Utente:Andrea And/Sandbox/3: differenze tra le versioni
Contenuto cancellato Contenuto aggiunto
←Pagina svuotata completamente |
Nessun oggetto della modifica |
||
Riga 1:
{|class="wikitable"
|-
! Descrizione || Figura || Momento di inerzia || Commento
|-
| Massa puntiforme ''m'' a distanza ''r'' dall'asse di rotazione.
|align="center"|
| <math> I = m r^2</math>
|— A point mass does not have a moment of inertia round its own axis, but by using the Parallel axis theorem a moment of inertia around a distant axis of rotation is achieved.
|-
| Two point masses, ''M'' and ''m'', with [[reduced mass]] ''<math> \mu </math>'' and separated by a distance, ''x''.
|align="center"|
| <math> I = \frac{ M m }{ M \! + \! m } x^2 = \mu x^2 </math>
|—
|-
| [[Rod (geometry)|Rod]] of length ''L'' and mass ''m'' <br>(Axis of rotation at the end of the rod)
| align="center"|[[Image:moment of inertia rod end.png]]
| <math>I_{\mathrm{end}} = \frac{m L^2}{3} \,\!</math> <ref name="serway"/>
| This expression assumes that the rod is an infinitely thin (but rigid) wire. This is also a special case of the thin rectangular plate with axis of rotation at the end of the plate, with ''h'' = ''L'' and ''w'' = ''0''.
|-
| [[Rod (geometry)|Rod]] of length ''L'' and mass ''m''
| align="center"|[[Image:moment of inertia rod center.png]]
| <math>I_{\mathrm{center}} = \frac{m L^2}{12} \,\!</math> <ref name="serway"/>
| This expression assumes that the rod is an infinitely thin (but rigid) wire. This is a special case of the thin rectangular plate with axis of rotation at the center of the plate, with ''w'' = ''L'' and ''h'' = ''0''.
|-
| Thin circular [[hoop]] of radius ''r'' and mass ''m''
| align="center"|[[Image:moment of inertia hoop.svg|170px]]
| <math>I_z = m r^2\!</math><br><math>I_x = I_y = \frac{m r^2}{2}\,\!</math>
| This is a special case of a [[torus]] for ''b''=0. (See below.), as well as of a thick-walled cylindrical tube with open ends, with ''r''<sub>1</sub>=''r''<sub>2</sub> and ''h''=0.
|-
| Thin, solid [[disk (mathematics)|disk]] of radius ''r'' and mass ''m''
|align="center"| [[Image:moment of inertia disc.svg|170px]]
| <math>I_z = \frac{m r^2}{2}\,\!</math><br><math>I_x = I_y = \frac{m r^2}{4}\,\!</math>
| This is a special case of the solid cylinder, with ''h''=0.
|-
| Thin [[cylinder (geometry)|cylindrical]] shell with open ends, of radius ''r'' and mass ''m''
|align="center"| [[Image:moment of inertia thin cylinder.png]]
| <math>I = m r^2 \,\!</math> <ref name="serway">{{cite book
|title=Physics for Scientists and Engineers, second ed.
|author=Raymond A. Serway
|page=202
|publisher=Saunders College Publishing
|isbn=0-03-004534-7
|year=1986
}}</ref>
| This expression assumes the shell thickness is negligible. It is a special case of the thick-walled cylindrical tube for ''r''<sub>1</sub>=''r<sub>2</sub>.
Also, a point mass (''m'') at the end of a rod of length ''r'' has this same moment of inertia and the value ''r'' is called the [[radius of gyration]].
|-
|Solid cylinder of radius ''r'', height ''h'' and mass ''m''
|align="center"| [[Image:moment of inertia solid cylinder.svg|170px]]
|<math>I_z = \frac{m r^2}{2}\,\!</math> <ref name="serway"/><br/><math>I_x = I_y = \frac{1}{12} m\left(3r^2+h^2\right)</math>
| This is a special case of the thick-walled cylindrical tube, with ''r''<sub>1</sub>=0. (Note: X-Y axis should be swapped for a standard right handed frame)
|-
| Thick-walled cylindrical tube with open ends, of inner radius ''r''<sub>1</sub>, outer radius ''r''<sub>2</sub>, length ''h'' and mass ''m''
|align="center"| [[Image:moment of inertia thick cylinder h.png]]
| <!-- Please read the discussion on the talk page and the cited source before changing the sign to a minus. --><math>I_z = \frac{1}{2} m\left({r_1}^2 + {r_2}^2\right)</math> <ref name="serway"/><ref>[http://www.livephysics.com/problems-and-answers/classical-mechanics/find-moment-of-inertia-of-a-uniform-hollow-cylinder.html Classical Mechanics - Moment of inertia of a uniform hollow cylinder]. LivePhysics.com. Retrieved on 2008-01-31.</ref><br><math>I_x = I_y = \frac{1}{12} m\left[3\left({r_2}^2 + {r_1}^2\right)+h^2\right]</math><br>or when defining the normalized thickness ''t<sub>n</sub>'' = ''t''/''r'' and letting ''r'' = ''r''<sub>2</sub>, <br>then <math>I_z = mr^2\left(1-t_n+\frac{1}{2}{t_n}^2\right) </math>
| With a density of ''ρ'' and the same geometry <math>I_z = \frac{1}{2} \pi\rho h\left({r_2}^4 - {r_1}^4\right)</math> <math>I_x = I_y = \frac{1}{12} \pi\rho h\left(3({r_2}^4 - {r_1}^4)+h^2({r_2}^2 - {r_1}^2)\right)</math>
|-
| [[Sphere]] (hollow) of radius ''r'' and mass ''m''
|align="center"| [[Image:moment of inertia hollow sphere.svg|170px]]
|<math>I = \frac{2 m r^2}{3}\,\!</math> <ref name="serway"/>
| A hollow sphere can be taken to be made up of two stacks of infinitesimally thin, circular hoops, where the radius differs from ''0'' to ''r'' (or a single stack, , where the radius differs from ''-r'' to ''r'').
|-
| [[ball (mathematics)|Ball]] (solid) of radius ''r'' and mass ''m''
|align="center"| [[Image:moment of inertia solid sphere.svg|170px]]
|<math>I = \frac{2 m r^2}{5}\,\!</math> <ref name="serway"/>
| A sphere can be taken to be made up of two stacks of infinitesimally thin, solid discs, where the radius differs from 0 to ''r'' (or a single stack, where the radius differs from ''-r'' to ''r'').
Also, it can be taken to be made up of infinitesimally thin, hollow spheres, where the radius differs from 0 to ''r''.
|-
| [[right angle|Right]] circular [[cone (geometry)|cone]] with radius ''r'', height ''h'' and mass ''m''
|align="center"| [[Image:moment of inertia cone.svg|120px]]
|<math>I_z = \frac{3}{10}mr^2 \,\!</math> <ref name="beer">{{cite book
|title=Vector Mechanics for Engineers, fourth ed.
|author=Ferdinand P. Beer and E. Russell Johnston, Jr
|page=911
|publisher=McGraw-Hill
|isbn=0-07-004389-2
|year=1984
}}</ref><br/><math>I_x = I_y = \frac{3}{5}m\left(\frac{r^2}{4}+h^2\right) \,\!</math> <ref name="beer"/>
|—
|-
| [[Torus]] of tube radius ''a'', cross-sectional radius ''b'' and mass ''m''.
|align="center"| [[Image:torus cycles.png|122px]]
| About a diameter: <math>\frac{1}{8}\left(4a^2 + 5b^2\right)m</math> <ref name="weisstein_torus">{{cite web
| url = http://scienceworld.wolfram.com/physics/MomentofInertiaRing.html
| title = Moment of Inertia — Ring
| author = [[Eric W. Weisstein]]
| publisher = [[Wolfram Research]]
| accessdate = 2010-03-25
}}</ref><br/>
About the vertical axis: <math>\left(a^2 + \frac{3}{4}b^2\right)m</math> <ref name="weisstein_torus"/>
|—
|-
| [[Ellipsoid]] (solid) of semiaxes ''a'', ''b'', and ''c'' with axis of rotation ''a'' and mass ''m''
| [[Image:Ellipsoid_321.png|170px]]
|<math>I_a = \frac{m (b^2+c^2)}{5}\,\!</math>
|—
|-
| Thin rectangular plate of height ''h'' and of width ''w'' and mass ''m'' <br>(Axis of rotation at the end of the plate)
|align="center"| [[Image:Recplaneoff.svg]]
|<math>I_e = \frac {m h^2}{3}+\frac {m w^2}{12}\,\!</math>
|—
|-
| Thin rectangular plate of height ''h'' and of width ''w'' and mass ''m''
|align="center"| [[Image:Recplane.svg]]
|<math>I_c = \frac {m(h^2 + w^2)}{12}\,\!</math> <ref name="serway"/>
|—
|-
| Solid [[cuboid]] of height ''h'', width ''w'', and depth ''d'', and mass ''m''
|align="center"| [[Image:moment of inertia solid rectangular prism.png]]
|<math>I_h = \frac{1}{12} m\left(w^2+d^2\right)</math><br><math>I_w = \frac{1}{12} m\left(h^2+d^2\right)</math><br><math>I_d = \frac{1}{12} m\left(h^2+w^2\right)</math>
| For a similarly oriented [[cube (geometry)|cube]] with sides of length <math>s</math>, <math>I_{CM} = \frac{m s^2}{6}\,\!</math>.
|-
| Solid [[cuboid]] of height ''D'', width ''W'', and length ''L'', and mass ''m'' with the longest diagonal as the axis.
|align="center"| [[Image: Moment of Inertia Cuboid.jpg|140px]]
|<math>I = \frac{m\left(W^2D^2+L^2D^2+L^2W^2\right)}{6\left(L^2+W^2+D^2\right)}</math>
| For a cube with sides <math>s</math>, <math>I = \frac{m s^2}{6}\,\!</math>.
|-
| Plane [[polygon]] with vertices <math>\vec{P}_{1}</math>, <math>\vec{P}_{2}</math>, <math>\vec{P}_{3}</math>, ..., <math>\vec{P}_{N}</math> and
mass <math>m</math> uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin.
|align="center"| [[Image:Polygon moment of inertia.png|130px]]
|<math>I=\frac{m}{6}\frac{\sum\limits_{n=1}^{N-1}\|\vec{P}_{n+1}\times\vec{P}_{n}\|((\vec{P}_{n+1}\cdot\vec{P}_{n+1})+(\vec{P}_{n+1}\cdot\vec{P}_{n})+(\vec{P}_{n}\cdot\vec{P}_{n}))}{\sum\limits_{n=1}^{N-1}\|\vec{P}_{n+1}\times\vec{P}_{n}\|}</math>
|This expression assumes that the polygon is [[star-shaped polygon|star-shaped]]. The vectors <math>\vec{P}_{1}</math>, <math>\vec{P}_{2}</math>, <math>\vec{P}_{3}</math>, ..., <math>\vec{P}_{N}</math> are [[position vector|position vectors]] of the vertices.
|-
| Infinite [[disk (mathematics)|disk]] with mass [[normally distributed]] on two axes around the axis of rotation
(i.e. <math> \rho(x,y) = \tfrac{m}{2\pi ab}\, e^{-((x/a)^2+(y/b)^2)/2} </math>
Where : <math> \rho(x,y) </math> is the mass-density as a function of x and y).
|align="center"| [[File:Gaussian 2D.png|130px]]
| <math>I = m (a^2+b^2) \,\!</math>
|—
|}
<!-- There is no such thing as an illegal set of axes. They may be invalid for some purposes but the x, y and z may just be labels. The right-hand rule has no bearing here.
the x-y-z axis for the solid cylinder does not follow the right-hand rule and is an illegal set of axis. -->
==See also==
*[[Parallel axis theorem]]
*[[Perpendicular axis theorem]]
*[[List of area moments of inertia]]
*[[List of moment of inertia tensors]]
==References==
<references/>
[[Category:Mechanics|Moment of inertia]]
[[Category:Physics lists|Moments of inertia]]
[[Category:Introductory physics]]
[[ar:ملحق:قائمة عزم القصور الذاتي]]
[[ko:관성모멘트의 목록]]
[[hi:जड़त्वाघूर्णों की सूची]]
[[id:Daftar momen inersia]]
[[hu:Tehetetlenségi nyomatékok listája]]
[[ms:Senarai momen inersia]]
[[pl:Lista momentów bezwładności]]
[[ru:Список моментов инерции]]
[[sq:Lista e momenteve të inercisë]]
[[sk:Vzorce na výpočet momentu zotrvačnosti]]
[[uk:Список моментів інерції]]
[[zh:轉動慣量列表]]
| |||