Pendulum: lokacin da hanzari na dabara

The inji tsarin da ta ƙunshi wani abu nufi (jiki), wanda aka rataye a weightless inextensible filament (ta taro ne negligible idan aka kwatanta da nauyin jiki) a cikin uniform gravitational filin, da ake kira ilmin lissafi Pendulum (wani sunan - da oscillator). Akwai sauran iri na'urorin. Maimakon a filament weightless sanda za a iya amfani da su. Pendulum iya fili bayyanãwa jigon da yawa ban sha'awa mamaki. Lokacin da kananan mawadãta vibrations na motsi ne ake kira masu jituwa.

Janar bayani game da inji tsarin

Da dabara na oscillation lokaci na Pendulum aka bred Dutch masanin kimiyya Huygens (1629-1695 gg.). Wannan zamani na Isaac Newton ya sosai m na inji tsarin. A 1656 da ya halitta na farko agogon tare da wani Pendulum inji. Su auna lokaci da matsananci daidaici ga waɗanda sau. Wannan sabuwar dabara ne mai babbar mataki a cikin ci gaban na jiki gwaje-gwajen da m ayyukan.

Idan Pendulum ne a wani ma'auni matsayin (rataye vertically), da karfi na nauyi za a adalci da yarn tashin hankali da karfi. Flat Pendulum on wani ba mike yarns ne a tsarin da biyu digiri na da 'yanci na sadarwa. Lokacin da canza kawai daya bangaren na canza halaye na dukan sassa. Alal misali, idan wani thread aka maye gurbinsu da sanda, to, wannan inji tsarin ne kawai 1 digiri na 'yanci. Abin da, sa'an nan, da kaddarorin a ilmin lissafi Pendulum? A wannan sauki tsarin, a ƙarƙashin rinjayar da wani lokaci-lokaci perturbation, hargitsi bayyana. A wannan yanayin, a lokacin da dakatar da ma'ana ba motsi, da kuma oscillates a Pendulum akwai wani sabon ma'auni matsayi. Idan m hawa da sauka har da saukar wannan inji tsarin zama barga matsayin "juye." Shi ma yana da da sunan da. Shi ne ake kira da Kapitza Pendulum.

A Properties daga cikin Pendulum

Pendulum yana da matukar ban sha'awa Properties. Dukkan su suna da goyan bayan sanannun jiki dokokin. A lokacin da oscillation na Pendulum wani dogara a kan daban-daban yanayi irin girman da siffar jiki, da nisa tsakanin batu na dakatar da cibiyar nauyi, nauyi rarraba game da wannan batu. Wannan shi ne dalilin da ya sa definition na jiki rataye lokaci ne quite kalubale. Shi ne sauƙin yin lissafi da tsawon wani sauki Pendulum, da dabara na wanda aka bai kasa. A sakamakon lura da wadannan alamu za a iya kafa a kan irin wannan inji tsarin:

• Idan, yayin da rike wannan tsawon na Pendulum, dakatar daga da dama lodi, da lokacin da oscillation samun wannan, ko da yake su nauyi zai bambanta ƙwarai. Saboda haka, lokacin da Pendulum ba ya dogara ne a kan nauyin da load.

• Idan tsarin farawa da samun koma baya a cikin Pendulum ba ma babban, amma da kusurwoyi mabambanta, shi zai canza tare da daidai wannan lokacin, amma a daban-daban amplitudes. Duk da yake sabawa daga tsakiyar balance ba ma manyan hawa da sauka a cikin irin zai zama kusa isa masu jituwa. A lokacin da irin wannan Pendulum ba ya dogara ne a kan vibrational mawadãta. Wannan dukiya na inji tsarin da aka kira isochronism (a Girkanci "chronos" - lokaci "Izosov" - daidaita).

A lokacin da wani sauki Pendulum

Wannan adadi wakiltar da na halitta lokaci na oscillation. Duk da hadaddun halitta, da aiwatar da kanta ne mai sauqi qwarai. Idan tsawon na yarn ilmin lissafi Pendulum L, da kuma gravitational hanzari g, wannan darajar ne daidai:

T = 2π√L / g

Kananan lokaci na halitta oscillations a wata hanya ba, ba ya dogara ne a kan taro na Pendulum da oscillation amplitude. A wannan yanayin, a matsayin ilmin lissafi Pendulum motsa tare da rage tsawon.

Oscillations na wani ilmin lissafi Pendulum

Ilmin Lissafi Pendulum oscillates, wanda za a iya bayyana ta a sauki bambanci lissafi:

x + ω2 zunubi x = 0,

inda x (t) - unknown aiki (wannan kwana na deflection daga cikin ƙananan matsayi na ma'auni a lokacin t, bayyana, a radians). ω - mai kyau m wanda aka ƙaddara daga sigogi na Pendulum (ω = √g / L, inda g - cikin hanzari da nauyi, da kuma L - da tsawon wani sauki Pendulum (dakatar).

Jadawalin kananan oscillations kusa ma'auni matsayin (masu jituwa lissafi) kamar haka:

x + ω2 zunubi x = 0

Oscillatory motsi na Pendulum

Pendulum, abin da ya sa kananan oscillations, motsi sinusoid. Biyu domin bambanci lissafi gana dukan bukatun da sigogi da irin wannan yunkuri. Don sanin hanyar kana bukatar ka saita gudun da kuma tsarawa, wanda daga baya m m constants:

x = A zunubi (θ ₀ + ωt),

inda θ ₀ - farko lokaci, A - mawadãta daga oscillation, ω - cyclic mita m daga lissafai na motsi.

Pendulum (dabara domin manyan amplitudes)

Wannan inji tsarin, yi su oscillations tare da babban amplitude, shi ne batun karin hadaddun dokokin tafiye-tafiye. su aka lasafta bisa ga dabara domin irin wannan Pendulum:

zunubi x / 2 = u * SN (ωt / u),

inda SN - ba tare da Jacobi, wanda for u <1 ne a lokaci-lokaci aiki, da kuma ga kananan u da shi ya yi daidai da sauki trigonometric ba tare da. The darajar u ne m da wadannan magana:

u = (ε + ω2) / 2ω2,

inda ε = E / mL2 (mL2 - makamashi na Pendulum).

Tabbatar da dalilin da maras mike oscillation lokaci na Pendulum da wadannan dabara:

T = 2π / Ω,

inda Ω = π / 2 * ω / 2k (u), K - elliptic integral, π - 3,14.

da Pendulum yunkuri na separatrix

Yana kira separatrix yanayin da tsauri tsarin, wanda mai biyu-girma lokaci sarari. Pendulum motsa a kan wani ba lokaci zuwa lokaci. A izuwa yanzu batu na lokaci shi saukad daga cikin matsananci babba matsayi a wajen wani sifili gudun, sa'an nan shi ne a hankali kamfata. Ya ƙarshe tsaya, ya dawo zuwa ta asali matsayi.

Idan mawadãta daga oscillation na Pendulum halarci lambar pi, aka ce cewa motsi a cikin lokaci jirgin sama ne kusa da separatrix. A wannan yanayin, a karkashin mataki na karamin lokaci-lokaci tuki da karfi na inji tsarin farfado m hali.

A taron na sauki Pendulum daga ma'auni matsayi da wani kwana Cp auku tangential karfi Fτ = -mg zunubi φ nauyi. "Debe" alamar nufin cewa tangential bangaren directed a gaban shugabanci daga shugabanci na sabawa daga cikin Pendulum. Lokacin da nufin via Pendulum hijirar x tare da wani madauwari baka da wani radius L ne daidai ta kusurwa kawar da φ = x / L. Na biyu dokar Isaaka Nyutona, tsara don tsinkaya daga cikin hanzari vector da ƙarfi ba da ake so darajar:

MG τ = Fτ = -mg zunubi x / L

Bisa ga wannan rabo ne, bayyananne cewa Pendulum ne maras mike tsarin, kamar yadda wani karfi da cewa o ƙarin tabbatar da dawo da ma'auni matsayi, ba ko da yaushe gwargwado ga kawar da x, da zunubi x / L.

Kawai a lokacin da ilmin lissafi Pendulum aikin kananan vibrations, shi ne masu jituwa oscillator. A wasu kalmomin, shi ya zama mai inji tsarin iya yin masu jituwa oscillations. Wannan kimantawa da yake da inganci domin kusan kusassari 15-20 °. Pendulum tare da manyan amplitudes ba jitu.

Newton ta doka ga kananan oscillations na wani Pendulum

Idan inji tsarin aikin kananan oscillations, 2nd Newton ta dokar za ta yi kama da wannan:

MG τ = Fτ = -m * g / L * x.

A wannan hasashe, za mu iya cewa da tangential hanzari da wani sauki Pendulum ne kunnen doki ta kawar da tare da alamar "debe". Wannan shi ne yanayin inda da tsarin zama masu jituwa oscillator. Module proportionality factor tsakanin hijirar da kuma cikin hanzari daidai da square daga cikin kusurwa mita:

ω02 = g / L; ω0 = √ g / L.

Wannan dabara nuna halitta mita na kananan oscillations na da irin wannan Pendulum. A wannan hasashe,

T = 2π / ω0 = 2π√ g / L.

Lissafi dangane da dokar da kiyayewa da makamashi

Kadarorin oscillating Pendulum ƙungiyoyi za a iya bayyana tare da taimakon dokokin kiyayewa da makamashi. Ya kamata a haifa tuna cewa da yuwuwar makamashi na da Pendulum a gravitational filin ne:

E = mgΔh = mgL (1 - cos α) = mgL2sin2 α / 2

Cikakken inji makamashi daidai sanadin motsi da kuma iyakar yuwuwar: Epmax = Ekmsx = E

Bayan ka rubuta dokar kiyayewa da makamashi, da shan wanda aka samu da hagu da dama tarnaƙi na lissafi:

Ep + Ek = const

Tun da wanda aka samu daga cikin constants ne daidai 0, sa'an nan (EP + Ek) '= 0. The wanda aka samu daga cikin Naira Miliyan Xari daidai da Naira Miliyan Xari da Kalam:

Ep '= (MG / L * x2 / 2)' = MG / 2L * 2x * x '= MG / L * v + Ek' = (mv2 / 2) = m / 2 (v2) '= m / 2 * 2v * v '= mv * α,

saboda haka:

MG / L * XV + mva = v (MG / L * x + m α) = 0.

Bisa ga karshe dabara, mun sami: α = - g / L * x.

Practical aikace-aikace na ilmin lissafi Pendulum

Hanzari daga free fall dabam tare da latitud, saboda da yawa daga cikin ɓawon burodi a kewayen duniya ba m. Inda kankara faruwa tare da wata babbar yawa, shi zai zama dan kadan mafi girma. Hanzari daga ilmin lissafi Pendulum ne sau da yawa amfani ga bincike. A ta taimako look for daban-daban ma'adanai. Kawai kirgawa yawan oscillations na wani Pendulum, yana yiwuwa a gano kwal ko tama a cikin hanjinsu na Duniya. Wannan shi ne saboda cewa wadannan albarkatun da yawa da kuma nauyi na fiye da kwance a ƙasa da sako-sako da kankara.

Ilmin Lissafi Pendulum amfani da irin wannan shahararren malamai kamar Socrates, Aristotle, Plato, Plutarch, Archimedes. Mutane da yawa su na da imani cewa inji tsarin iya shafar makomar da rai. Archimedes amfani da ilmin lissafi Pendulum tare da lissafin. Yau, da yawa occultists da psychics amfani da wannan inji tsarin ga aiwatar da ta annabce-annabce, ko search for bace mutane.

Shahararren Faransa falaki da masanin kimiyya, Flammarion domin su gudanar da bincike kuma yi amfani da wani ilmin lissafi Pendulum. Ya da'awar cewa tare da taimakon ya ya iya hango ko hasashen gano wani sabon duniya, da fitowan na Tunguska meteorite, da kuma sauran muhimman abubuwan da suka faru. A lokacin yakin duniya na biyu a Jamus (Berlin) yi aiki a matsayin na musamman institute na Pendulum. A zamanin yau, wannan bincike ne ba samuwa Munich Cibiyar Parapsychology. Aikinsa tare da Pendulum da ma'aikatan wannan ma'aikata da ake kira "radiesteziey".