qstruct:teoria:qsection:integrali_rotazione
Differenze
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qstruct:teoria:qsection:integrali_rotazione [2014/11/26 13:31] mickele |
qstruct:teoria:qsection:integrali_rotazione [2021/06/13 13:10] (versione attuale) |
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| Le formule che ci permettono di calcolare i momenti di terzo ordine in caro di rotazione di angolo $\theta$ sono | Le formule che ci permettono di calcolare i momenti di terzo ordine in caro di rotazione di angolo $\theta$ sono | ||
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| + | $$\iint \limits_{S} y_{\odot}^3 \; \mathrm{d}y \mathrm{d}z = \\ | ||
| + | \cos^3 \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z + 3 \cos^2 \theta \, \sin \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z + 3 \cos \theta \, \sin^2 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \sin^3 \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z $$ | ||
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| + | $$\iint \limits_{S} z_{\odot}^3 \; \mathrm{d}y \mathrm{d}z = \\ | ||
| + | - \sin^3 \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z + 3 \sin^2 \theta \, \cos \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z - 3 \sin \theta \, \cos^2 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \cos^3 \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z $$ | ||
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| + | $$\iint \limits_{S} y_{\odot}^2 \, z_{\odot} \; \mathrm{d}y \mathrm{d}z = \\ | ||
| + | - \sin \theta \, \cos^2 \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z - 2 \sin^2 \theta \, \cos \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z - \sin^3 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z \\ | ||
| + | + \cos^3 \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z + 2 \sin \theta \, \cos^2 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \sin^2 \theta \, \cos \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z $$ | ||
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| + | $$\iint \limits_{S} y_{\odot} \, z_{\odot}^2 \; \mathrm{d}y \mathrm{d}z = \\ | ||
| + | \sin^2 \theta \, \cos \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z - 2 \sin \theta \, \cos^2 \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z + \cos^3 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z \\ | ||
| + | + \sin^3 \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z - 2 \sin^2 \theta \, \cos \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \sin \theta \, \cos^2 \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z $$ | ||
| Si riporta di seguito il dettaglio dei calcoli | Si riporta di seguito il dettaglio dei calcoli | ||
| $$\iint \limits_{S} y_{\odot}^3 \; \mathrm{d}y \mathrm{d}z = \iint \limits_{S} \left( \cos \theta \, y + \sin \theta \, z \right)^3 \; \mathrm{d}y \mathrm{d}z = \\ | $$\iint \limits_{S} y_{\odot}^3 \; \mathrm{d}y \mathrm{d}z = \iint \limits_{S} \left( \cos \theta \, y + \sin \theta \, z \right)^3 \; \mathrm{d}y \mathrm{d}z = \\ | ||
| - | $$ | + | \iint \limits_{S} \left( \cos^3 \theta \, y^3 + 3 \cos^2 \theta \, \sin \theta \, y^2 \, z + 3 \cos \theta \, \sin^2 \theta \, y \, z^2 + \sin^3 \theta \, z^3 \right) \; \mathrm{d}y \mathrm{d}z = \\ |
| + | \cos^3 \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z + 3 \cos^2 \theta \, \sin \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z + 3 \cos \theta \, \sin^2 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \sin^3 \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z | ||
| $$\iint \limits_{S} z_{\odot}^3 \; \mathrm{d}y \mathrm{d}z = \iint \limits_{S} \left( - \sin \theta \, y + \cos \theta \, z \right)^3 \; \mathrm{d}y \mathrm{d}z = \\ | $$\iint \limits_{S} z_{\odot}^3 \; \mathrm{d}y \mathrm{d}z = \iint \limits_{S} \left( - \sin \theta \, y + \cos \theta \, z \right)^3 \; \mathrm{d}y \mathrm{d}z = \\ | ||
| - | $$ | + | \iint \limits_{S} \left( - \sin^3 \theta \, y^3 + 3 \sin^2 \theta \, \cos \theta \, y^2 \, z - 3 \sin \theta \, \cos^2 \theta \, y \, z^2 + \cos^3 \theta \, z^3 \right) \; \mathrm{d}y \mathrm{d}z = \\ |
| + | - \sin^3 \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z + 3 \sin^2 \theta \, \cos \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z - 3 \sin \theta \, \cos^2 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \cos^3 \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z | ||
| $$\iint \limits_{S} y_{\odot}^2 \, z_{\odot} \; \mathrm{d}y \mathrm{d}z = \iint \limits_{S} \left( \cos \theta \, y + \sin \theta \, z \right)^2 \, \left( - \sin \theta \, y + \cos \theta \, z \right) \; \mathrm{d}y \mathrm{d}z = \\ | $$\iint \limits_{S} y_{\odot}^2 \, z_{\odot} \; \mathrm{d}y \mathrm{d}z = \iint \limits_{S} \left( \cos \theta \, y + \sin \theta \, z \right)^2 \, \left( - \sin \theta \, y + \cos \theta \, z \right) \; \mathrm{d}y \mathrm{d}z = \\ | ||
| - | $$ | + | \iint \limits_{S} \left( \cos^2 \theta \, y^2 + 2 \sin \theta \, \cos \theta \, y \, z + \sin^2 \theta \, z^2 \right) \, \left( - \sin \theta \, y + \cos \theta \, z \right) \; \mathrm{d}y \mathrm{d}z = \\ |
| + | \iint \limits_{S} \left( - \sin \theta \, \cos^2 \theta \, y^3 - 2 \sin^2 \theta \, \cos \theta \, y^2 \, z - \sin^3 \theta \, y \, z^2 + \cos^3 \theta \, y^2 \, z + 2 \sin \theta \, \cos^2 \theta \, y \, z^2 + \sin^2 \theta \, \cos \theta \, z^3 \right) \; \mathrm{d}y \mathrm{d}z = \\ | ||
| + | - \sin \theta \, \cos^2 \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z - 2 \sin^2 \theta \, \cos \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z - \sin^3 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z \\ | ||
| + | + \cos^3 \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z + 2 \sin \theta \, \cos^2 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \sin^2 \theta \, \cos \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z | ||
| $$\iint \limits_{S} y_{\odot} \, z_{\odot}^2 \; \mathrm{d}y \mathrm{d}z = \iint \limits_{S} \left( \cos \theta \, y + \sin \theta \, z \right) \, \left( - \sin \theta \, y + \cos \theta \, z \right)^2 \; \mathrm{d}y \mathrm{d}z = \\ | $$\iint \limits_{S} y_{\odot} \, z_{\odot}^2 \; \mathrm{d}y \mathrm{d}z = \iint \limits_{S} \left( \cos \theta \, y + \sin \theta \, z \right) \, \left( - \sin \theta \, y + \cos \theta \, z \right)^2 \; \mathrm{d}y \mathrm{d}z = \\ | ||
| - | $$ | + | \iint \limits_{S} \left( \cos \theta \, y + \sin \theta \, z \right) \, \left( \sin^2 \theta \, y^2 - 2 \sin \theta \, \cos \theta \, y \, z + \cos^2 \theta \, z^2 \right) \; \mathrm{d}y \mathrm{d}z = \\ |
| + | \iint \limits_{S} \left( \sin^2 \theta \, \cos \theta \, y^3 - 2 \sin \theta \, \cos^2 \theta \, y^2 \, z + \cos^3 \theta \, y \, z^2 + \sin^3 \theta \, y^2 \, z - 2 \sin^2 \theta \, \cos \theta \, y \, z^2 + \sin \theta \, \cos^2 \theta \, z^3 \right) \; \mathrm{d}y \mathrm{d}z = \\ | ||
| + | \sin^2 \theta \, \cos \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z - 2 \sin \theta \, \cos^2 \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z + \cos^3 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z \\ | ||
| + | + \sin^3 \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z - 2 \sin^2 \theta \, \cos \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \sin \theta \, \cos^2 \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z | ||
qstruct/teoria/qsection/integrali_rotazione.1417005086.txt.gz · Ultima modifica: 2021/06/13 13:10 (modifica esterna)
