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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク List of moments of inertia ： ウィキペディア英語版 List of moments of inertia In physics and applied mathematics, the mass moment of inertia, usually denoted by , measures the extent to which an object resists rotational acceleration about a particular axis, and is the rotational analogue to mass. Mass moments of inertia have units of dimension ML2(() × ()2). It should not be confused with the second moment of area, which is used in bending calculations. The mass moment of inertia is often also known as the rotational inertia, and sometimes as the angular mass. For simple objects with geometric symmetry, one can often determine the moment of inertia in an exact closed-form expression. Typically this occurs when the mass density is constant, but in some cases the density can vary throughout the object as well. In general, it may not be straightforward to symbolically express the moment of inertia of shapes with more complicated mass distributions and lacking symmetry. When calculating moments of inertia, it is useful to remember that it is an additive function and exploit the parallel axis and perpendicular axis theorems. This article mainly considers symmetric mass distributions, with constant density throughout the object, and the axis of rotation is taken to be through the center of mass unless otherwise specified. ==Moments of inertia== Following are scalar moments of inertia. In general, the moment of inertia is a tensor, see below. x^2 = \mu x^2 |- | Rod of length ''L'' and mass ''m'', rotating about its center. 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''. | align="center"|170px | $I\_\; \backslash ,\backslash !$  〔 |- | Rod of length ''L'' and mass ''m'', rotating about one end. 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''. | align="center"|170px | $I\_\; \backslash ,\backslash !$  〔 |- | Thin circular hoop of radius ''r'' and mass ''m''. 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''1 = ''r''2 and ''h'' = 0. | align="center"|170px | $I\_z\; =\; m\; r^2\backslash !$ $I\_x\; =\; I\_y\; =\; \backslash frac\backslash ,\backslash !$ |- | Thin, solid disk of radius ''r'' and mass ''m''. This is a special case of the solid cylinder, with ''h'' = 0. That $I\_x\; =\; I\_y\; =\; \backslash frac\backslash ,$ is a consequence of the Perpendicular axis theorem. |align="center"| 170px | $I\_z\; =\; \backslash frac\backslash ,\backslash !$ $I\_x\; =\; I\_y\; =\; \backslash frac\backslash ,\backslash !$ |- | Thin cylindrical shell with open ends, of radius ''r'' and mass ''m''. This expression assumes that 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 rod of length ''r'' has this same moment of inertia and the value ''r'' is called the radius of gyration. |align="center"| File:moment of inertia thin cylinder.png | $I\; =\; m\; r^2\; \backslash ,\backslash !$   |- |Solid cylinder of radius ''r'', height ''h'' and mass ''m''. This is a special case of the thick-walled cylindrical tube, with ''r''1 = 0. (Note: X-Y axis should be swapped for a standard right handed frame). |align="center"| 170px |$I\_z\; =\; \backslash frac\backslash ,\backslash !$  〔 $I\_x\; =\; I\_y\; =\; \backslash frac\; \backslash left(3r^2+h^2\backslash right)$ |- | Thick-walled cylindrical tube with open ends, of inner radius ''r''1, outer radius ''r''2, length ''h'' and mass ''m''. |align="center" rowspan="2" | File:moment of inertia thick cylinder h.svg | $I\_z\; =\; \backslash frac\; \backslash left(r\_1^2\; +\; r\_2^2\backslash right)\; =\; m\; r\_2^2\; \backslash left(1-t+\backslash frac\backslash right)$   〔 〔(Classical Mechanics - Moment of inertia of a uniform hollow cylinder ). LivePhysics.com. Retrieved on 2008-01-31.〕 where ''t'' = (''r2–r1'')/''r2'' is a normalized thickness ratio; $I\_x\; =\; I\_y\; =\; \backslash frac\; \backslash left(3\backslash left(^2\; +\; ^2\backslash right)+h^2\backslash right)$ |- | With a density of ''ρ'' and the same geometry | $I\_z\; =\; \backslash frac\; \backslash left(^4\; -\; ^4\backslash right)$ $I\_x\; =\; I\_y\; =\; \backslash frac\; \backslash left(3(^4\; -\; ^4)+h^2(^2\; -\; ^2)\backslash right)$ |- | Regular tetrahedron of side ''s'' and mass ''m'' |align="center"| 170px | $I\_\; =\; \backslash frac\backslash ,\backslash !$ $I\_\; =\; \backslash frac\backslash ,\backslash !$ 〔 |- | Regular octahedron of side ''s'' and mass ''m'' |align="center"| 170px | $I\_=I\_=I\_\; =\; \backslash frac\backslash ,\backslash !$ $I\_=I\_=I\_\; =\; \backslash frac\backslash ,\backslash !$ 〔 |- | Regular dodecahedron of side ''s'' and mass ''m'' |align="center"| | $I\_=I\_=I\_\; =\; \backslash frac$ $I\_=I\_=I\_\; =\; \backslash frac\backslash ,\backslash !$ (where $\backslash phi=\backslash frac$) |- | Regular icosahedron of side ''s'' and mass ''m'' |align="center"| | $I\_=I\_=I\_\; =\; \backslash frac$ $I\_=I\_=I\_\; =\; \backslash frac\backslash ,\backslash !$ |- | Hollow sphere of radius ''r'' and mass ''m''. 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''). |align="center"| 170px |$I\; =\; \backslash frac\backslash ,\backslash !$  〔 |- | Solid sphere (ball) of radius ''r'' and mass ''m''. 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''). |align="center"| 170px |$I\; =\; \backslash frac\backslash ,\backslash !$  〔 |- | Sphere (shell) of radius ''r''2, with centered spherical cavity of radius ''r''1 and mass ''m''. When the cavity radius ''r''1 = 0, the object is a solid ball (above). When ''r''1 = ''r''2, $\backslash left(\backslash frac\; \backslash left(b^2+c^2\backslash right)\backslash ,\backslash !$ $I\_b\; =\; \backslash frac\; \backslash left(a^2+c^2\backslash right)\backslash ,\backslash !$ $I\_c\; =\; \backslash frac\; \backslash left(a^2+b^2\backslash right)\backslash ,\backslash !$ |- | Thin rectangular plate of height ''h'', width ''w'' and mass ''m'' (Axis of rotation at the end of the plate) |align="center"| 170px |$I\_e\; =\; \backslash frac\; \backslash left(4h^2\; +\; w^2\backslash right)\backslash ,\backslash !$ |- | Thin rectangular plate of height ''h'', width ''w'' and mass ''m'' (Axis of rotation at the center) |align="center"| 170px |$I\_c\; =\; \backslash frac\; \backslash left(h^2\; +\; w^2\backslash right)\backslash ,\backslash !$  〔 |- | Solid cuboid of height ''h'', width ''w'', and depth ''d'', and mass ''m''. For a similarly oriented cube with sides of length $s$, $I\_\; =\; \backslash frac\backslash ,\backslash !$ |align="center"| File:moment of inertia solid rectangular prism.png |$I\_h\; =\; \backslash frac\; \backslash left(w^2+d^2\backslash right)$ $I\_w\; =\; \backslash frac\; \backslash left(d^2+h^2\backslash right)$ $I\_d\; =\; \backslash frac\; \backslash left(w^2+h^2\backslash right)$ |- | Solid cuboid of height ''D'', width ''W'', and length ''L'', and mass ''m'', rotating about the longest diagonal. For a cube with sides $s$, $I\; =\; \backslash frac\backslash ,\backslash !$. |align="center"| 140px |$I\; =\; \backslash frac\backslash left(\backslash frac\backslash right)$ |- | Triangle with vertices at the origin and at P and Q, with mass ''m'', rotating about an axis perpendicular to the plane and passing through the origin. | |$I=\backslash frac(\backslash mathbf\backslash cdot\backslash mathbf+\backslash mathbf\backslash cdot\backslash mathbf+\backslash mathbf\backslash cdot\backslash mathbf)$ |- | Plane polygon with vertices P1, P2, P3, ..., P''N'' and mass ''m'' uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin. |align="center"| 130px |$I=\backslash frac\backslash left(\backslash frac\backslash |\backslash mathbf\_\backslash times\backslash mathbf\_\backslash |\backslash left(\backslash left(\backslash mathbf\_\backslash cdot\backslash mathbf\_\backslash right)+\backslash left(\backslash mathbf\_\backslash cdot\backslash mathbf\_\backslash right)+\backslash left(\backslash mathbf\_\backslash cdot\backslash mathbf\_\backslash right)\backslash right)\}\backslash |\backslash mathbf\_\backslash times\backslash mathbf\_\backslash |\}\backslash right)$ |- | Plane regular polygon with ''n''-vertices and mass ''m'' uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin. ''a'' stands for side length. |align="center"| |$I=\backslash frac\backslash left(1\; +\; 3\backslash cot^2\backslash left(\backslash tfrac\backslash right)\backslash right)$  〔}〕 |- | Infinite disk with mass normally distributed on two axes around the axis of rotation with mass-density as a function of ''x'' and ''y'': :$\backslash rho(x,y)\; =\; \backslash tfrac\backslash ,\; e^\backslash ,,$ |align="center"| 130px | $I\; =\; m\; (a^2+b^2)\; \backslash ,\backslash !$ |- | Uniform disk about an axis perpendicular to its edge. |align="center"| | $I\; =\; \backslash frac$〔http://www.pas.rochester.edu/~ygao/phy141/Lecture15/sld010.htm〕 |} 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「List of moments of inertia」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.