Pendulum: nyengo ndi mathamangitsidwe wa chilinganizo

Dongosolo lopangidwa kuti imakhala ndi mfundo chuma (thupi), zomwe chagona pa ndiwachabe inextensible filament (misa ndi negligible poyerekeza ndi kulemera kwa thupi) yunifomu munda yokoka, wotchedwa pendulum masamu (dzina lina - ndi oscillator). Pali mitundu ina ya zipangizo. M'malo filament ndiwachabe ndodo angagwiritsidwe ntchito. Pendulum akhoza limasonyeza akamanena za zochitika zambiri zosangalatsa. Pamene yaing'ono Phokoso matalikidwe a zoyenda wake wotchedwa amgwirizano.

General zokhudza chitetezo cha mawotchi

Chilinganizo cha nyengo oscillation wa pendulum anali ayipanga Dutch wasayansi Huygens (1629-1695 gg.). Izi m'nthawi ya Isaac Newton anali kutikonda kwambiri dongosolo mawotchi. Mu 1656 analenga ulonda woyamba ndi limagwirira pendulum. Iwo anayeza nthawi mwandondomeko kwambiri nthawi imeneyo. Kutulukira ili linali lofunika kwambiri mu kukula kwa zatsopano thupi ndi ntchito zothandiza.

Ngati pendulum ndi mu mgwirizano malo (atapachikidwa vertically), ndi mphamvu yokoka adzakhala moyenera ndi thonje mavuto mphamvu. Lathyathyathya pendulum pa yarns sanali stretchable ndi dongosolo ndi madigiri awiri ufulu kulankhulana. Pamene kusintha imodzi yokha zikuluzikulu za kusintha makhalidwe a ziwalo zanga zonse. Mwachitsanzo, ngati ulusi m'malo ndi ndodo, ndiye dongosolo makina 1 kokha ufulu. Nanga katundu wa pendulum masamu? Mu dongosolo losavuta, mchikakamizo cha perturbation nthawi, chisokonezeko akuwonekera. Zikatero, pamene kuyimitsidwa mfundo chikubwera, ndipo oscillates ndi pendulum pali latsopano mgwirizano malo. Ngati kusinthasintha mofulumira mmwamba ndi pansi dongosolo makina akukhala malo okhazikika "mozondoka." Lilinso ndi dzina lake. Njirayi imatchedwa Kapitza pendulum.

The katundu wa pendulum ndi

Pendulum ali katundu chidwi kwambiri. Onse amapereka mwa malamulo odziwika bwino thupi. Nthawi ya oscillation wa pendulum ina zimadalira zosiyanasiyana monga kukula ndi mawonekedwe a thupi, mtunda pakati pa mfundo kuyimitsidwa ndi pakati yokoka, kugawa kulemera ndi kulemekeza mfundo imeneyi. N'chifukwa chake tanthauzo la thupi atapachikidwa nthawi ndi zolimbikitsa. Ndi zosavuta kuwerengera nthawi ya pendulum yosavuta, chilinganizo cha lopatsidwa pansipa. Chifukwa kutsatira dongosolo awa akhoza akayiyike pa machitidwe ofanana makina:

• Ngati popitiriza kutalika kwa pendulum ndi, inaimitsidwa zosiyanasiyana katundu, nthawi oscillation alandira yomweyo, ngakhale thupi lawo zimasiyana kwambiri. Choncho, nthawi pendulum sadalira pa kulemera kwa katundu.

• Ngati dongosolo akuyamba kuchepa mu pendulum ndi aakulu kwambiri, koma kumathandiza kupeza ngodya zosiyanasiyana, izo kwanthawi ndi nthawi yomweyo, koma pa amplitudes osiyana. Ngakhale zolakwika kuchokera pakati pa muyezo si kusinthasintha waukulu kwambiri mu mawonekedwe awo adzakhala pafupi mokwanila amgwirizano. Nthawi imeneyi pendulum sikudalira matalikidwe vibrational. Mundawu a dongosolo mawotchi amatchedwa isochronism (mu Greek "chronos" - nthawi "Izosov" - ofanana).

Nthawi ya pendulum yosavuta

chithunzi umenewu umatanthauza nthawi zachilengedwe a oscillation. Ngakhale chiphunzitso zovuta, njira yokha lophweka. Ngati kutalika kwa thonje masamu pendulum L, ndi Mphamvu mathamangitsidwe ga, phindu zimenezi n'zofanana:

T = 2π√L / g

Small nthawi oscillations chibadwidwe zosatheka sikudalira unyinji wa pendulum ndi oscillation matalikidwe. Pankhaniyi, monga pendulum masamu imayenda wotsikirapo kutalika.

Oscillations wa pendulum masamu

Masamu pendulum oscillates, sipangakhalenso pofotokoza masiyanidwe aone kuti pali chidule:

× + ω2 tchimo × = 0,

kumene × (T) - ntchito osadziwika (ili ngodya za kuchimwa kwa malo apansi mgwirizano nthawi T, m'kukoma radians); ω - zonse zabwino zimene anatsimikiza kwa magawo a pendulum (ω = √g / L, kumene ga - mathamangitsidwe yokoka, ndipo L - kutalika kwa pendulum chosavuta (kuyimitsidwa).

Aone kuti pali oscillations waung'ono pafupi ndi mgwirizano malo (amgwirizano aone kuti pali) motere:

× + ω2 tchimo × = 0

Oscillatory zoyenda wa pendulum ndi

Pendulum, zomwe zimachititsa oscillations yaing'ono, kusunthira sinusoid. Chachiwiri kuti masiyanidwe aone kuti pali likukwaniritsa zofunikira ndi magawo a gulu amenewa. Kudziwa njira muyenera kudziikira liwiro ndi ndondomeko chomwe pambuyo anatsimikiza zokhazikika odzilamulira okha:

× = A tchimo (θ ₀ + ωt),

kumene θ ₀ - koyamba gawo, A - matalikidwe a oscillation, ω - ozungulira pafupipafupi anatsimikiza ku maikwezhoni za kayendedwe.

Pendulum (tikutsata ndondomeko za amplitudes chachikulu)

dongosolo makina, kuchita oscillations awo ndi matalikidwe lalikulu, ndi kuti azitsatira malamulo ovuta kwambiri magalimoto. iwo mopitirira malinga ndi chilinganizo umenewu pendulum:

tchimo × / 2 = u * SN (ωt / u),

kumene SN - sine Jacobi, amene u <1 ndi ntchito nthawi, ndi u yaing'ono izo chikugwirizana ndi trigonometric sine yosavuta. Mtengo wa u anatsimikiza ndi mawu awa:

u = (ε + ω2) / 2ω2,

kumene ε = E / mL2 (mL2 - mphamvu ya pendulum a).

Khama la osalunjika oscillation nthawi ya pendulum ndi chilinganizo zotsatirazi:

T = 2π / Ω,

kumene Ω = π / 2 * ω / 2K (u), K - dzira integral, π - 3,14.

ndi pendulum kayendedwe ka separatrix ndi

Likutchedwa separatrix njira yodutsamo a dongosolo mphamvu, imene ziwiri ooneka enieni gawo danga. Pendulum amasuntha sanali nthawi. Mu mfundo wamphamvu kutali kwa nthawi akutsikira ku kwambiri chapamwamba udindo kwa liwiro ziro, ndipo ndiye pang'onopang'ono kupeza. Patapita nthawi, iye anayima, kubwerera ku malo ake apachiyambi.

Ngati matalikidwe a oscillation wa pendulum likuyandikira nambala PI, zikunenedwa kuti chisonyezo mu gawo ndege ali pafupi separatrix lapansi. Pankhaniyi, pansi zochita za yaing'ono zodziwitsa galimoto mphamvu ya dongosolo mawotchi akusonyeza khalidwe wachisokonezo.

Mu chochitika cha pendulum yosavuta kwa mgwirizano malo ndi kumanyazi ndi m'nyozo ngodya amapezeka tangential mphamvu Fτ = -mg tchimo φ yokoka. "Opanda" chizindikiro zikutanthauza kuti chigawo tangential anatilangizira zosiyana kuchokera kumbali ya kupatuka a pendulum lapansi. Potchula kudzera pendulum kusamutsidwa × pamodzi ndi Arc zozungulira ndi utali wozungulira L ndi wofanana kusamutsidwa ake okhota φ = × / L. Lamulo lachiwiri Isaaka Nyutona, anaikira ziyerekezo kwa mathamangitsidwe vekitala ndi mphamvu kupereka mtengo wolakalakika:

mg wa τ = Fτ = -mg tchimo × / L

Kuchokera pa chiŵerengero ichi, zikuonekeratu kuti pendulum ndi dongosolo osalunjika, ndi mphamvu zimene zimakupangitsani kuti abwerere kwa mgwirizano ililipo, si nthawi zonse njila kwa kusamutsidwa ×, tchimo × / L.

Only pamene pendulum masamu amachita Phokoso laling'ono, ndi oscillator amgwirizano. M'mawu ena, kumakhaladi dongosolo makina angathe kuchita oscillations amgwirizano. chiyerekezo zimenezi zili zoyenera kwa pafupifupi kumathandiza kupeza ngodya zabwino 15-20 °. Pendulum ndi amplitudes lalikulu si zogwirizana.

lamulo Newton kwa oscillations ka pendulum ndi

Ngati dongosolo makina amachita oscillations yaing'ono, malamulo 2 Newton lidzaonekera izi:

mg wa τ = Fτ = -m * g / L * x.

Pa mfundo imeneyi, tinganene kuti mathamangitsidwe tangential wa pendulum yosavuta ndi njila kwa kusamutsidwa ake ndi chizindikiro "opanda". Ichi ndi chikhalidwe limene dongosolo amakhala oscillator amgwirizano. Gawo Kugwirizana chinthu pakati pa kusamutsidwa ndi mathamangitsidwe ndi liri lalikulu la pafupipafupi okhota:

ω02 = g / L; ω0 = √ g / L.

chilinganizo amasonyeza pafupipafupi zachilengedwe a oscillations ka mtundu wa pendulum. Pa mfundo imeneyi,

T = 2π / ω0 = 2π√ g / L.

Asayansi zochokera m'chilamulo cha kusamala mphamvu

Katundu zokhala kayendedwe pendulum tinganene kuti ndi thandizo la chilamulo cha kusamala mphamvu. Ziyenera adzanyamula kukumbukira kuti mphamvu ndi kuthekera kwa pendulum mu munda Mphamvu ndi:

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

Full mawotchi mphamvu wofanana ndi kayendedwe ndi pazipita angathe: Epmax = Ekmsx = E

Mukamaliza kulemba malamulo a kusamala mphamvu, kutenga ndilochokera ku lamanzere ndi lamanja mbali ya aone kuti pali kuti:

EP + akuti ek = const

Popeza ndilochokera ku zokhazikika ndi wofanana 0, ndiye (EP + akuti ek) '= 0. The ndilochokera ku ndalama za liri Uwerenge opangidwa kuchokera kwa:

EP '= (mg / L * x2 / 2)' = mg / 2L * 2X * x '= mg / L * ndime + akuti ek' = (mv2 / 2) = m / 2 (V2) '= m / 2 * 2v * V '= MV * α,

Choncho:

Mg / L * xv + mva = vesi (mg / L * X + mamita α) = 0.

Kutengera pa chilinganizo otsiriza, ife tikupeza: α = - g / L * x.

Othandiza ntchito ya pendulum masamu

Mofulumira ufulu kugwa zimasiyanasiyana ndi kumpoto chifukwa kachulukidwe wa kutumphuka padziko lapansili si zofanana. Kodi miyala zimachitika ndi osalimba apamwamba, zidzakhala pang'ono apamwamba. Mathamangitsidwe wa pendulum masamu kawirikawiri amagwiritsidwa ntchito chifukwa kufufuza. Thandizo ake amafufuza mchere osiyana. Mwachidule aŵerenga oscillations wa pendulum ndi n'zotheka kudziwa khala kapena miyala mu matumbo a Dziko Lapansi. Ichi ndi chifukwa chakuti chuma ndi kachulukidwe ndi kulemera kwa kuposa atagona pansi pa miyala lotayirira.

Masamu pendulum ntchito ndi akatswiri ena otchuka monga Socrates, Aristotle, Plato, Plutarch, Archimedes. Ambiri a iwo adakhulupirira kuti dongosolo makina angachite tsoka ndi moyo. Archimedes ntchito pendulum masamu ndi kuwerengetsera wake. Masiku ano, occultists ambiri ndi amizimu ntchito dongosolo mawotchi kwa kukhazikitsa maulosi ake, kapena kufufuza kusowa anthu.

Wotchuka French zakuthambo ndi wasayansi, Flammarion pochita kafukufuku wawo anagwiritsanso ntchito pendulum masamu. Anaonetsa kuti ndi thandizo lake iye anali wokhoza kulosera atulukira ndi dziko latsopano, ndi zikamera wa Tunguska kugwa kwa miyala yochokera ndiponso zinthu zina zofunika. Pa Nkhondo Yachiwiri ya World mu Germany (Berlin) ankagwira ntchito anayambitsa apadera pendulum lapansi. Masiku ano, kufufuza ngati palibe Munich Institute of Parapsychology. ntchito yake pendulum ndi ndodo ya bungwe otchedwa "radiesteziey".