Pendulum: au a me ka hoʻonui o ka haʻilula

Ka mechanical e'ōnaehana i oia no ka mea wahi (ke kino), i kēkahi ma ka weightless inextensible filament (kona nuipa mea negligible hoohalikeia ia i ke kaupaona ana o ke kino) i loko o kaʻaʻahu gravitational kahua, i kapaʻia ka makemakika pendulum (kekahi inoa - ka oscillator). Aia nō 'ē aʻe ke ano o na manao ona. Ma kahi o ka filament weightless laau hiki ke hoʻohana. Pendulum hiki maopopo hōʻike i ka pā like ana o na mea he nui hana hoihoiʻike manaʻo. I ka wa uuku ikaika hawewe kuekueni o kona noi ua kapaia harmonic.

General 'ike e pili ana i ka me mechanical'ōnaehana

Ke haʻilula o ka oscillation wā o ka pendulum i piha iho Hōlani nä känaka 'epekema Huygens (1629-1695 GG.). Kēia Contemporary o Isaac Newton ua loa leʻaleʻa o ka me mechanical nenoaiu. I ka 1656, ua hana mai ka mua o ka wati me ka pendulumʻano papa hana. Ka mea ana i ka manawa me ka lōʻihi miomio no ia mau manawa. Kēia hana mea hou i hana muaʻole ua he nuiʻanuʻu i loko o ka ulu ana o ke kino ai 'ho okolohua me ka mea hiki hanana.

Inā ka pendulum mea i loko o ka equilibrium wahi (e kau hoʻopapakū), i ka ikaika o ka umekaumaha e e 'O ke kaulike ma ka huakaʻi hoʻopilikia max. I lalo pendulum ma ka pili-stretchable nā huakaʻi o ka'ōnaehana me nā degere o ke ku okoa ana o ka 'ōlelo. wale i kekahi ke keʻena o ka hoʻololi i nā 'ano o nā mea a pau kona wahi ka wā hoʻololi. No ka laʻana, ina he pae UAIAaIN ma ke koʻokoʻo, a laila, i kēia me mechanical'ōnaehana mea wale 1 degere o ke ku okoa ana. He aha, laila, i ka waiwai o ka makemakika pendulum? Ma keia mea'ōnaehana, ma lalo o ka aoao o ka papa'āweʻaweʻa kemika perturbation, chaos ikeia. Ma ia hihia, i ka wa o ka wehe wahi ua ole e neʻe, a oscillates he pendulum aia no he hou equilibrium kūlana. Inā kokololio fluctuations mai, a i keia me mechanical'ōnaehana i ka pilina paʻa wahi "i lalo ke alo." Ua hoi i kona inoa. Ua Ua kapaʻia ka Kapitza pendulum.

Nā waiwai o ka pendulum

Pendulum i loa hoihoi 'waiwai. A pau o ia i kākoʻo 'ia e pono-ike physical kānāwai. Ke au o ka oscillation o ka pendulum i kekahi'ē aʻe hilinaʻi nui ma luna o kekahi mau ka noho ana e like me ka nui a me ke kinona o ke kino, o ka mamao ma waena o ka wahi o ka wehe a me ka piko o ka umekaumaha, ke kaupaonaʻana o ka makana me ka pili ana i kēia wahi. Ia mea ke kumu o ka mana o ke kino e kau ana au mea he hana nui. Ua nui maʻalahi e hoʻomaulia i ka wā o ka poe noonoo ole pendulum, i ka haʻilula o i ua hāʻawiʻia ma lalo nei. E like me ka hopena o ka malama ana i kēia mau lauana hiki ke kau ma luna o like me mechanical nenoai:

• Inā, oiai e malama ana i ka like loa o ka pendulum, ke wehe 'ia mai keʻano o ka wahie,, i ka wā o ka oscillation kiʻi i ka ia, eia nae lākou ke kaupaonaʻana E oko a nui. Nolaila, i ka wā o ka pendulum, aole ia e hilinaʻi ma ka paona o ka haawe.

• Inā ka'ōnaehana hoʻomaka e huli i loko o ka pendulum mea,ʻaʻole he nui, akā, okoa pana pua, ka mea e fluctuate me ka ia wā, akā, ma kekahi mau amplitudes. Ia deviations, mai ka piko o ke kaulike mea,ʻaʻoleʻoi aku ko lākou nui fluctuations i loko o kā lākouʻano e e kokoke lawa harmonic. Ke au o ia ka pendulum 'aʻole e hilinaʻi ma ka vibrational ikaika hawewe. Kēia waiwai o ka me mechanical'ōnaehana ua kapaia isochronism (i loko o Greek "chronos" - manawa "Izosov" - like).

Ke au o ka poe noonoo ole pendulum

Kēia huahelu ho i ka wā maoli o oscillation. I loko nō o ka luna 'hoʻomanaʻo, ke kaʻina iho ka loa mea. Inā ka lōʻihi o ke kālepa ma makemakika pendulum L, a me ka gravitational hoʻonui g., Ua nui ka like:

T = 2π√L / g.

Small wā o maoli oscillations ma oleʻaoʻao, aole ia e hilinaʻi ma ka hū o ka pendulum, a me ka oscillation ikaika hawewe. Ma keia hihia, e like me ka makemakika pendulum höʻike me ka ho'ēmi i kā lōʻihi.

Oscillations o ka makemakika pendulum

Makemakika pendulum oscillates, a e ho'ākāka 'ia ma hiki ma ka mea palena pau helu kaulike.

m + ω2 hewa m = 0,

kahi m (Halau) - ike kuleana pili i (keia huina o deflection mai ka haʻahaʻa kulana o equilibrium i ka manawa t, kaheaia ma radians); ω - he maikaʻi ia a eia i ka nānā 'ana mai o nā mea kiko'î o ka pendulum (ω = √g / L, kahi g. - ka hoʻonui o ka umekaumaha, a me L - i ka lōʻihi o ka poe noonoo ole pendulum (mai paha).

Helu kaulike oscillations liʻiliʻi ma kahi o equilibrium kulana (harmonic helu kaulike) like penei:

m + ω2 hewa m = 0

Oscillatory ka noi o ka pendulum

Pendulum, aʻo uuku oscillations, neʻe maila sinusoid. Lua o ka papa palena pau helu kaulike me nā mea a pau i ke koi a me ka mea kiko'î o ia ka neʻeʻana. E ike i ke ala e pono ai, e hoʻonoho i ka mämä holo a me ka wahi, a ma hope hoʻoholo kuokoa ia a eia i:

He m = A hewa (θ ₀ + ωt),

kahi θ ₀ - loiloi mua pae, A - ikaika hawewe o oscillation, ω - cyclic alapine (frequency) hoʻoholo mai na nā helu kaulike o ke noi.

Pendulum (haʻilula no ka nui amplitudes)

Kēia mechanical e nenoaiu, hana lākou oscillations me ka ikaika hawewe nui, ia mea malalo iho o ke oi luna 'kalepa kānāwai. ka mea, e pōpilikia e like me ka haʻilula no ia i pendulum:

hewa m / 2 =? u * sn (ωt / u make),

kahi sn - sine Jacobi, ka mea no ka make <1 mea he papa'āweʻaweʻa kemika papa, a no ka uuku u make ka mea, coincides me ka noonoo ole trigonometric sine. I ka waiwai o ke one me ka nānā 'ana i kēia hōʻike:

? u = (ε + ω2) / 2ω2,

kahi ε = E / mL2 (mL2 - ikehu o ka pendulum).

Hoʻoholo 'ana o nonlinear oscillation au o ka pendulum ma ke kēia haʻilula:

T = 2π / Ω,

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

ka pendulum ana o ka separatrix

Ua kapa separatrix ala o ka hōʻeuʻeu'ōnaehana, ma a he mau-dimensional pae ka lewa. Pendulum kolo ana maluna o ka pili-kekahi manawa. Ma ka infinitely mamao wahi o ka manawa e kulu mai o ka loa maluna manao i ka 'Aʻohe māmā holo, a laila ka mea, ua kiʻekiʻe o ho onui ana. He e ho'ōla pani, hoi mai i kona kikokikona wahi.

Inā ka ikaika hawewe o oscillation o ka pendulum hoʻokokoke mai ke helu pi, ka mea, ua olelo mai ia i ke noi ana ma ka pae pelane mea kokoke i ka separatrix. Ma keia hihia, ma lalo o ka hana ana o kekahi mea uuku papa'āweʻaweʻa kemika holo ikaika ana o ka me mechanical'ōnaehana hōʻike chaotic kolohe.

Ma ka hanana o ka poe noonoo ole pendulum mai ka equilibrium wahi me ka huina CP ia lŘlŘ tangential ikaika Fτ = -mg hewa φ umekaumaha. "Hua" hoailona o ia hoʻi, i ka tangential ke keʻena kuhikuhi ma ka ku pono ana i ala e hiki aku, mai ka olelo mai o haiahū kūmau o ka pendulum. I ka Ua haawiia Via pendulum pohupani hale m ma ka hoʻopoepoe piʻo me ka ke kahahńnai o L mea like i kona angular pohupani hale φ = m / L. Ka lua o ka kānāwai Isaaka Nyutona, papahana no ka wānana o ka hoʻonui maʻi lele a me ka ikaika e haawi aku i ka makemake waiwai:

mg τ = Fτ = -mg hewa m / L

Ma muli o kēia lākiō, ka mea, ua ike ia ka pendulum mea he nonlinear'ōnaehana, like me ka mana e mālama, e hoʻi i kona equilibrium kulana, ua i mau hapaʻauhau i ka pohupani hale m, he hewa m / L.

Wale i ka wa a ka makemakika pendulum hanaʻia uuku kuekueni, ia mea he harmonic oscillator. Ma na olelo e ae, e lilo i me mechanical'ōnaehana hiki ke hoʻokō i harmonic oscillations. Kēia kokekau mea i pololei ia no ka aneane pana pua 15-20 °. Pendulum me ka nui amplitudes mea ole harmonious.

Newton ka kānāwai no ka oscillations uuku o ka pendulum

Inā ka mechanical e'ōnaehana hanaʻia uuku oscillations, 2nd Newton ka kānāwai, e nana me keia:

mg τ = Fτ = -m * g. / L * m.

Ma keia kumu, ua hiki ke manao nei makou, o ka tangential hoʻonui o ka poe noonoo ole pendulum ka hapaʻauhau i kona pohupani hale me ka hoailona "hua". Kēia mea he ano e hiki ai i ka'ōnaehana i ka harmonic oscillator. Module proportionality ololi ma waena o ka pohupani hale, a me ka hoʻonui lākou, i ke kahua o ka angular alapine (frequency).

ω02 = g. / L; ω0 = √ g. / L.

Kēia haʻilula Hōʻike 'i ke alapine (frequency) maoli o oscillationsʻuʻuku o kēia' ano o ka pendulum. Ma keia kumu,

T = 2π / ω0 = 2π√ g. / L.

Ho omaulia ka nānā 'ana ma luna o ke kānāwai o ka mālama o ka ikehu

Waiwai oscillating pendulum neʻe hiki ke ho'ākāka 'ia me ke kōkua o ke kānāwai o ka mālama o ka ikehu. It E e kaʻikaʻiʻia ma ka manao ana o ka ikehu Loaʻaʻia o ka pendulum i loko o ka gravitational kahua o:

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

Full mechanical e ikehu lākou, i ka kinetic a me kā mākou Loaʻaʻia: Epmax = Ekmsx = E

Ma hope o 'oe i palapala i ke kānāwai o ka mālama o ka ikehu, e lawe i ka derivative o ka hema, a me ka pololei aoao o ka helu kaulike.

Ep + Hai = const

Mai ka derivative o na ia a eia i mea like i 0, laila, (Ep + Hai) '= 0. lākou, Ke derivative o ka huina i ka huina o na nā mea loaʻa:

Ep '= (mg / L * X2 / 2)' = mg / 2L * 2x * m '= mg / L * V + Hai' = (mv2 / 2) = m / 2 (v2) '= m / 2 * 2v * V '= MV * α,

nolaila:

Mg / L * XV + MVA = V (mg / L * m + m α) = 0.

Ma muli o ka hope haʻilula, ua loaʻa: α = - g. / L * m.

E pili ana o ka makemakika pendulum

Hoʻonui o noa haule loli aʻe me ka latitu, no ka paʻapū o ka pńpa a puni ka Honua i'ālike. Where pöhaku hana ana me ka pa apŘ kiekie, ka mea e e iki kiʻekiʻe. Hoʻonui o ka makemakika pendulum ua pinepine hoʻohana no kaʻimi loa i ka lewa. I kona nana aku kōkua no kekahi mau nā minelala hao. Aloha helu i ka helu ana o oscillations o ka pendulum, he mea hiki, e huai ae i ka lānahu ahi a me ka mea e hoohehee ai i loko o ka'ōpū o ka Honua. O ke kumu ho i ka mea i kēia mau waiwai i ka pa apŘ a me ke kaumaha o ka oi ma mua o e moe ana ma lalo iho o ka moe kolohe pōhaku.

Makemakika pendulum hoʻohana 'ia e ka koʻikoʻi haumana me Socrates, Aristotle, Plato, Plutarch, Archimedes. He nui no lakou i manaoio mai i ka me mechanical'ōnaehana ke ho ololi i ka hopena a me ke ola. Archimedes hoʻohana i ka makemakika pendulum me kona mau ho. Kēia manawa, he nui occultists a me psychics e hoʻohana i kēia me mechanical'ōnaehana no ka manaʻo o kona mau wanana, a me ka huli 'ana no ka nalowale kanaka.

Ke kaulana Palani astronomer a me nä känaka 'epekema, Flammarion no kā lākou noiʻi i hoʻohana i ka makemakika pendulum. He koi ia ka mea, ua hiki ke kilokilo i ka loaʻa mai o ka hou Honua, ka muli o ka Tunguska meteorite, a me ka nui hanana'ē aʻe i kona kōkua. I loko o ka Apana o ke Kaua Honua ma Germany (Pelelina) hana like me ka hana kūikawā kula, o ka pendulum. Kēia manawa, ua noiʻi mea,ʻaʻole i loaʻa Munich Institute o Parapsychology. Kana hana me ka pendulum ke kookoo o keia kula i kapaia "radiesteziey".