Utente:Andrea And/Sandbox/3: differenze tra le versioni

Versione delle 18:00, 30 giu 2012

Versione delle 12:29, 4 lug 2012

Riga 1:

{| class="wikitable"

|-

! Descrizione || Figura || Momento di inerzia || Commento

Riga 7:

|align="center"|

| <math> I = m r^2</math>

| Un massa puntiforme non ha momento di inerzia intorno al proprio asse, ma usandouseo il Parallel axis theorem a moment of inertia around a distant axis of rotation is achieved.

|-

| Due masse puntiformi, ''M'' e ''m'', con [[massa ridotta]] ''<math> \mu </math>'' e separatoseparate da una distanza, ''x''.

|align="center"|

| <math> I = \frac{ M m }{ M \! + \! m } x^2 = \mu x^2 </math>

|—

|-

| [[Rod (geometry)|Rod]]Asta di lunghezza ''L'' e massa ''m'' <br>(Asse di rotazione alla fine del roddell'asta)

| align="center"|[[Image:moment of inertia rodAsta end.png]]

| <math>I_{\mathrm{end}} = \frac{m L^2}{3} \,\!</math>&nbsp;&nbsp;<ref name="serway"/>

| ThisQuesta expressionespressione assumesassume thatche thel'asta rodsia isun anfilo infinitelyinfinitamente thinsottile (butma rigid) wirerigido. ThisQuesto isè alsoanche aun specialcaso caseparticolare ofdella thepiastra thinrettangolare rectangularcon plateasse withdi axisrotazione ofalla rotationfine atdella thepiastra, end of the plate,e withcon ''h'' = ''L'' ande ''w'' = ''0''.

|-

| [[RodAsta (geometry)|Rod]]di of lengthlunghezza ''L'' ande massmassa ''m''

| align="center"|[[Image:moment of inertia rodAsta center.png]]

| <math>I_{\mathrm{center}} = \frac{m L^2}{12} \,\!</math>&nbsp;&nbsp;<ref name="serway"/>

| ThisQuesta expressionespressione assumesassume thatche thel'asta rodsia isun anfilo infinitelyinfinitamente thinsottile (butma rigid) wirerigido.Questo è This is a special case ofanche theun thincaso rectangularparticolare platedella withpiastra axisrettangolare ofcon rotationasse atdi therotazione centeral ofcentro thedella platepiastra, withcon ''w'' = ''L'' ande ''h'' = ''0''.

|-

| ThinCerchio circularsottile [[hoop]]di of radiusraggio ''r'' ande massmassa ''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>

| ThisQuesto isè aanche specialun casecaso ofparticolare adel [[torusToro (geometria)|toro]] forper ''b''=0. (Seevedi più in belowbasso.), as well as of a thick-walled cylindrical tube withcon open ends, withcon ''r''₁=''r''₂ ande ''h''=0.

|-

| Thin, solid [[disk (mathematics)|disk]] ofdi radiusraggio ''r'' ande massmassa ''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, withcon ''h''=0.

|-

| Thin [[cylinder (geometry)|cylindrical]] shell withcon open ends, ofdi radiusraggio ''r'' ande massmassa ''m''

|align="center"| [[Image:moment of inertia thin cylinder.png]]

| <math>I = m r^2 \,\!</math>&nbsp;&nbsp;<ref name="serway">{{citecita booklibro

|titletitolo=Physics for Scientists ande Engineers, second ed.

|authorautore=Raymond A. Serway

|pagepagina=202

|publishereditore=Saunders College Publishing

|isbn=0-03-004534-7

|yearanno=1986

}}</ref>

| This expression assumes the shell thickness is negligible. It is a special case of the thick-walled cylindrical tube for ''r''₁=''r₂.

Also, a point mass (''m'') at the end of a rodAsta of length ''r'' has this same moment of inertia ande the value ''r'' is called the [[radius of gyration]].

|-

|Solid cylinder ofdi radiusraggio ''r'', height ''h'' ande massmassa ''m''

|align="center"| [[Image:moment of inertia solid cylinder.svg|170px]]

|<math>I_z = \frac{m r^2}{2}\,\!</math>&nbsp;&nbsp;<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, withcon ''r''₁=0. (Note: X-Y axis should be swapped for a standardsteard right handedheed frame)

|-

| Thick-walled cylindrical tube withcon open ends, of inner radius ''r''₁, outer radius ''r''₂, length ''h'' ande massmassa ''m''

|align="center"| [[Image:moment of inertia thick cylinder h.png]]

| <!-- Please read the discussion on the talk pagepagina ande the citedcitad source before changing the sign to a minus. --><math>I_z = \frac{1}{2} m\left({r_1}^2 + {r_2}^2\right)</math>&nbsp;&nbsp;<ref name="serway"/><ref>[http://www.livephysics.com/problems-ande-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_n''&nbsp;=&nbsp;''t''/''r'' ande letting ''r''&nbsp;=&nbsp;''r''₂, <br>then <math>I_z = mr^2\left(1-t_n+\frac{1}{2}{t_n}^2\right) </math>

| Withcon a density of ''ρ'' ande 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) ofdi radiusraggio ''r'' ande massmassa ''m''

|align="center"| [[Image:moment of inertia hollow sphere.svg|170px]]

|<math>I = \frac{2 m r^2}{3}\,\!</math>&nbsp;&nbsp;<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) ofdi radiusraggio ''r'' ande massmassa ''m''

|align="center"| [[Image:moment of inertia solid sphere.svg|170px]]

|<math>I = \frac{2 m r^2}{5}\,\!</math>&nbsp;&nbsp;<ref name="serway"/>

Riga 68:

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]] withcon radius ''r'', height ''h'' ande massmassa ''m''

|align="center"| [[Image:moment of inertia cone.svg|120px]]

|<math>I_z = \frac{3}{10}mr^2 \,\!</math>&nbsp;&nbsp;<ref name="beer">{{citecita booklibro

|titletitolo=Vector Mechanics for Engineers, fourth ed.

|authorautore=FerdinandFerdine P. Beer ande E. Russell Johnston, Jr

|pagepagina=911

|publishereditore=McGraw-Hill

|isbn=0-07-004389-2

|yearanno=1984

}}</ref><br/><math>I_x = I_y = \frac{3}{5}m\left(\frac{r^2}{4}+h^2\right) \,\!</math>&nbsp;&nbsp;<ref name="beer"/>

|—

|-

| [[TorusToro (geometria)|toro]] of tube radius ''a'', cross-sectional radius ''b'' ande massmassa ''m''.

|align="center"| [[Image:torustoro cycles.png|122px]]

| About a diameter: <math>\frac{1}{8}\left(4a^2 + 5b^2\right)m</math>&nbsp;&nbsp;<ref name="weisstein_torusweisstein_toro">{{citecita web

| url = http://scienceworld.wolfram.com/physics/MomentofInertiaRing.html

| titletitolo = Moment of Inertia &mdash; Ring

| authorautore = [[Eric W. Weisstein]]

| publishereditore = [[Wolfram Research]]

| accessdateaccesso = 2010-03-25

}}</ref><br/>

About the vertical axis: <math>\left(a^2 + \frac{3}{4}b^2\right)m</math>&nbsp;&nbsp;<ref name="weisstein_torusweisstein_toro"/>

|—

|-

| [[Ellipsoid]] (solid) of semiaxes ''a'', ''b'', ande ''c'' withcon axis of rotation ''a'' ande massmassa ''m''

| [[Image:Ellipsoid_321.png‎|170px]]

|<math>I_a = \frac{m (b^2+c^2)}{5}\,\!</math>

|—

|-

| Thin rectangular plate ofdi heightaltezza ''h'' ande ofdi widthlarghezza ''w'' ande massmassa ''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 ofdi heightaltezza ''h'' ande ofdi widthlarghezza ''w'' ande massmassa ''m''

|align="center"| [[Image:Recplane.svg]]

|<math>I_c = \frac {m(h^2 + w^2)}{12}\,\!</math>&nbsp;&nbsp;<ref name="serway"/>

|—

|-

| Solid [[cuboid]] ofdi heightaltezza ''h'', widthlarghezza ''w'', ande depth ''d'', ande massmassa ''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]] withcon sides of length <math>s</math>, <math>I_{CM} = \frac{m s^2}{6}\,\!</math>.

|-

| Solid [[cuboid]] ofdi heightaltezza ''D'', widthlarghezza ''W'', ande length ''L'', ande massmassa ''m'' withcon 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 withcon sides <math>s</math>, <math>I = \frac{m s^2}{6}\,\!</math>.

|-

| Plane [[polygon]] withcon vertices <math>\vec{P}_{1}</math>, <math>\vec{P}_{2}</math>, <math>\vec{P}_{3}</math>, ..., <math>\vec{P}_{N}</math> ande

mass <math>m</math> uniformly distributed on its interior, rotating about an axis perpendicular to the plane ande 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>

|ThisQuesta expressionespressione assumesassume thatche 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]] withcon 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 ande y).

|align="center"| [[File:Gaussian 2D.png|130px]]

| <math>I = m (a^2+b^2) \,\!</math>

Riga 131:

|}

<!-- There is no such thing as an illegal set of axes. They may be invalid for some purposes but the x, y ande z may just be labels. The right-handhe rule has no bearing here.

the x-y-z axis for the solid cylinder does not follow the right-handhe rule ande is an illegal set of axis. -->

==See also==

Riga 140:

*[[List of moment of inertia tensors]]

==ReferencesNote==

<references/>

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[[Category:Mechanics|Moment of inertia]]

[[Category:Physics lists|Moments of inertia]]

[[Category:IntroductoryIntAstauctory physics]]

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