Pendulum: Isikhathi nokusheshisa ifomula

Uhlelo mechanical ukuthi siqukethe iphuzu impahla (umzimba), okuyinto ulenga phezu ube intambo isisindo inextensible (isisindo salo azinakwa kuqhathaniswa isisindo emzimbeni) ensimini iyunifomu okudonsa, ebizwa ngokuthi i-pendulum zezibalo (elinye igama - the oscillator). Zikhona nezinye izinhlobo amadivaysi. Esikhundleni ube intambo induku isisindo ingasetshenziswa. Pendulum ezingabonakala ngokucacile zembula okushiwo izenzakalo eziningi ezithakazelisayo. Lapho kunyakaza encane amplitude motion yayo libizwa harmonic.

ulwazi olujwayelekile mayelana uhlelo mechanical

Ifomula kwesikhathi oscillation we pendulum u ezifuywa Dutch usosayensi Huygens (1629-1695 GG.). Lokhu wesimanje ka-Isaac Newton, wayejabula kakhulu ohlelweni lokukhanda. Ngo 1656 edala Ngomlindo wokuqala nge indlela okuthiwa i-pendulum. Bona kulinganiswa isikhathi ngobuciko obukhulu kulabo izikhathi. Lokhu okusungulile kwaba igxathu enkulu ekuthuthukiseni zokuhlolwa ngokomzimba futhi nemisebenti lephatsekako.

Uma pendulum kuyinto isikhundla yakhoibrium (ulenga thwi), i- amandla adonsela phansi ngeke sibe abalinganiselayo yi ntambo ukungezwani amandla. pendulum Flat kwi lokuluhlaza non-stretchable uhlelo amabili lamadigri enkululeko yokuxhumana. Uma uguqula yengxenye eyodwa zokushintsha izimfanelo zonke izingxenye zawo. Ngokwesibonelo, uma intambo esikhundleni induku ke lesi simiso awayindwayo 1 kuphela degree enkululeko. Khona-ke, izakhiwo a pendulum zezibalo? Kulesi simiso elula, ngaphansi kwethonya a perturbation gqwa, chaos livela. Kuleso simo, lapho iphuzu ukumiswa asibahambeli, futhi oscillates a pendulum kukhona yakhoibrium isikhundla esisha. Uma iyehla iyenyuka okusheshayo up and down lesi simiso mechanical iba isikhundla ozinzile "esiphendula esibhekisa phansi." It Ubuye igama layo. Ibizwa ngokuthi Kapitza pendulum.

Izakhiwo pendulum

Pendulum anezakhi ethakazelisa kakhulu. Bonke zisekelwa kwemithetho yemvelo owaziwa. Inkathi oscillation we pendulum yimuphi omunye incike izimo ezehlukene ezifana ubukhulu kanye ukuma komzimba, ibanga phakathi iphuzu ukumiswa kanye maphakathi gravity, ukusatshalaliswa isisindo maqondana leli phuzu. Yingakho kwencazelo umzimba uhlanganyela inkathi kuyinselele ngempela. Ingabe kulula kakhulu ukubala inkathi a pendulum elula, formula okuyinto ngezansi. Ngenxa yalokho lokugcina lamaphethini lingasethwa ku izinhlelo ezifanayo mechanical:

• Uma, ngesikhathi esifanayo ulondoloze isimo ubude efanayo pendulum, umiswe ezihlukahlukene imithwalo, inkathi oscillation uthole abafanayo, nakuba ayengafani isisindo sabo kuzohluka kakhulu. Ngenxa yalokho, inkathi pendulum akuxhomekile isisindo umthwalo.

• Uma uhlelo iqala ukunqabela ku pendulum akuyona likhulu kakhulu, kodwa ngezindlela ezihlukahlukene, ke izokhuphuka nenani lesikhathi esifanayo, kodwa ngezinye amplitudes ezahlukene. Nakuba ngeziphambeko kusukela maphakathi ibhalansi akuyona iyehla iyenyuka inkulu kakhulu ngefomu yabo iyoba beseduze harmonic. Inkathi a pendulum enjalo akuxhomekile amplitude vibrational. Le mpahla kohlelo mechanical libizwa isochronism (e "Chronos" ngesiGreki - isikhathi "Izosov" - alinganayo).

Inkathi a pendulum elula

Lesi sibalo imelela inkathi engokwemvelo oscillation. Naphezu ekwakhiweni eziyinkimbinkimbi, inqubo ngokwalo silula. Uma ubude ntambo zezibalo pendulum L, kanye adonsela phansi ukusheshisa g, lesi silinganiso ilingana:

T = 2π√L / g

isikhathi elincane oscillations zemvelo neze akuxhomekile kunobukhulu bendawo pendulum kanye amplitude oscillation. Kulokhu, njengoba pendulum zezibalo oluhamba obuphelele esincishisiwe.

Oscillations of a pendulum zezibalo

pendulum Mathematical oscillates, okungase esachazwa ezothando elula umehluko:

x + ω2 isono x = 0,

lapho x (t) - umsebenzi alaziwa (lolu hlobo engela kuka kusukela isikhundla engezansi yakhoibrium ngesikhathi t, obonakaliswe radians); ω - njalo omuhle okuyinto kunqunywa kusukela kwemigomo pendulum (ω = √g / L, lapho g - ukusheshisa amandla adonsela phansi, kanye no-L - ubude pendulum elula (ukumiswa).

Equation oscillations elincane eduze yakhoibrium isikhundla (kwesibalo harmonic) kanje:

x + ω2 isono x = 0

ezinyakazayo Oscillatory we pendulum

Pendulum, okwenza oscillations encane, ezihambayo sinusoid. Okwesibili oda umehluko kwesibalo elihlangabezana nazo zonke izidingo kanye nemingcele wenhlangano ezinjalo. Ukuze sithole indlela udinga ukusetha ijubane futhi izixhumanisi, okuyinto kamuva kunqunywa constants ezimele:

x = A isono (θ ₀ + ωt),

lapho θ ₀ - esigabeni sokuqala, A - amplitude oscillation, ω - imvamisa eyindilinga kwanqunywa kusukela zibalo motion.

Pendulum (formula for amplitudes elikhulu)

Lolu hlelo mechanical, enze oscillations zabo amplitude enkulu, ingaphansi kwemithetho eziyinkimbinkimbi traffic. basuke ibalwe ngokwendlela formula for a pendulum ezifana:

isono x / 2 = u * sn (ωt / u),

lapho sn - sine Jacobi, ngubani ngoba u <1 umsebenzi ngezikhathi ezithile, futhi u elincane ke kuvumelana sine elula Trigonometric. Inani u kunqunywa Inkulumo elandelayo:

u = (ε + ω2) / 2ω2,

lapho ε = E / mL2 (mL2 - lukagesi pendulum).

Ukunqunywa nonlinear oscillation inkathi pendulum by le ndlela elandelayo:

T = 2π / Ω,

lapho Ω = π / 2 * ω / 2K (u), K - elliptic ebalulekile, π - 3,14.

ukuhamba pendulum we separatrix

It ngokuthi separatrix trajectory uhlelo ashukumisayo, lapho isigaba isikhala mgudumbili. Pendulum ihamba okungeyona njalo. In iphuzu ngokungenamkhawulo kude isikhathi ke lehla ukusuka kakhulu isikhundla engenhla ngase isivinini zero, bese-ke kancane kancane ukuthola. Wagcina wayeka, abuyele isikhundla yayo yasekuqaleni.

Uma amplitude oscillation we pendulum osondela pi inombolo, kuthiwa ukunyakaza indiza isigaba useduze separatrix. Kulokhu, ngaphansi isenzo esincane gqwa ukushayela force kohlelo mechanical ubukisa ukuziphatha kunezinxushunxushu.

Esimeni a pendulum elula kusukela isikhundla lizama ukuzivumelanisa nendawo i cp engela kwenzeka amandla tangential Fτ = isono -mg φ adonsela phansi. uphawu "Minus" kusho ukuthi ingxenye tangential iqondiswe kolunye uhlangothi kusukela isiqondiso ukuphambuka we-pendulum. Lapho ebhekisela nge-pendulum ukufuduka x kanye nge-engela elingama isiyingi nge L engaba ilingana yayo emincane ukufuduka φ = x / L. Eyesibili umthetho Isaaka Nyutona, eyenzelwe Iwumbono vector ukusheshisa namandla ukunikeza ukubaluleka oyifunayo:

mg τ = Fτ = -mg isono x / L

Kususelwa kule isilinganiso, kuyacaca ukuthi pendulum uhlelo nonlinear, njengoba ibutho ukuthi ivame ukuba ubuyele yakhoibrium yayo isikhundla, akunjalo ngaso sonke isikhathi wakhona x ukufuduka, isono x / L.

Kuphela lapho i-pendulum zezibalo wenza kunyakaza encane, kuba oscillator harmonic. Ngamanye amazwi, kuba uhlelo mechanical ongenza oscillations harmonic. Lokhu isilinganiselo isebenza cishe angles 15-20 °. Pendulum nge amplitudes esikhulu akuyona obunobunye.

umthetho Newton analo oscillations elincane a pendulum

Uma uhlelo mechanical wenza oscillations encane, umthetho 2nd Newton izobukeka kanje:

mg τ = Fτ = -M * g / L * x.

Kulesi sisekelo, singaphetha ngokuthi ukusheshisa tangential of a pendulum elula wakhona ukufuduka yayo uphawu "lokususa". Lesi isimo lapho uhlelo iba oscillator harmonic. Module proportionality isici phakathi ukufuduka kanye ukusheshisa lilingana skwele imvamisa emincane:

ω02 = g / L; ω0 = √ g / L.

Lokhu ifomula ibonisa imvamisa engokwemvelo oscillations elincane lolu hlobo pendulum. Kulesi sisekelo,

T = 2π / ω0 = 2π√ g / L.

Izibalo zisekelwe umthetho ukulondolozwa amandla

Izakhiwo oscillating pendulum ukunyakaza lingachazwa ngosizo umthetho ukulondolozwa amandla. Kumele kukhunjulwe ukuthi amandla bezikhali le pendulum ensimini okudonsa yilesi:

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

Okugcwele amandla mechanical lilingana ezingaba kinetic futhi esiphezulu: Epmax = Ekmsx = E

Ngemva usubhale lesi umthetho ukulondolozwa amandla, ngokuthatha esuselwe ezinhlangothini kwesokunxele kanye kwesokudla kwesibalo:

Ep + Ek = const

Kusukela esuselwe constants ilingana 0 ke (Ep + Ek) '= 0. I esuselwe isamba lilingana isamba nemikhiqizo:

Ep '= (mg / L * X2 / 2)' = mg / 2l * 2x * x '= mg / L * v + Ek' = (mv2 / 2) = m / 2 (v2) '= m / 2 * 2v * v '= mv * α,

Ngakho-ke:

Mg / L * xv + MVA = v (mg / L * x + m α) = 0.

Kususelwa ifomula zokugcina, sithola: α = - g / L * x.

isicelo Engokoqobo Yezindaba pendulum zezibalo

Ukusheshisa ngokuwa khulula siyahlukahluka latitude, ngoba kwabantu uqweqwe emhlabeni hhayi ezifanayo. Lapho amadwala zenzeka nge kwabantu ephakeme, kuyoba kancane ephakeme. Ukusheshisa pendulum zezibalo livame ukusetshenziswa ukuhlola. Ngo Ngosizo lwawo ukubukeka namaminerali ezahlukene. Vele Ubala oscillations of a pendulum, kungenzeka ukuthola ilahle noma i-ore kuleli yembathani Earth. Lokhu kungenxa yokuthi la imithombo abe ukuminyana nesisindo ezingaphezu kuka elele ngaphansi kwamadwala okuxekethile.

pendulum Mathematical esetshenziswa enjalo izazi ezivelele njengoba uSocrates, u-Aristotle, Plato UPlutarch, Archimedes. Abaningi babo babekholelwa ukuthi uhlelo mechanical ingaba nomthelela ukudalelwa futhi ukuphila. Archimedes wasebenzisa pendulum zezibalo kanye izibalo zakhe. Namuhla, occultists eziningi izangoma ukusebenzisa lolu uhlelo mechanical ngokuba ukuqaliswa iziprofetho zalo, noma ucinge ulahlekile abantu.

Adume sezinkanyezi French kanye nososayensi, Flammarion ucwaningo lwabo futhi wasebenzisa a pendulum zezibalo. Sathi ngosizo lwakhe wakwazi ukubikezela kokutholakala iplanethi entsha, ukuvela kwezinhlobo zalo Tunguska kwemvula yezinkanyezi ezinomsila, nezinye izenzakalo ezibalulekile. Phakathi kwempi yezwe yesibili ngo-Germany (Berlin) wasebenza njengendlela sikhungo okukhethekile pendulum. Namuhla, cwaningo ayitholakali Munich Isikhungo Parapsychology. Umsebenzi wakhe kanye pendulum ubhoko kulesi sikhungo ngokuthi "radiesteziey".