Kyonig teoremasi (kinetika)

Kinetikada Kyonig teoremasi yoki Kyonigning parchalanishi Iogann Samuel König tomonidan olingan matematik munosabat bo'lib, jismlar va zarralar tizimlarining burchak momenti va kinetik energiyalarini hisoblashda yordam beradi.

Zarrachalar tizimi uchun

Teorema ikki qismga bo'lingan.

Kyonig teoremasining birinchi qismi

Birinchi qism tizimning burchak momenti massa markazining burchak momenti va massa markaziga nisbatan zarrachalarga tatbiq etilgan burchak momentining yig'indisi sifatida ifodalaydi.

$\overrightarrow{L}={\overrightarrow{r}}_{CoM}\times \sum _i{m}_i{\overrightarrow{v}}_{CoM}+{\overrightarrow{L}}^{\prime }={\overrightarrow{L}}_{CoM}+{\overrightarrow{L}}^{\prime }$

Isbot

Koordinata boshi O bo'lgan inersial sanoq sistemasini hisobga olsak, tizimning burchak momentini quyidagicha aniqlash mumkin:

$\overrightarrow{L}=\sum _i({\overrightarrow{r}}_i\times m_i{\overrightarrow{v}}_i)$

Bitta zarrachaning joylashishini quyidagicha ifodalash mumkin:

${\overrightarrow{r}}_i={\overrightarrow{r}}_{CoM}+\overrightarrow{r}{\text{'}}_i$

Shunday qilib, biz bitta zarrachaning tezligini aniqlashimiz mumkin:

${\overrightarrow{v}}_i={\overrightarrow{v}}_{CoM}+\overrightarrow{v}{\text{'}}_i$

Birinchi tenglama quyidagiga keladi:

$\overrightarrow{L}=\sum _i({\overrightarrow{r}}_{CoM}+\overrightarrow{r}{\text{'}}_i)\times m_i({\overrightarrow{v}}_{CoM}+\overrightarrow{v}{\text{'}}_i)$

$\overrightarrow{L}=\sum _i\overrightarrow{r}{\text{'}}_i\times m_i\overrightarrow{v}{\text{'}}_i+\left(\sum _i{m}_i\overrightarrow{r}{\text{'}}_i\right)\times {\overrightarrow{v}}_{CoM}+{\overrightarrow{r}}_{CoM}\times \sum _i{m}_i\overrightarrow{v}{\text{'}}_i+\sum _i{\overrightarrow{r}}_{CoM}\times m_i{\overrightarrow{v}}_{CoM}$

Ammo quyidagi shartlar nolga teng:

$\sum _i{m}_i\overrightarrow{r}{\text{'}}_i=0$

$\sum _i{m}_i\overrightarrow{v}{\text{'}}_i=0$

Shunday qilib, isbotlandi:

$\overrightarrow{L}=\sum _i\overrightarrow{r}{\text{'}}_i\times m_i\overrightarrow{v}{\text{'}}_i+M{\overrightarrow{r}}_{CoM}\times {\overrightarrow{v}}_{CoM}$

bu yerda M - tizimning umumiy massasi .

Kyonig teoremasining ikkinchi qismi

Ikkinchi qism zarralar tizimining kinetik energiyasini alohida zarrachalarning tezligi va massa markazi nuqtai nazaridan ifodalaydi.

Xususan, unga ko'ra, zarralar tizimining kinetik energiyasi massa markazining harakati bilan bog'liq kinetik energiya va zarralarning massa markaziga nisbatan harakati bilan bog'liq kinetik energiya yig'indisidir.

$K=K^{\prime }+K_{\text{CoM}}$

Isbot

Tizimning umumiy kinetik energiyasi :

$K=\sum _i\frac{1}{2}{m}_i{v}_i^2$

Birinchi qismda bo'lgani kabi, biz tezlikni almashtiramiz:

$K=\sum _i\frac{1}{2}{m}_i|\overline{v}{\text{'}}_i+{\overline{v}}_{\text{CoM}}{|}^2$

$K=\sum _i\frac{1}{2}{m}_i(\overline{v}{\text{'}}_i+{\overline{v}}_{\text{CoM}})\cdot (\overline{v}{\text{'}}_i+{\overline{v}}_{\text{CoM}})=\sum _i\frac{1}{2}{m}_i{v{\text{'}}_i}^2+{\overline{v}}_{\text{CoM}}\cdot \sum _i{m}_i\overline{v}{\text{'}}_i+\sum _i\frac{1}{2}{m}_i{v}_{\text{CoM}}^2$

Biz bilamizki ${\overline{v}}_{CoM}\cdot \sum _i{m}_i\overline{v}{\text{'}}_i=0,$ Shunday qilib, agar biz aniqlasak:

$K^{\prime }=\sum _i\frac{1}{2}{m}_i{v{\text{'}}_i}^2$

$K_{\text{CoM}}=\sum _i\frac{1}{2}{m}_i{v}_{\text{CoM}}^2=\frac{1}{2}M{v}_{\text{CoM}}^2$

bizda qoldi:

$K=K^{\prime }+K_{\text{CoM}}$

Qattiq jism uchun

Teoremani qattiq jismlarga ham qo'llash mumkin, bunda ba'zi bir inersial sanoq sistemasida o'rnatilgan N kuzatuvchi tomonidan ko'rilganidek qattiq jismning kinetik energiyasi K, quyidagicha yozilishi mumkin:

$N^{}K=\frac{1}{2}m\cdot {}^N\overline{𝐯}\cdot {}^N\overline{𝐯}+\frac{1}{2}{}^N\overline{𝐇}{\cdot }^N{\mathit{\omega }}^R$

bu yerda $m$ - qattiq jismning massasi; ${}^N\overline{𝐯}$ - inersial sanoq sistemasida N o'rnatilgan kuzatuvchi tomonidan ko'rilgan qattiq jismning massa markazining tezligi; ${}^N\overline{𝐇}$ - N inersial sanoq sistemasidagi qattiq jismning massa markaziga nisbatan burchak impulsi; va $N^{}{\mathit{\omega }}^R$ - N inersial sanoq sistemasiga nisbatan qattiq jismning burchak tezligi R

Foydalanilgan adabiyotlar ro'yxati

• Hanno Essen: Oʻrtacha burchak tezligi (1992), Qirollik texnologiya instituti mexanika boʻlimi, S-100 44 Stokgolm, Shvetsiya.
• Samuel König (Sam. Koenigio): De universali principio æquilibrii & motus, in viva reperto, deque nexu inter vim vivam & actionem, utriusque minimo, dissertatio, Nova acta eruditorum (1751) 125-135, 162-176 ( Arxivlangan ).
• Pol A. Tipler va Gen Moska (2003), Olimlar va muhandislar uchun fizika (qog'oz): 1A jild: Mexanika (Olimlar va muhandislar uchun fizika), WH Freeman Ed.,Andoza:ISBN