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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. | | The other day I woke up and realized - Now I have also been solitary for a little while and after much bullying from pals I now find myself signed up for online dating. They promised me that there are lots of sweet, regular and enjoyable [http://lukebryantickets.iczmpbangladesh.org luke bryan past tour dates] folks to meet, therefore here goes the toss!<br>My friends and fam are awesome and hanging out with them at bar gigabytes or dishes is definitely imperative. [http://minioasis.com Tickets To Luke Bryan Concert] As I locate that you can do not own a nice dialog against the sound I have never been into night clubs. I likewise got 2 undoubtedly cheeky and quite [http://adorable.org/ adorable] dogs who are consistently ready to meet up fresh individuals.<br>I attempt to stay as physically healthy as potential being at the [http://Www.Bing.com/search?q=gymnasium&form=MSNNWS&mkt=en-us&pq=gymnasium gymnasium] several times per week. I love my sports and attempt to perform or view while many a possible. I will regularly at Hawthorn suits being winter. Note: I have experienced the carnage of fumbling fits at stocktake revenue, If you really contemplated purchasing a sport I don't mind.<br><br>Visit my homepage ... [http://lukebryantickets.neodga.com luke bryan luke bryan luke bryan] |
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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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| | [[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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| | 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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| | [[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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| | [[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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| | 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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| | 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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| | 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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| |}
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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/>
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| [[Category:Mechanics|Moment of inertia]]
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| [[Category:Physics-related lists|Moments of inertia]]
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