List of moments of inertia

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The mass moment of inertia, usually denoted 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 units of dimension mass × length2. It should not be confused with the second moment of area, which is used in bending calculations.

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.

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.

Moments of inertia[edit]

DescriptionFigureMoment(s) of inertiaComment
Point mass m at a distance r from the axis of rotation. I = m r^2A 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.
Two point masses, M and m, with reduced mass  \mu and separated by a distance, x. I = \frac{ M m }{ M \! + \! m } x^2 = \mu x^2
Rod of length L and mass m
(Axis of rotation at the end of the rod)
Moment of inertia rod end.svgI_{\mathrm{end}} = \frac{m L^2}{3} \,\!  [1]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 of length L and mass mMoment of inertia rod center.svgI_{\mathrm{center}} = \frac{m L^2}{12} \,\!  [1]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 mMoment of inertia hoop.svgI_z = m r^2\!
I_x = I_y = \frac{m r^2}{2}\,\!
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 r1 = r2 and h = 0.
Thin, solid disk of radius r and mass mMoment of inertia disc.svgI_z = \frac{m r^2}{2}\,\!
I_x = I_y = \frac{m r^2}{4}\,\!
This is a special case of the solid cylinder, with h = 0. That I_x = I_y = \frac{I_z}{2}\, is a consequence of the Perpendicular axis theorem.
Thin cylindrical shell with open ends, of radius r and mass mMoment of inertia thin cylinder.pngI = m r^2 \,\!  [1]This expression assumes the shell thickness is negligible. It is a special case of the thick-walled cylindrical tube for r1 = 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.

Solid cylinder of radius r, height h and mass mMoment of inertia solid cylinder.svgI_z = \frac{m r^2}{2}\,\!  [1]
I_x = I_y = \frac{1}{12} m\left(3r^2+h^2\right)
This is a special case of the thick-walled cylindrical tube, with r1 = 0. (Note: X-Y axis should be swapped for a standard right handed frame)
Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass mMoment of inertia thick cylinder h.svgI_z = \frac{1}{2} m\left({r_2}^2 + {r_1}^2\right)  [1][2]
I_x = I_y = \frac{1}{12} m\left[3\left({r_2}^2 + {r_1}^2\right)+h^2\right]
or when defining the normalized thickness tn = t/r and letting r = r2,
then I_z = mr^2\left(1-t_n+\frac{1}{2}{t_n}^2\right)
With a density of ρ and the same geometry I_z = \frac{1}{2} \pi\rho h\left({r_2}^4 - {r_1}^4\right) 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)
Tetrahedron of side s and mass mTetraaxial.gifI_{solid} = \frac{3m s^2}{7}\,\!

I_{hollow} = \frac{4m s^2}{7}\,\!

Octahedron (hollow) of side s and mass mOctahedral axis.gifI_z=I_x=I_y = \frac{5m s^2}{9}\,\!
Octahedron (solid) of side s and mass mOctahedral axis.gifI_z=I_x=I_y = \frac{m s^2}{6}\,\!
Sphere (hollow) of radius r and mass mMoment of inertia hollow sphere.svgI = \frac{2 m r^2}{3}\,\!  [1]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 (solid) of radius r and mass mMoment of inertia solid sphere.svgI = \frac{2 m r^2}{5}\,\!  [1]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.

Sphere (shell) of radius r2, with centered spherical cavity of radius r1 and mass mSpherical shell moment of inertia.pngI = \frac{2 m}{5}\left[\frac{{r_2}^5-{r_1}^5}{{r_2}^3-{r_1}^3}\right]\,\!  [1]When the cavity radius r1 = 0, the object is a solid ball (above).

When r1 = r2, \left[\frac{{r_2}^5-{r_1}^5}{{r_2}^3-{r_1}^3}\right]=\frac{5}{3}{r_2}^2 , and the object is a hollow sphere.

Right circular cone with radius r, height h and mass mMoment of inertia cone.svgI_z = \frac{3}{10}mr^2 \,\!  [3]
I_x = I_y = \frac{3}{5}m\left(\frac{r^2}{4}+h^2\right) \,\!  [3]
Torus of tube radius a, cross-sectional radius b and mass m.Torus cycles.pngAbout a diameter: \frac{1}{8}\left(4a^2 + 5b^2\right)m  [4]

About the vertical axis: \left(a^2 + \frac{3}{4}b^2\right)m  [4]

Ellipsoid (solid) of semiaxes a, b, and c with axis of rotation a and mass mEllipsoid 321.pngI_a = \frac{m (b^2+c^2)}{5}\,\!

I_b = \frac{m (a^2+c^2)}{5}\,\!

I_c = \frac{m (a^2+b^2)}{5}\,\!
Thin rectangular plate of height h and of width w and mass m
(Axis of rotation at the end of the plate)
Recplaneoff.svgI_e = \frac {m h^2}{3}+\frac {m w^2}{12}\,\!
Thin rectangular plate of height h and of width w and mass mRecplane.svgI_c = \frac {m(h^2 + w^2)}{12}\,\!  [1]
Solid cuboid of height h, width w, and depth d, and mass mMoment of inertia solid rectangular prism.pngI_h = \frac{1}{12} m\left(w^2+d^2\right)
I_w = \frac{1}{12} m\left(h^2+d^2\right)
I_d = \frac{1}{12} m\left(h^2+w^2\right)
For a similarly oriented cube with sides of length s, I_{CM} = \frac{m s^2}{6}\,\!.
Solid cuboid of height D, width W, and length L, and mass m with the longest diagonal as the axis.Moment of Inertia Cuboid.svgI =  \frac{m\left(W^2D^2+L^2D^2+L^2W^2\right)}{6\left(L^2+W^2+D^2\right)}For a cube with sides s, I = \frac{m s^2}{6}\,\!.
Plane polygon with vertices \vec{P}_{1}, \vec{P}_{2}, \vec{P}_{3}, ..., \vec{P}_{N} and

mass m uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin.

Polygon moment of inertia.pngI=\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}\|}This expression assumes that the polygon is star-shaped. The vectors \vec{P}_{1}, \vec{P}_{2}, \vec{P}_{3}, ..., \vec{P}_{N} are position vectors of the vertices.
Infinite disk with mass normally distributed on two axes around the axis of rotation

(i.e.  \rho(x,y) = \tfrac{m}{2\pi ab}\, e^{-((x/a)^2+(y/b)^2)/2} Where :  \rho(x,y) is the mass-density as a function of x and y).

Gaussian 2D.pngI = m (a^2+b^2) \,\!


See also[edit]

References[edit]

  1. ^ a b c d e f g h i Raymond A. Serway (1986). Physics for Scientists and Engineers, second ed. Saunders College Publishing. p. 202. ISBN 0-03-004534-7. 
  2. ^ Classical Mechanics - Moment of inertia of a uniform hollow cylinder. LivePhysics.com. Retrieved on 2008-01-31.
  3. ^ a b Ferdinand P. Beer and E. Russell Johnston, Jr (1984). Vector Mechanics for Engineers, fourth ed. McGraw-Hill. p. 911. ISBN 0-07-004389-2. 
  4. ^ a b Eric W. Weisstein. "Moment of Inertia — Ring". Wolfram Research. Retrieved 2010-03-25.