Momenti i Inercisë: Dallime mes rishikimesh

[redaktim i pashqyrtuar][redaktim i pashqyrtuar]
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Rreshti 60:
 
=== Percaktimi ===
Per nje object te ngurte te perbere nga <math>N</math> pika lendore <math>m_{k}</math>, [[tensor]] i momentit te inercise jepet nga
:<math>
\mathbf{I} = \begin{bmatrix}
I_{xx} & I_{xy} & I_{xz} \\
I_{yx} & I_{yy} & I_{yz} \\
I_{zx} & I_{zy} & I_{zz}
\end{bmatrix}
</math>.
 
Komponentet e saj percaktohen si
 
:<math>I_{ij} \ \stackrel{\mathrm{def}}{=}\ \sum_{k=1}^{N} m_{k} (r_k^{2}\delta_{ij} - r_{ki}r_{kj})\,\!</math>
 
ku
 
:''i'', ''j'' jane te barabarta me 1, 2, or 3 per x, y, and z, respektivisht,
:''r''<sub>''k''</sub> eshte distanca e mases ''k'' rreth pikes nga e cila llogaritet tensori, dhe
:''<math>\delta_{ij}</math>'' eshte [[delta e Kronekerit]].
 
Elementet e diagonals mund te shkruhen ne menyre me te permbledhur si
 
:<math>I_{xx} \ \stackrel{\mathrm{def}}{=}\ \sum_{k=1}^{N} m_{k} (y_{k}^{2}+z_{k}^{2}),\,\! </math>
:<math>I_{yy} \ \stackrel{\mathrm{def}}{=}\ \sum_{k=1}^{N} m_{k} (x_{k}^{2}+z_{k}^{2}),\,\!</math>
:<math>I_{zz} \ \stackrel{\mathrm{def}}{=}\ \sum_{k=1}^{N} m_{k} (x_{k}^{2}+y_{k}^{2}),\,\!</math>
 
Kurse elementet jashte diagonales, qe njihen si '''produktet e inercise''', jane
 
:<math>I_{xy} = I_{yx} \ \stackrel{\mathrm{def}}{=}\ -\sum_{k=1}^{N} m_{k} x_{k} y_{k},\,\!</math>
:<math>I_{xz} = I_{zx} \ \stackrel{\mathrm{def}}{=}\ -\sum_{k=1}^{N} m_{k} x_{k} z_{k},\,\!</math> and
:<math>I_{yz} = I_{zy} \ \stackrel{\mathrm{def}}{=}\ -\sum_{k=1}^{N} m_{k} y_{k} z_{k},\,\!</math>
 
Ketu <math>I_{xx}</math> jep momentin e inercise rreth bushtit-<math>x</math> kur objektet rrotullohen rreth aksit-x, <math>I_{xy}</math> tregon momentin e inercise rreth aksit-<math>y</math> kur objektet rrotullohen rreth aksit-<math>x</math>, e keshtu me rradhe.
 
Keto madhesi mund te pergjithesohen tek nje object me nje densitet constant ne nje menyre te ngjashme me momentin skalar te inercise. Tani marrim
:<math>\mathbf{I}=\iiint_V \rho(x,y,z)\left( r^2 \mathbf{E}_{3} - \mathbf{r}\otimes \mathbf{r}\right)\, dx\,dy\,dz,</math>
 
ku <math>\mathbf{r}\otimes \mathbf{r}</math> eshte [[produkti i jashtem]], '''E'''<sub>3</sub> eshte 3 &here; 3 [[matrica njesi]], dhe ''V'' eshte nje rajon i hapesires qe e permban komplet objektin.
 
=== Derivimi i komponenteve te tensorit ===