Pendulum: muddada iyo dardargelinta ee caanaha

nidaamka farsamo ah in ay ka kooban tahay hal dhibic wax (jirka), taas oo ka laalaadshaa a buuxieyey inextensible weightless (ay mass waa mid laga samri marka loo eego miisaanka jidhka) ee beer uniform gravitational, loo yaqaan pendulum xisaabta (magac kale - oscillator ah). Waxaa jira noocyo kale oo ka mid ah qalabka. Halkii a buuxieyey ul weightless waxaa loo isticmaali karaa. Pendulum si cad u muujin karaa nuxurka ifafaale badan oo xiiso leh. Marka vibrations Ilodanimada yar oo ay mooshin waxaa loo yaqaan harmonic.

macluumaad guud oo ku saabsan nidaamka farsamo

formula muddada oscillation of pendulum ah la rijeynaya in Huygens saynisyahan Dutch (1629-1695 GG.). Tani casriga ah ee Isaac Newton ahaa mid aad u jeclaan of habka farsamo. In 1656 wuxuu abuuray watch hore farsamo pendulum ah. Waxay qiyaasay waqtiga usii xad-dhaaf ah, waayo, waayahaas. war abuurashadoodii Tani waxa ay ahayd tallaabo weyn ee horumarinta tijaabo jireed iyo hawlo wax ku ool ah.

Haddii pendulum waa in booska nuqdad (laalaadeen vertical), ee awood of cuf waxaa lagu dheelitiri doonaa xoog dun xiisadda. pendulum Flat on a non-daah stretchable waa nidaam leh laba darajo xoriyada isgaarsiinta. Marka aad ka bedelanayso kaliya hal qayb ka mid ah la beddelo dabeecadaha dhammaan qaybaha ay. Tusaale ahaan, haddii dun waxaa lagu bedelaa ul, ka dibna nidaamka this farsamo waa degree of xorriyadda 1 keliya. Waa maxay, ka dibna, sifooyinka pendulum xisaabeed? In nidaamka fudud, sarkhaansan perturbation a wareegsan, fowdo muuqataa. Haddii ay sidaa tahay, marka dhibic ganaax uusan u dhaqaaqin, iyo oscillates pendulum a waxaa jira meel hoganaya cusub. Haddii isbedbedelka degdeg ah kor iyo hoos nidaamka this farsamo noqdo meel xasiloon "foorari." Waxa kale oo ay leedahay magaca. Waxaa lagu magacaabaa pendulum Kapitza ah.

The sifooyinka pendulum ah

Pendulum wuxuu leeyahay sifooyin aad u xiiso badan. Oo kulligoodna waxaa lagu taageeray by sharciyada jirka si fiican u yaqaan. Muddada oscillation of pendulum mar kale ku xiran tahay xaaladaha kala duwan sida xajmiga iyo qaabka jirka ah, masaafada u dhexeysa dhibic ka joojinta iyo xarunta of cuf, miisaanka qaybinta si ixtiraam leh in ay marayso. Taasi waa sababta qeexidda muddo jidhka ku deldesheen oo waa arrin adag. Waa u fududdahay in la xisaabiyo muddada pendulum fudud, formula ee kaas oo hoos ku siiyey. Sidaas darteed, sida u fiirsashada nidaamyada waxaa lagu dhigay karaa nidaamyada si farsamaysan la mid ah,

• Haddii, iyadoo la ilaalinayo isla mudada pendulum ah, noocyo kala duwan oo naf dhibkeed ganaax, muddada oscillation isla heli, in kasta oo ay miisaan aad bay u kala duwan yihiin. Sidaa darteed, muddada pendulum uusan ku xiran tahay miisaanka load ah.

• Haddii nidaamka bilaabo inuu hoos u pendulum ma yahay mid aad u weyn, laakiin xaglo kala duwan, waxaa la isla muddadaas is beddelid doonaa, laakiin at amplitudes kala duwan. Iyadoo leexashooyinkaasi xarunta of dheelitirka ma aha isbedbedelka aad u weyn in ay qaab ahaan doonaa dhow ku filan harmonic. Muddada pendulum noocan oo kale ah kuma xirna Ilodanimada inayan vibrational. Tani waxay hanti ah ee nidaamka farsamo waxaa lagu magacaabaa isochronism (ee "chronos" Greek - time "Izosov" - siman).

Muddada pendulum fudud

Tiradan waxaa ay ka dhigan tahay muddo dabiiciga ah ee oscillation. Inkasta oo dejinta ah oo adag, geeddi-socodka laftiisa waa mid aad u fudud. Haddii dhererka dun L pendulum xisaabeed ah, iyo xawaaraha g gravitational, qiimaha this waa loo siman yahay:

T = 2π√L / g

muddo yar oo oscillations dabiiciga ah sinaba kuma xirna mass ee pendulum iyo Ilodanimada inayan oscillation ah. Xaaladdan oo kale, sida pendulum xisaabeed guuro dherer hoos.

Oscillations of pendulum fudud

pendulum Xisaabeed oscillates, taas oo lagu tilmaami karo by a isla'egta kala duwan oo fudud:

x + ω2 dembi x = 0,

halkaas oo x (t) - shaqo la garanayn (xagal this ee kubad ka booska hoose ee hoganaya waqti t ah, ayaa muujiyay in halbeegga); ω - joogta wanaagsan wixii la goostay ayaa ka soo koobin ee pendulum ah (ω = √g / L, halkaas oo g - xawaaraha ee cuf, iyo L - dhererka pendulum fudud (ganaax).

Isla'egta oscillations yar oo u dhow meel hoganaya (isla'egta harmonic) sida soo socota:

x + ω2 dembi x = 0

mooshin Oscillatory of pendulum ah

Pendulum, taas oo ka dhigaysa oscillations yar, dhaqaaqin sinusoid. isla'egta si kala duwan Second kulmo shuruudaha iyo xuduudaheedu dhaqdhaqaaqa noocan oo kale ah. Si loo go'aamiyo Jidka aad u baahan tahay si ay u dejiyaan xawaaraha iyo wadataa, taasi oo markii dambe go'aamisay geysid madax banaan:

x = dembi A (θ ₀ + ωt),

halkaas oo θ ₀ - wajiga hore, A - Ilodanimada inayan oscillation, ω - noqnoqoshada xilliile ka isleegyo ee mooshinka ka go'an.

Pendulum (formula for amplitudes weyn)

Nidaamkan farsamo, oofin oscillations ay la Ilodanimada weyn, waxa ay ku xiran yahay sharciyada gaadiidka dheeraad ah oo adag. ay xisaabiyaa sida ay caanaha ee pendulum noocan oo kale ah:

dembi x / 2 = u SN * (ωt / u),

halkaas oo SN - Culumida Jacobi, kii u <1 waa shaqo wareegsan, iyo u yar waxa ay ku beegan tahay Culumida fudud trigonometric. Qiimaha u waxaa lagu go'aamiyaa ra'yi soo socda:

u = (ε + ω2) / 2ω2,

halkaas oo ε = E / mL2 (mL2 - tamarta ee pendulum ah).

Determination of muddo oscillation Fansaarada of pendulum ah by formula soo socda:

T = 2π / Ω,

halkaas oo Ω = π / 2 * ω / 2k (u), K - elliptic muhiim, π - 3,14.

dhaqdhaqaaqa pendulum of separatrix ah

Waxaa loo yaqaan dhabbihii separatrix nidaamka firfircoon, taas oo meel wajiga laba dhinac-cabbir. Pendulum guuro on a non-xilliyo. In barta Rowan fog waqtiga waxaa lagu dhibciyo ka booska xad dhaaf ah sare xagga xawaare eber, ka dibna waxaa si tartiib tartiib ah ayaa la helo. Waxa uu ugu danbeyn joojin, in ay markii hore soo laabtay.

Haddii Ilodanimada inayan ee oscillation of pendulum ku dhowaado pi tirada, waxa la sheegay in mooshinka in diyaarad wajiga u dhow separatrix yahay. Xaaladdan oo kale, hoos tallaabo xoog yar oo wareegsan wadida nidaamka farsamo muujinayaa dhaqanka qasan.

Haddii ay dhacdo in pendulum fudud laga qabto, hoganaya la CP xagal dhacdaa xoog tangential Fτ = dembi -mg φ cuf. "Jaray" calaamad ka dhigan tahay in qayb ka tangential faray dhinaca ka soo horjeeda ka jihada of weecasho ee pendulum ah. Marka laga via barakac pendulum x weheliyaan ARC wareeg ah la L gacan ka waa loo siman yahay si ay φ barakac angular = x / L. Sharcigu labaad Isaaka Nyutona, loogu talagalay qiyaas ee dulinka xawaaraha iyo xoog siin qiimaha la doonayo:

mg τ = Fτ = -mg dembi x / L

Iyada oo ku saleysan ratio this, waxaa cad in pendulum waa nidaam Fansaarada, sida ciidan in u egtahay inuu ku soo laabto booskiisa hoganaya ay, ma aha had iyo jeer saami x barakaca, a dembi x / L.

Kaliya marka pendulum xisaabta qabata gariirka yar, waa oscillator a harmonic. In si kale loo dhigo, waxa uu noqonayaa nidaam farsamo awood ugu leeyihiin qabashada oscillations harmonic. qiyaas Tani waa ansax ah muddo ku dhow xaglaha 15-20 °. Pendulum la amplitudes weyn ma aha habboonaanta.

sharciga Newton ee oscillations yar oo pendulum a

Haddii nidaamka si farsamaysan u qabata oscillations yar, sharciga 2aad Newton ayaa eegi doonaa sida tan:

mg τ = Fτ = D * g / L * x.

Iyadoo ku saleysan, waxaannu ku tirinnaa karaa in xawaaraha tangential of pendulum fudud saami ay barakac calaamada "jaray". Tani waa xaalad ku dheehan yahay nidaamka noqdo oscillator a harmonic. factor dheellitirnaanta Module dhexeeya barakac iyo xawaaraha uyeelaysaan kuwo square ee inta jeer ee angular:

ω02 = g / L; ω0 = √ g / L.

formula Tani waxay ka tarjumaysaa inta jeer ee dabiiciga ah ee oscillations yar oo noocan ah pendulum. Iyadoo ku saleysan,

T = 2π / ω0 = 2π√ g / L.

Xisaabinta ku saleysan sharciga ilaalinta ee tamarta

Guryaha oscillating dhaqdhaqaaqa pendulum lagu tilmaami karaa iyadoo la kaashanayo of sharciga ilaalinta ee tamarta. Waa in la dhasha maskaxda ku haysaa in tamarta ka iman kara pendulum ee beer gravitational waa:

E = mgΔh = MGLCutubka (1 - sababtoo ah α) = mgL2sin2 α / 2

Full tamarta si farsamaysan uyeelaysaan kuwo awood u ogeed iyo ugu badnaan: Epmax = Ekmsx = E

Ka dib marka aad soo qoray sharciga ilaalinta ee tamarta, qaadashada derivative oo ka mid ah dhinac bidix iyo midig ee isla'egta:

EP + Ek = const

Tan iyo derivative of geysid waa loo siman yahay si 0, ka dibna (EP + Ek) = 0. derivative ee wadarta ay le'eg tahay wadarta ee taagayo ah:

EP '= (mg / L * x2 / 2)' = mg / 2L * 2x * x '= mg / L * v + Ek' = (mv2 / 2) = m / 2 (v2) = m / 2 * 2V * v '= α * MV,

Sidaa darteed:

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

Iyada oo ku saleysan formula la soo dhaafay, waxaan ka heli: α = - g / L * x.

codsiga la taaban karo ee pendulum xisaabta

Acceleration of dayrta free kala duwan la loolka, maxaa yeelay, cufnaanta ee qolof ah ee daafaha adduunka oo aan isku mid. Halkee dhagaxyada dhacaan cufnaanta sare, waxa ay noqon doontaa wax yar ka sareeya. Acceleration of pendulum xisaabta waxaa badanaa loo isticmaalaa baaritaanka. In ay fiirin caawimaad macdanta kala duwan. Si fudud u tirinta tirada oscillations of pendulum ah, waxaa suurtagal ah in la ogaado sida dhuxul ama ore ee uurkii Dhulka. Tan waxa u sabab xaqiiqda ah in ilaha waxay leeyihiin cufnaanta ah iyo miisaan ka badan been dhagaxyada dabacsan dureeri.

pendulum xisaabta loo isticmaalo by culimada caanka ah sida Socrates, Aristotle, Plato, Plutarch, Archimedes. Qaar badan oo iyaga ka mid ah la rumeysan yahay in nidaamka farsamo saamayn karaan masiirka iyo nolosha. Archimedes isticmaalo pendulum xisaabta la xisaabo uu. Maalmahan, dhaqamada iyo psychics badan isticmaalaan nidaamka this farsamo for hirgelinta waxsii ay, ama raadinta maqan dadka.

The astronomer Faransiis caan ah iyo saynisyahan, Flammarion ay cilmi sidoo kale loo isticmaalaa pendulum xisaabeed. Waxa uu sheegay in isaga iyo caawimaad uu awooday in la saadaaliyo helitaanka meeraha cusub, soo bixitaanka of meteorite Tunguska ah, iyo dhacdooyin kale oo muhiim ah. Intii lagu guda jiray dagaalkii labaad ee adduunka ee Germany (Berlin) shaqeeyay machadka gaar ah pendulum ah. Maalmahan, cilmi sida aan la heli karin Munich Institute of Parapsychology. Shuqulkiisa la pendulum shaqaalaha ee machadkan loo yaqaan "radiesteziey".