Tāwēwē: wā me te whakatere o te tātai

Ko te pūnaha pūkaha e ngā o te ira rauemi (te tinana), i whakairia ki runga ki te kakawaea inextensible weightless (he hokiate tona papatipu whakaritea ki te taimaha o te tinana) i roto i te mara tō kākahu, ka karanga te tautau pāngarau (te ingoa i tetahi atu - te oscillator). He atu momo o pūrere. Engari o te tokotoko weightless kakawaea taea te whakamahi. Tāwēwē taea puta marama te uho o maha ari ngā. A, no te huaina wiri tīwera iti o tona nekehanga te hawarite.

ngā mōhiohio whānui e pā ana te pūnaha aunoa

I muia te tātai o te wā oscillation o te tāwēwē Huygens pūtaiao Tati (1629-1695 gg.). Tenei nāianei o Ihaka Newton i tino aroha o te pūnaha aunoa. I te 1656 i hanga e ia te tiaki tuatahi ki te tikanga tāwēwē. whanganga ratou te wa ki te hanga pū tino mo aua wa. Tenei tenenga ko he taahiraa nui i roto i te whanaketanga o ngā tinana me ngā mahi ā-.

Ki te ko te tāwēwē i roto i te tūranga taurite (iri poutū), te kaha o te mahara , ka ke mafamafatatau e te miro mānukanuka kaha. tautau Flat i runga i te ngangana kore-stretchable ko te pūnaha ki e rua nekehanga o te haere noa o kōrero. A, no te huri i tetahi noa wāhanga o te huri i te āhuatanga o ona wahi katoa. Hei tauira, ki te whakakapi i te miro e te tokotoko, ka tenei pūnaha aunoa ko 1 anake tohu o te haere noa. He aha, Na, nga āhuatanga o te tāwēwē pāngarau? I roto i tenei pūnaha ohie, i raro i te mana o te perturbation wā, puta hepohepo. I roto i taua take, ka kore e neke i te wāhi whakatārewatanga, a ka oscillates he tāwēwē i reira ko te tūranga taurite hou. Ki te rerekē tere ake me te ki raro i tenei pūnaha aunoa riro tūranga pūmau "ka huripokitia e." Kei reira hoki tona ingoa. Kei te karanga a te reira i te Kapitza tāwēwē.

Ko te āhuatanga o te tāwēwē

He āhuatanga tino ngā tāwēwē. E tautokona ana katoa o ratou e ture tinana pai-mohiotia. Ko te wā o te oscillation o te tāwēwē tetahi atu tei runga i ngā tūkunga pērā i te rahi me te āhua o te tinana, te tawhiti i waenganui i te wāhi o te whakatārewatanga, me te pokapū o te mahara, tohatoha taimaha ki te faatura ki tenei wāhi. Ko Ko te aha te mea tino wero te whakamāramatanga o te wā iri tinana. He nui māmā ki te tātai i te wā o te tāwēwē ohie, te tātai o nei homai te raro nei. Ka rite ki te hua o te mātakitaki enei tauira taea te whakaturia i runga i ngā pūnaha mīhini rite:

• Ki te, i te tiaki i te roa taua o te tautau, iri i te momo o pikaunga, te wā o te oscillation tiki te taua, ahakoa ka rerekē nui ratou taimaha. Nō reira, e kore e te wā o te tāwēwē whakawhirinaki i runga i te taimaha o te kawenga.

• Ki te tīmata te pūnaha ki te peka ke i roto i te tautau e kore he nui rawa, engari rerekē koki, ka kapewhiti reira ki te wā taua, engari i amplitudes rerekē. Ahakoa kore ine i te pokapū o te toenga he rerekē nui rawa i roto i to ratou puka ka kia tata nui hawarite. E kore e te wā o te tāwēwē taua whakawhirinaki i runga i te kaha tīwera vibrational. ka karanga tenei taonga o te pūnaha aunoa te isochronism (i roto i te Kariki "chronos" - wā "Izosov" - rite).

Ko te wā o te tāwēwē ohie

Tenei ahua tohu te wā tūturu o te oscillation. Ahakoa te waihanga matatini, te tukanga iho he tino māmā. Ki te te roa o te miro L tautau pāngarau, me te whakatere g tō, he rite tenei uara:

T = 2π√L / g

e kore e wā iti o oscillations tūturu i roto i kore ara whakawhirinaki i runga i te papatipu o te tāwēwē me te kaha tīwera oscillation. I roto i tenei take, kia rite ki te tāwēwē pāngarau neke ki te roa iti.

Kōpiupiutanga o te tāwēwē pāngarau ko

oscillates tautau Mathematical, e taea te whakaahuatia e te whārite pārōnaki ohie:

x + ω2 hara x = 0,

te wahi x (t) - mahi unknown (tenei koki o deflection i te tūranga o raro o taurite i te t wa, faaite i roto i tātoro); ω - he taimau pai whakaritea e te i te tawhā o te tāwēwē (ω = √g / L, te wahi g - te whakatere o te mahara, me te L - te roa o te tāwēwē ohie (whakatārewatanga).

Whārite oscillations iti e tata ana tūranga taurite (whārite hawarite) e whai ake:

x + ω2 hara x = 0

nekehanga Oscillatory o te tāwēwē

Tautau, e hanga oscillations iti, neke sinusoid. tutaki Tuarua whārite kia pārōnaki nga whakaritenga katoa me ngā tawhā o te kaupapa taua. Hei whakatau i te huarahi e hiahia ana koe ki te whakaturia te tere me ngā taunga, i whakaritea i muri taimau motuhake:

x = He hara (θ ₀ + ωt),

te wahi θ ₀ - wāhanga tuatahi, A - tīwera o oscillation, ω - auau huringa whakaritea i te whārite o te nekehanga.

Tāwēwē (tātai mō amplitudes nui)

Tenei pūnaha aunoa, te mahi ratou oscillations ki te kaha tīwera nui, nei hoki e ngawari ki te ture waka matatini ake. E tatau ratou rite ki te tātai mō te tāwēwē taua:

hara x / 2 = u * sn (ωt / u),

te wahi sn - Jacobi hara nei ki a u <1 Ko te mahi wā, me mo te u iti hāngai reira ki te aho pākoki ohie. whakaritea te uara o te u te e te kīanga e whai ake nei:

u = (ε + ω2) / 2ω2,

te wahi ε = E / mL2 (mL2 - pūngao o te tāwēwē).

Whakatau i te wā o te oscillation o te tāwēwē nonlinear i te tātai e whai ake nei:

T = 2π / Ω,

i reira Ω = π / 2 * ω / 2K (u), K - Ēripitiki wāhi, π - 3,14.

te kaupapa tautau o te separatrix

ka karanga te reira i separatrix poroāwhio o te pūnaha hihiri, i roto i nei te wāhi wā e rua-ahu. neke tāwēwē i runga i te kore-wā. I roto i te wāhi ore tawhiti o te wā makaka reira i te tūranga tino o runga ki te kore tere, a ka te āta reira te whiwhi. Ia te mutunga mutu, e hoki ana ki tona tūranga taketake.

Ki te whakatata te kaha tīwera o te oscillation o te tāwēwē te pi maha, kua mea i te reira e te motini i roto i te manureva wāhanga he tata ki te separatrix. I roto i tenei take, i raro i te mahi o te ope taraiwa wā iti o te pūnaha aunoa whakaaturia whanonga huru.

I roto i te hui o te tāwēwē ohie i te tūranga taurite ki te CP koki puta kaha tangential Fτ = mahara hara -mg φ. tikanga "Tango" tohu e te wāhanga tangential whakahaua i roto i te huarahi i te ritenga i te aronga o te ine o te tāwēwē. A, no te faahiti mā tautau displacement x haere te pewa porohita ki te L pūtoro he rite ki tona φ displacement koki = x / L. Ko te ture tuarua Isaaka Nyutona, hangaia hoki ngä o te pere whakaterenga me te kaha e hoatu te uara e hiahiatia ana:

mg τ = Fτ = -mg hara x / L

I runga i tenei ōwehenga, ko reira mārama e te tautau ko te pūnaha nonlinear, rite te ope e whangai ana ki te hoki ki tona tūranga taurite, he wa katoa hautanga kore ki te displacement x, i te hara x / L.

Anake ka mahi te tautau pāngarau wiri iti, he mea he oscillator hawarite. I roto i te mau parau te tahi atu, riro te reira i te pūnaha aunoa taea o te raveraa i oscillations hawarite. Ko te tika tenei āwhiwhitanga mō tata koki 15-20 °. e kore e mea au maite tāwēwē ki amplitudes nui.

ture o Newton mo oscillations iti o te tāwēwē

Ki te mahi te pūnaha mīhini oscillations iti, ka titiro rite tenei ture 2 o Newton:

mg τ = Fτ = -m * g / L * x.

I tenei kaupapa, ka taea e tatou e faaoti e te whakaterenga tangential o te tāwēwē ohie he hautanga ki tona displacement ki te tohu "tango". Ko te huru ai riro te pūnaha he oscillator hawarite tenei. Kōwae tauwehe proportionality i waenganui i te displacement me te whakaterenga ōrite te tapawha o te auau koki:

ω02 = g / L; ω0 = √ g / L.

whakaata ana tēnei tātai te auau tūturu o oscillations iti o tenei momo o te tāwēwē. I tenei kaupapa,

T = 2π / ω0 = 2π√ g / L.

Tātaitanga e hāngai ana i runga i te ture o te te tiaki o te pūngao

Āhuatanga kōpiupiu nekehanga tautau taea te whakaahuatia ki te awhina o te ture o te tiaki o te pūngao. kia whanau te reira i roto i te ngakau e te pūngao moe o te tāwēwē i roto i te mara tō ko:

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

Full pūngao pūkaha ōrite te pūmanawa neke, me te mōrahi: Epmax = Ekmsx = E

I muri kua tuhituhia e koe i te ture o te atawhai o te pūngao, tango i te pārōnaki o mahue me tika nga taha o te whārite:

Ep + Ek = const

Mai te mea e rite ana ki te 0 te pärönaki o te taimau, ka (Ep + Ek) '= 0. te pärönaki o te moni ōrite te moni o nga pärönaki:

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

reira:

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

I runga i te tātai whakamutunga, kitea tatou: α = - g / L * x.

tono Practical o te tāwēwē pāngarau

Whakatere o hinga noa rerekē ki ahopae, no te mea kahore te kiato o te kirinuku huri noa te ao ōrite. Hea puta toka ki te kiato teitei, ka waiho i te reira paku teitei. Kei te maha whakamahia whakatere o tautau pāngarau mō te tūhura. I roto i tona titiro awhina mo kohuke rerekē. Noa tatau i te maha o oscillations o te tāwēwē, he reira taea ki te kitea te waro ore ranei i nga whekau o te whenua. Ko e tika ana ki te meka i enei rauemi i te mātotoru me te taimaha o neke atu i te e takoto ana i raro i te toka wewete tenei.

tautau Mathematical whakamahia e taua mataotao rongonui rite Socrates, Aristotle, Platon, Plutarch, Archimedes. He tokomaha o ratou i whakapono e kia whakaawe i te pūnaha mīhini te whakarau, me te ora. whakamahia Archimedes te tautau pāngarau ki ana tātaitanga. I tčnei wā, he maha occultists me mata maori whakamahi i tēnei pūnaha pūkaha mo te whakatinanatanga o ona kikite, te rapu hoki ngaro te iwi ranei.

Famous arorangi Wīwī, me te pūtaiao, whakamahia ano Flammarion mo ratou rangahau he tāwēwē pāngarau. kī ia i taea ki te matapae i te kitea o te aorangi hou, te putanga o te meteorite Tunguska, me era atu meatanga nui ia ki tona awhina. I roto i te Pakanga Tuarua o te Ao i roto i Germany (Berlin) he'inisititiutí motuhake o te tāwēwē. I tčnei wā, e kore te mea e wātea ana Institute o Parapsychology Munich taua rangahau. Tona mahi ki te tautau te tokotoko o tenei institution i huaina "radiesteziey".