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Mehanika tverdogo tela. Lekcii.

V.A.Aleshkevich, L.G.Dedenko, V.A.Karavaev (Fizicheskii fakul'tet MGU)
Izdatel'stvo Fizicheskogo fakul'teta MGU, 1997 g. Soderzhanie

Postupatel'noe dvizhenie.

Postupatel'noe dvizhenie - eto takoe dvizhenie, pri kotorom lyuboi vydelennyi v tele otrezok ostaetsya parallel'nym samomu sebe.

Klassicheskim primerom na etu temu yavlyaetsya dvizhenie kabinok kolesa obozreniya (ris. 1.4). Etot primer naglyadno pokazyvaet, chto postupatel'noe dvizhenie - sovsem ne obyazatel'no pryamolineinoe. Ochevidno, chto chislo stepenei svobody tela v etom sluchae ravno trem, tak kak dostatochno opisat' dvizhenie kakoi-nibud' odnoi tochki tela (naprimer, tochki A na ris. 1.5). Traektorii vseh ostal'nyh tochek (naprimer, tochki V na ris. 1.5) mogut byt' polucheny putem "parallel'nogo" perenosa.


Ris. 1.4.	Ris. 1.5.

Dopustim, zakon dvizheniya tochki A zadan v vide

${\displaystyle \bf r}_{A} = {\displaystyle \bf r}_{A} \left( {\displaystyle t} \right)$

(1.2)

Togda zakon dvizheniya tochki V budet imet' vid

${\displaystyle \bf r}_{B} = {\displaystyle \bf r}_{A} + {\displaystyle \bf r}_{AB} ,$

(1.3)

gde $\bf r}_{AB$ - vektor, provedennyi ot tochki A k tochke V.

Skorost' tochki A

${\displaystyle \bf v}_{A} = {\displaystyle \frac{\displaystyle {\displaystyle d{\displaystyle \bf r}_{A} }}{\displaystyle {\displaystyle dt}}},$

(1.4)

skorost' tochki V

${\displaystyle \bf v}_{B} = {\displaystyle \frac{\displaystyle {\displaystyle d{\displaystyle \bf r}_{B} }}{\displaystyle {\displaystyle dt}}} = {\displaystyle \bf v}_{A} ,$

(1.5)

tak kak $\bf r}_{AB$ - vektor, postoyannyi po velichine (absolyutno tverdoe telo) i napravleniyu (postupatel'noe dvizhenie).

Uskoreniya tochek A i V takzhe ravny mezhdu soboi:

${\displaystyle \bf a}_{A} = {\displaystyle \frac{\displaystyle {\displaystyle d{\displaystyle \bf v}_{A} }}{\displaystyle {\displaystyle dt}}} = {\displaystyle \frac{\displaystyle {\displaystyle d{\displaystyle \bf v}_{B} }}{\displaystyle {\displaystyle dt}}} = {\displaystyle \bf a}_{B} .$

(1.6)

Takim obrazom, kinematika postupatel'nogo dvizheniya tverdogo tela v principe nichem ne otlichaetsya ot kinematiki material'noi tochki.

Vrashenie vokrug nepodvizhnoi osi.

Esli pri dvizhenii tverdogo tela kakie-libo dve ego tochki vse vremya ostayutsya nepodvizhnymi, to cherez eti tochki mozhno provesti pryamuyu, yavlyayushuyusya nepodvizhnoi os'yu vrasheniya. S takim dvizheniem my stalkivaemsya ezhednevno, otkryvaya i zakryvaya dver' v komnatu. Ochevidno, chto v etom sluchae telo obladaet lish' odnoi stepen'yu svobody, svyazannoi s uglom ego povorota vokrug osi. Pri etom vse tochki tela dvizhutsya po okruzhnostyam, lezhashim v ploskostyah, kotorye perpendikulyarny osi vrasheniya; centry okruzhnostei lezhat na etoi osi.

Sushestvenno, chto lineinye skorosti tochek, nahodyashihsya na raznom rasstoyanii ot osi vrasheniya, raznye. V etom mozhno ubedit'sya, kasayas' stal'noi provolokoi vrashayushegosya diska tochila (ris. 1.6): chem dal'she ot osi, tem dlinnee snop iskr - tem bol'she skorost' sootvetstvuyushei tochki diska. Pri etom takzhe vidno, chto iskry letyat po kasatel'noi k okruzhnosti, opisyvaemoi dannoi tochkoi diska.

Ris. 1.6.

Yasno, chto uglovoe peremeshenie vseh tochek tverdogo tela za odno i to zhe vremya budet odinakovym. Eto obstoyatel'stvo pozvolyaet vvesti obshuyu kinematicheskuyu harakteristiku - uglovuyu skorost'

$\omega = {\displaystyle \mathop {\displaystyle \lim }\limits_{\Delta t \to 0} }{\displaystyle \frac{\displaystyle {\displaystyle \Delta \varphi }}{\displaystyle {\displaystyle \Delta t}}} = {\displaystyle \frac{\displaystyle {\displaystyle d\varphi }}{\displaystyle {\displaystyle dt}}},$

(1.7)

gde $\Delta \varphi$ - ugol povorota tela za vremya $\Delta t.$

Mozhno vvesti vektor elementarnogo uglovogo peremesheniya $\Delta j,$ napravlennyi vdol' osi vrasheniya v sootvetstvii s pravilom pravogo buravchika: esli rukoyatku buravchika povorachivat' v napravlenii vrasheniya tela, to postupatel'noe peremeshenie buravchika dast napravlenie $\Delta j.$ Ustremlyaya interval vremeni $\Delta t,$ za kotoroe proizoshlo uglovoe peremeshenie $\Delta j,$ k nulyu, my poluchim vektor uglovoi skorosti

$\omega = {\displaystyle \frac{\displaystyle {\displaystyle dj}}{\displaystyle {\displaystyle dt}}},$

(1.8)

kotoryi opredelyaet, vo-pervyh, modul' uglovoi skorosti tela, vo-vtoryh, - orientaciyu osi vrasheniya v prostranstve, a v-tret'ih, - napravlenie vrasheniya tela. Sleduet podcherknut', chto $\omega$ - vektor skol'zyashii v tom smysle, chto ego nachalo mozhno sovmestit' s lyuboi tochkoi, prinadlezhashei osi vrasheniya.

Naprimer, dlya Zemli, vrashayusheisya vokrug svoei osi s zapada na vostok, vektor $\omega$ imeet napravlenie ot yuzhnogo polyusa k severnomu.

Velichina uglovoi skorosti

$\omega _{Zemli} = {\displaystyle \frac{\displaystyle {\displaystyle 2\pi }}{\displaystyle {\displaystyle 24 \cdot 3600 \ {\displaystyle n}}}} \approx 7,3 \cdot 10^{ - 5}c^{ - 1}.$

Dlya sravneniya: uglovaya skorost' orbital'nogo dvizheniya Zemli sostavlyaet

$\omega _{orb} \approx {\displaystyle \frac{\displaystyle {\displaystyle \omega _{Zemli} }}{\displaystyle {\displaystyle 365}}} \approx 2,0 \cdot 10^{ - 7}\ {\displaystyle n}^{- 1}.$

Zametim, chto period orbital'nogo vrasheniya ne kraten prodolzhitel'nosti sutok, chto sozdaet izvestnye trudnosti v postroenii kalendarya (neobhodimo vvodit' visokosnye gody i proch.)

Znaya $\omega,$ legko opredelit' lineinuyu skorost' lyuboi tochki tverdogo tela. Vvedem radius-vektor $\bf r}_{A$ nekotoroi tochki A tverdogo tela, pomestiv ego nachalo v tochku O na osi vrasheniya (ris. 1 .7). $\rho$ - vektor, provedennyi v tochku A ot osi vrasheniya, to est' perpendikulyarno osi.

Ris. 1.7.

Vektor skorosti $\bf v}_{A$ mozhno svyazat' s vektorami $\bf r}_{A$ i $\omega$ :

$\bf v}_{A} = \omega\times {\displaystyle \bf r}_{A$

(1.9)

(formula Eilera). Pri etom velichina skorosti

$v_{A} = \omega r_{A} \cdot \sin \alpha = \omega \rho$

(1.10)

Yasno, chto tochku O na osi vrasheniya mozhno vybrat' proizvol'no - znachenie $\rho = r_{A} \sin \alpha$ budet odnim i tem zhe.

Uskorenie tochki A

${\displaystyle \bf a}_{A} = {\displaystyle \frac{\displaystyle {\displaystyle d\omega}}{\displaystyle {\displaystyle dt}}}\times {\displaystyle \bf r}_{A} + \omega\times {\displaystyle \frac{\displaystyle {\displaystyle d{\displaystyle \bf r}_{A} }}{\displaystyle {\displaystyle dt}}} = \varepsilon\times {\displaystyle \bf r}_{A} + \omega\times {\displaystyle \bf v}_{A} .$

(1.11)

Zdes' $\varepsilon = {\displaystyle \frac{\displaystyle {\displaystyle d\omega}}{\displaystyle {\displaystyle dt}}}$ uglovoe uskorenie tela. Eto aksial'nyi vektor, napravlennyi v tu zhe storonu, chto i $\omega,$ esli vrashenie uskorennoe, i protivopolozhno $\omega,$ esli vrashenie zamedlennoe.

Takim obrazom, uskorenie $\bf a}_{A$ yavlyaetsya summoi dvuh velichin:

${\displaystyle \bf a}_{A} = {\displaystyle \bf a}_{\tau } + {\displaystyle \bf a}_{n} ,$

(1.12)

(ris. 1.8), prichem vse tri vektora $\bf a}_{A} , {\displaystyle \bf a}_{\tau$ i $\bf a}_{n$ lezhat v ploskosti, perpendikulyarnoi osi vrasheniya.

$a_{\tau } = \varepsilon\times r_{A} = \varepsilon \rho \tau$

(1.13)

- eto tangencial'noe uskorenie ( $\tau$ - edinichnyi vektor v napravlenii $\bf v}_{A$ ).

$\bf a}_{n} = \omega\times {\displaystyle \bf v}_{A} = \omega\times \left( {\displaystyle \omega\times {\displaystyle \bf r}_{A} } \right) = \omega ^{2}\rho {\displaystyle \bf n$

(1.14)

- eto osestremitel'noe uskorenie (n - edinichnyi vektor v napravlenii k osi vrasheniya). Eti sostavlyayushie polnogo uskoreniya horosho izvestny iz kinematiki vrashatel'nogo dvizheniya material'noi tochki.

Ris. 1.8.

Nazad| Vpered

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