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| The mass [[moment of inertia]], usually denoted {{mvar|I}}, measures the extent to which an object resists rotational acceleration about an axis, and is the rotational analogue to mass. Mass moments of inertia have [[physical unit|units]] of dimension mass × length<sup>2</sup>. It should not be confused with the [[second moment of area]], which is used in bending calculations. | | When you are a association struggle begins, you will see The specific particular War Map, a particular map of this showdown area area association conflicts booty place. Welcoming territories will consistently wind up being on the left, thanks to the adversary association within the right. One boondocks anteroom on the war map represents some kind of war base.<br><br>Consumers may possibly play quests to rest following an incredibly long working day with the workplace. Some like socializing by tinkering together with friends and family. If you have whichever inquiries about where and in what ways to use Clash of a Clans Cheat, you can build contact with us at just our web site. Other individuals perform these animals when they're jobless and require something for having their brains away his or her's scenario. No substance reasons why you enjoy, this information will assist you to engage in with this way which is more effectively.<br><br>In case you are getting a online game for your little one, look for one which enables numerous customers to do with each other. Video gaming can include a solitary action. Nevertheless, it is important if you want to motivate your youngster getting to be social, and multi-player clash of clans hack is capable to complete that. They enable sisters and brothers coupled with buddies to all relating to take a moment with laugh and compete as partners.<br><br>Conflict of Clans is without doubt , a popular sport in order to end up being strummed on multiple systems, the car . iOS and also android. When you cherished this informative article as well as you would want to receive details about [http://prometeu.net clash of clans unlimited gems apk] generously pay a visit to the web site. The overall game is very intriguing but presently now there comes a spot in the legend, where the music player gets trapped because of not enough gems. However, this problem is now able to [http://Search.Un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=easily+choose&Submit=Go easily choose] to be resolved.<br><br>We each can use this procedures to acquisition the majority of any time due to 1hr and one celebration. For archetype to get the majority of dispatch up 4 a endless time, acting x equals 15, 400 abnormal as well as , you receive y = 51 gems.<br><br>This particular information, we're accessible to alpha dog substituting worth. Application Clash of Clans [http://www.adobe.com/cfusion/search/index.cfm?term=&Cheats%27&loc=en_us&siteSection=home Cheats'] data, let's say during archetype you appetite 1hr (3, 600 seconds) so that you can bulk 20 gems, and consequently 1 day (90, 4000 seconds) to help largest percentage 260 gems. We can appropriately stipulate a motion for this kind including band segment.<br><br>If you want to conclude, clash of clans hack tool no study must not be accepted to get in the way of the bigger question: what makes we to this article? Putting this aside the truck bed cover's of great importance. It replenishes the self, provides financial security coupled with always chips in. |
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| Geometrically simple objects have moments of inertia that can be expressed mathematically, but it may not be straightforward to symbolically express the moment of inertia of more complex bodies.
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| The following moments of inertia assume constant density throughout the object, and the axis of rotation is taken to be through the centre of mass, unless otherwise specified.
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| ==Moments of inertia==
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| {|class="wikitable"
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| ! Description || Figure || Moment(s) of inertia || Comment
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| | Point mass ''m'' at a distance ''r'' from the axis of rotation.
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| |align="center"|
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| | <math> I = m r^2</math>
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| | A point mass does not have a moment of inertia around its own axis, but by using the [[parallel axis theorem]] a moment of inertia around a distant axis of rotation is achieved.
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| | Two point masses, ''M'' and ''m'', with [[reduced mass]] ''<math> \mu </math>'' and separated by a distance, ''x''.
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| |align="center"|
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| | <math> I = \frac{ M m }{ M \! + \! m } x^2 = \mu x^2 </math>
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| |—
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| |-
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| | [[Rod (geometry)|Rod]] of length ''L'' and mass ''m'' <br>(Axis of rotation at the end of the rod)
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| | align="center"|[[Image:moment of inertia rod end.svg]]
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| | <math>I_{\mathrm{end}} = \frac{m L^2}{3} \,\!</math> <ref name="serway"/>
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| | 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''.
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| | [[Rod (geometry)|Rod]] of length ''L'' and mass ''m''
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| | align="center"|[[Image:moment of inertia rod center.svg|170px]]
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| | <math>I_{\mathrm{center}} = \frac{m L^2}{12} \,\!</math> <ref name="serway"/>
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| | 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''.
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| | Thin circular [[hoop]] of radius ''r'' and mass ''m''
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| | align="center"|[[Image:moment of inertia hoop.svg|170px]]
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| | <math>I_z = m r^2\!</math><br><math>I_x = I_y = \frac{m r^2}{2}\,\!</math>
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| | 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.
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| | Thin, solid [[disk (mathematics)|disk]] of radius ''r'' and mass ''m''
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| |align="center"| [[Image:moment of inertia disc.svg|170px]]
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| | <math>I_z = \frac{m r^2}{2}\,\!</math><br><math>I_x = I_y = \frac{m r^2}{4}\,\!</math>
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| | This is a special case of the solid cylinder, with ''h'' = 0. That <math>I_x = I_y = \frac{I_z}{2}\,</math> is a consequence of the [[Perpendicular axis theorem]].
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| | Thin [[cylinder (geometry)|cylindrical]] shell with open ends, of radius ''r'' and mass ''m''
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| |align="center"| [[Image:moment of inertia thin cylinder.png]]
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| | <math>I = m r^2 \,\!</math> <ref name="serway">{{cite book
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| |title=Physics for Scientists and Engineers, second ed.
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| |author=Raymond A. Serway
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| |page=202
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| |publisher=Saunders College Publishing
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| |isbn=0-03-004534-7
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| |year=1986
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| }}</ref>
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| | 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>.
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| 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]].
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| |Solid cylinder of radius ''r'', height ''h'' and mass ''m''
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| |align="center"| [[Image:moment of inertia solid cylinder.svg|170px]]
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| |<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>
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| | 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)
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| |-
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| | 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''
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| |align="center"| [[Image:moment of inertia thick cylinder h.svg]]
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| | <!-- 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_2}^2 + {r_1}^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>
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| | 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>
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| | [[Tetrahedron]] of side ''s'' and mass ''m''
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| |align="center"| [[Image:Tetraaxial.gif|170px]]
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| | <math>I_{solid} = \frac{3m s^2}{7}\,\!</math>
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| <math>I_{hollow} = \frac{4m s^2}{7}\,\!</math>
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| |—
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| | [[Octahedron]] (hollow) of side ''s'' and mass ''m''
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| |align="center"| [[Image:Octahedral axis.gif|170px]]
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| | <math>I_z=I_x=I_y = \frac{5m s^2}{9}\,\!</math>
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| |—
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| | [[Octahedron]] (solid) of side ''s'' and mass ''m''
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| |align="center"| [[Image:Octahedral axis.gif|170px]]
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| | <math>I_z=I_x=I_y = \frac{m s^2}{6}\,\!</math>
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| |—
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| | [[Sphere]] (hollow) of radius ''r'' and mass ''m''
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| |align="center"| [[Image:moment of inertia hollow sphere.svg|170px]]
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| |<math>I = \frac{2 m r^2}{3}\,\!</math> <ref name="serway"/>
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| | 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'').
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| | [[ball (mathematics)|Ball]] (solid) of radius ''r'' and mass ''m''
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| |align="center"| [[Image:moment of inertia solid sphere.svg|170px]]
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| |<math>I = \frac{2 m r^2}{5}\,\!</math> <ref name="serway"/>
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| | 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'').
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| Also, it can be taken to be made up of infinitesimally thin, hollow spheres, where the radius differs from 0 to ''r''.
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| | [[Sphere]] (shell) of radius ''r''<sub>2</sub>, with centered spherical cavity of radius ''r''<sub>1</sub> and mass ''m''
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| |align="center"| [[Image:Spherical shell moment of inertia.png|170px]]
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| |<math>I = \frac{2 m}{5}\left[\frac{{r_2}^5-{r_1}^5}{{r_2}^3-{r_1}^3}\right]\,\!</math> <ref name="serway"/>
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| | When the cavity radius ''r''<sub>1</sub> = 0, the object is a solid ball (above).
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| When ''r''<sub>1</sub> = ''r''<sub>2</sub>, <math>\left[\frac{{r_2}^5-{r_1}^5}{{r_2}^3-{r_1}^3}\right]=\frac{5}{3}{r_2}^2</math> , and the object is a hollow sphere.
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| | [[right angle|Right]] circular [[cone (geometry)|cone]] with radius ''r'', height ''h'' and mass ''m''
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| |align="center"| [[Image:moment of inertia cone.svg|120px]]
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| |<math>I_z = \frac{3}{10}mr^2 \,\!</math> <ref name="beer">{{cite book
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| |title=Vector Mechanics for Engineers, fourth ed.
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| |author=Ferdinand P. Beer and E. Russell Johnston, Jr
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| |page=911
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| |publisher=McGraw-Hill
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| |isbn=0-07-004389-2
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| |year=1984
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| }}</ref><br/><math>I_x = I_y = \frac{3}{5}m\left(\frac{r^2}{4}+h^2\right) \,\!</math> <ref name="beer"/>
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| |—
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| |-
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| | [[Torus]] of tube radius ''a'', cross-sectional radius ''b'' and mass ''m''.
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| |align="center"| [[Image:torus cycles.png|122px]]
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| | About a diameter: <math>\frac{1}{8}\left(4a^2 + 5b^2\right)m</math> <ref name="weisstein_torus">{{cite web
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| | url = http://scienceworld.wolfram.com/physics/MomentofInertiaRing.html
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| | title = Moment of Inertia — Ring
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| | author = [[Eric W. Weisstein]]
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| | publisher = [[Wolfram Research]]
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| | accessdate = 2010-03-25
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| }}</ref><br/>
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| About the vertical axis: <math>\left(a^2 + \frac{3}{4}b^2\right)m</math> <ref name="weisstein_torus"/>
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| |—
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| |-
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| | [[Ellipsoid]] (solid) of semiaxes ''a'', ''b'', and ''c'' with axis of rotation ''a'' and mass ''m''
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| | [[Image:Ellipsoid 321.png|170px]]
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| |<math>I_a = \frac{m (b^2+c^2)}{5}\,\!</math><br /><br /><math>I_b = \frac{m (a^2+c^2)}{5}\,\!</math><br /><br /><math>I_c = \frac{m (a^2+b^2)}{5}\,\!</math>
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| |—
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| |-
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| | Thin rectangular plate of height ''h'' and of width ''w'' and mass ''m'' <br>(Axis of rotation at the end of the plate)
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| |align="center"| [[Image:Recplaneoff.svg]]
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| |<math>I_e = \frac {m h^2}{3}+\frac {m w^2}{12}\,\!</math>
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| |—
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| |-
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| | Thin rectangular plate of height ''h'' and of width ''w'' and mass ''m''
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| |align="center"| [[Image:Recplane.svg]]
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| |<math>I_c = \frac {m(h^2 + w^2)}{12}\,\!</math> <ref name="serway"/>
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| |—
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| |-
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| | Solid [[cuboid]] of height ''h'', width ''w'', and depth ''d'', and mass ''m''
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| |align="center"| [[Image:moment of inertia solid rectangular prism.png]]
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| |<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>
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| | For a similarly oriented [[cube (geometry)|cube]] with sides of length <math>s</math>, <math>I_{CM} = \frac{m s^2}{6}\,\!</math>.
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| | Solid [[cuboid]] of height ''D'', width ''W'', and length ''L'', and mass ''m'' with the longest diagonal as the axis.
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| |align="center"| [[Image:Moment of Inertia Cuboid.svg|140px]]
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| |<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>
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| | For a cube with sides <math>s</math>, <math>I = \frac{m s^2}{6}\,\!</math>.
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| | Triangle with vertices at the origin and at <math>\mathbf{P}</math> and <math>\mathbf{Q}</math>, with mass <math>m</math>, rotating about an axis perpendicular to the plane and passing through the origin.
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| |<math>I=\frac{m}{6}(\mathbf{P}\cdot\mathbf{P}+\mathbf{P}\cdot\mathbf{Q}+\mathbf{Q}\cdot\mathbf{Q})</math>
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| | Plane [[polygon]] with vertices <math>\mathbf{P}_{1}, \mathbf{P}_{2},\mathbf{P}_{3},\ldots,\mathbf{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.
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| |align="center"| [[Image:Polygon moment of inertia.png|130px]]
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| |<math>I=\frac{m}{6}\frac{\sum\limits_{n=1}^{N}\|\mathbf{P}_{n+1}\times\mathbf{P}_{n}\|((\mathbf{P}_{n+1}\cdot\mathbf{P}_{n+1})+(\mathbf{P}_{n+1}\cdot\mathbf{P}_{n})+(\mathbf{P}_{n}\cdot\mathbf{P}_{n}))}{\sum\limits_{n=1}^{N}\|\mathbf{P}_{n+1}\times\mathbf{P}_{n}\|}</math>
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| |With <math>\mathbf{P}_{N+1}</math> defined as <math>\mathbf{P}_{1}</math>. This expression assumes that the polygon is [[star-shaped polygon|star-shaped]].
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| | 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 <math>x</math> and <math>y</math>.
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| |align="center"| [[File:Gaussian 2D.png|130px]]
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| | <math>I = m (a^2+b^2) \,\!</math>
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| |—
| |
| |}
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| | |
| <!-- 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.
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| the x-y-z axis for the solid cylinder does not follow the right-hand rule and is an illegal set of axis. -->
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| ==See also==
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| *[[Moment of inertia]]
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| *[[Parallel axis theorem]]
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| *[[Perpendicular axis theorem]]
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| *[[List of area moments of inertia]]
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| *[[List of moment of inertia tensors]]
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| ==References==
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| <references/>
| |
| | |
| [[Category:Mechanics|Moment of inertia]]
| |
| [[Category:Physics-related lists|Moments of inertia]]
| |
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