi ne ɗaya ƙarshen: bugun hannun ya fi abin tsayi, sauyin siffa nan take ne, kuma gefunan kullum suna zagayewa.
Samfurin bai haɗa da gogayya (friction), matsin hannaye na zahiri mara daidaito da kayan aikin pitch mai laushi da ake amfani da shi wajen goge na ƙarshe ba, kuma tantance shi an yi shi ne a girman adadi, ba ta hanyar gwaji na musamman ba. Takardar, wadda American Journal of Physics ta karɓa bisa sharaɗi, an gabatar da ita a matsayin misali na aji: daidaito mai tabbas da mara tabbas (stable and unstable equilibria), raguwar exponential da yaɗuwa, duk daga goga guntun gilashi biyu tare.