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Catégorie :Category: nCreator TI-Nspire
Auteur Author: brihkj
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 2.37 Ko KB
Mis en ligne Uploaded: 26/11/2024 - 05:48:38
Uploadeur Uploader: brihkj (Profil)
Téléchargements Downloads: 2
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4343409
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 2.37 Ko KB
Mis en ligne Uploaded: 26/11/2024 - 05:48:38
Uploadeur Uploader: brihkj (Profil)
Téléchargements Downloads: 2
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4343409
Description
Fichier Nspire généré sur TI-Planet.org.
Compatible OS 3.0 et ultérieurs.
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Moment of Inertia: Integral Equations for Common Physics Problems ================================================================= 1. **General Formula**: I = + r² dm - I: Moment of inertia (kg·m²) - r: Perpendicular distance from the axis of rotation to the element of mass (m) - dm: Small mass element 2. **Uniform Rod (about center)**: I = +_{-L/2}^{L/2} x² » dx - »: Linear mass density (M / L) - Result: I = (1/12) M L² 3. **Uniform Rod (about one end)**: I = +_{0}^{L} x² » dx - »: Linear mass density (M / L) - Result: I = (1/3) M L² 4. **Solid Disk (about center)**: I = + r² Ã dA - Ã: Surface mass density (M / ÀR²) - dA: Elemental area in polar coordinates (dA = 2Àr dr) - Result: I = (1/2) M R² 5. **Solid Sphere (about center)**: I = + r² Á dV - Á: Volume mass density (M / (4/3)ÀR³) - dV: Elemental volume in spherical coordinates (dV = 4Àr² dr) - Result: I = (2/5) M R² 6. **Hollow Sphere (about center)**: I = + r² Á dV - Á: Surface mass density (M / 4ÀR²) - Only surface contributes (constant radius (R)). - Result: I = (2/3) M R² 7. **Thin Ring or Hoop (about center)**: I = + r² dm - All mass at radius (R). - Result: I = M R² 8. **Thin Rectangular Plate (about center)**: I = +_{-L/2}^{L/2} +_{-w/2}^{w/2} (x² + y²) Ã dx dy - Ã: Surface mass density (M / (Lw)) - Result (for rotation about center and perpendicular to plane): I = (1/12) M (L² + w²) 9. **Cylindrical Shell (about center)**: I = + r² Á dV - (r): Constant radius of the shell. - Result: I = M R² 10. **Solid Cylinder (about central axis)**: I = + r² Ã dA - Same process as solid disk. - Result: I = (1/2) M R² 11. **Parallel Axis Theorem**: I = I_cm + M d² - Adds the moment of inertia about the center of mass ((I_cm)) and the term (M d²), where (d) is the perpendicular distance between the axes. Made with nCreator - tiplanet.org
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Compatible OS 3.0 et ultérieurs.
<<
Moment of Inertia: Integral Equations for Common Physics Problems ================================================================= 1. **General Formula**: I = + r² dm - I: Moment of inertia (kg·m²) - r: Perpendicular distance from the axis of rotation to the element of mass (m) - dm: Small mass element 2. **Uniform Rod (about center)**: I = +_{-L/2}^{L/2} x² » dx - »: Linear mass density (M / L) - Result: I = (1/12) M L² 3. **Uniform Rod (about one end)**: I = +_{0}^{L} x² » dx - »: Linear mass density (M / L) - Result: I = (1/3) M L² 4. **Solid Disk (about center)**: I = + r² Ã dA - Ã: Surface mass density (M / ÀR²) - dA: Elemental area in polar coordinates (dA = 2Àr dr) - Result: I = (1/2) M R² 5. **Solid Sphere (about center)**: I = + r² Á dV - Á: Volume mass density (M / (4/3)ÀR³) - dV: Elemental volume in spherical coordinates (dV = 4Àr² dr) - Result: I = (2/5) M R² 6. **Hollow Sphere (about center)**: I = + r² Á dV - Á: Surface mass density (M / 4ÀR²) - Only surface contributes (constant radius (R)). - Result: I = (2/3) M R² 7. **Thin Ring or Hoop (about center)**: I = + r² dm - All mass at radius (R). - Result: I = M R² 8. **Thin Rectangular Plate (about center)**: I = +_{-L/2}^{L/2} +_{-w/2}^{w/2} (x² + y²) Ã dx dy - Ã: Surface mass density (M / (Lw)) - Result (for rotation about center and perpendicular to plane): I = (1/12) M (L² + w²) 9. **Cylindrical Shell (about center)**: I = + r² Á dV - (r): Constant radius of the shell. - Result: I = M R² 10. **Solid Cylinder (about central axis)**: I = + r² Ã dA - Same process as solid disk. - Result: I = (1/2) M R² 11. **Parallel Axis Theorem**: I = I_cm + M d² - Adds the moment of inertia about the center of mass ((I_cm)) and the term (M d²), where (d) is the perpendicular distance between the axes. Made with nCreator - tiplanet.org
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