Kyoning teoremasi (kinetika)

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Kinetikada König teoremasi yoki Königning 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

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Teorema ikki qismga bo'lingan.

Kyoning teoremasining birinchi qismi

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








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m

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L
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{\displaystyle \displaystyle {\vec {L}}={\vec {r}}_{CoM}\times \sum \limits _{i}m_{i}{\vec {v}}_{CoM}+{\vec {L}}'={\vec {L}}_{CoM}+{\vec {L}}'}


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







L
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i


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r
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i


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m

i






v
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i


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{\displaystyle {\vec {L}}=\sum \limits _{i}({\vec {r}}_{i}\times m_{i}{\vec {v}}_{i})}


Bitta zarrachaning joylashishini quyidagicha ifodalash mumkin:








r
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i


=




r
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C
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M


+




r
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i

′



{\displaystyle {\vec {r}}_{i}={\vec {r}}_{CoM}+{\vec {r}}'_{i}}


Shunday qilib, biz bitta zarrachaning tezligini aniqlashimiz mumkin:








v
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i


=




v
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C
o
M


+




v
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i

′



{\displaystyle {\vec {v}}_{i}={\vec {v}}_{CoM}+{\vec {v}}'_{i}}


Birinchi tenglama quyidagiga keladi:







L
→



=

∑

i


(




r
→




C
o
M


+




r
→




i

′

)
×

m

i


(




v
→




C
o
M


+




v
→




i

′

)


{\displaystyle {\vec {L}}=\sum \limits _{i}({\vec {r}}_{CoM}+{\vec {r}}'_{i})\times m_{i}({\vec {v}}_{CoM}+{\vec {v}}'_{i})}








L
→



=

∑

i






r
→




i

′

×

m

i






v
→




i

′

+

(


∑

i



m

i






r
→




i

′


)

×




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C
o
M


+




r
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C
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M


×

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m

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v
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i

′

+

∑

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r
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C
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M


×

m

i






v
→




C
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M




{\displaystyle {\vec {L}}=\sum \limits _{i}{\vec {r}}'_{i}\times m_{i}{\vec {v}}'_{i}+\left(\sum \limits _{i}m_{i}{\vec {r}}'_{i}\right)\times {\vec {v}}_{CoM}+{\vec {r}}_{CoM}\times \sum \limits _{i}m_{i}{\vec {v}}'_{i}+\sum \limits _{i}{\vec {r}}_{CoM}\times m_{i}{\vec {v}}_{CoM}}


Ammo quyidagi shartlar nolga teng:





∑

i



m

i






r
→




i

′

=
0


{\displaystyle \sum \limits _{i}m_{i}{\vec {r}}'_{i}=0}






∑

i



m

i






v
→




i

′

=
0


{\displaystyle \sum \limits _{i}m_{i}{\vec {v}}'_{i}=0}


Shunday qilib, isbotlandi:







L
→



=

∑

i






r
→




i

′

×

m

i






v
→




i

′

+
M




r
→




C
o
M


×




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C
o
M




{\displaystyle {\vec {L}}=\sum \limits _{i}{\vec {r}}'_{i}\times m_{i}{\vec {v}}'_{i}+M{\vec {r}}_{CoM}\times {\vec {v}}_{CoM}}


bu yerda M - tizimning umumiy massasi .

Qattiq jism uchun

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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
=


1
2


m
⋅




N





v
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⋅




N





v
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+


1
2






N






H
¯





⋅

N







ω



R




{\displaystyle ^{N}K={\frac {1}{2}}m\cdot {^{N}\mathbf {\bar {v}} }\cdot {^{N}\mathbf {\bar {v}} }+{\frac {1}{2}}{^{N}\!\mathbf {\bar {H}} }\cdot ^{N}{\!\!\mathbf {\omega } }^{R}}


bu yerda




m



{\displaystyle {m}}

- qattiq jismning massasi;







N





v
¯






{\displaystyle {^{N}\mathbf {\bar {v}} }}

- inersial sanoq sistemasida N o'rnatilgan kuzatuvchi tomonidan ko'rilgan qattiq jismning massa markazining tezligi;







N






H
¯






{\displaystyle {^{N}\!\mathbf {\bar {H}} }}

- N inersial sanoq sistemasidagi qattiq jismning massa markaziga nisbatan burchak impulsi; va






N







ω



R




{\displaystyle ^{N}{\!\!\mathbf {\omega } }^{R}}

- N inersial sanoq sistemasiga nisbatan qattiq jismning burchak tezligi R

Foydalanilgan adabiyotlar ro'yxati

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Iqtiboslar

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uz.wikipedia.org