List of moments of inertia

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In physics and applied mathematics, 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] × [length]2. It should not be confused with the second moment of area, which is used in bending calculations.

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 helpful to exploit the properties of the moment of inertia, namely it is an additive quantity and the parallel axis theorem, and perpendicular axis theorem.

This article considers mainly 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[edit]

Following are scalar moments of inertia. In general, the moment of inertia is a tensor, see below.

DescriptionFigureMoment(s) of inertia
Point mass m at a distance r from the axis of rotation.

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.

 I = m r^2
Two point masses, M and m, with reduced mass μ 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.

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.

Moment of inertia rod end.svgI_{\mathrm{end}} = \frac{m L^2}{3} \,\!  [1]
Rod of length L and mass m.

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.

Moment of inertia rod center.svgI_{\mathrm{center}} = \frac{m L^2}{12} \,\!  [1]
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 r1 = r2 and h = 0.

Moment of inertia hoop.svgI_z = m r^2\!
I_x = I_y = \frac{m r^2}{2}\,\!
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 = \frac{I_z}{2}\, is a consequence of the Perpendicular axis theorem.

Moment of inertia disc.svgI_z = \frac{m r^2}{2}\,\!
I_x = I_y = \frac{m r^2}{4}\,\!
Thin cylindrical shell with open ends, of radius r and mass m.

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.
Moment of inertia thin cylinder.pngI = m r^2 \,\!  [1]
Solid cylinder of radius r, height h and mass m.

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).

Moment 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)
Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass m.

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)

Moment 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)
Tetrahedron of side s and mass mTetraaxial.gifI_{solid} = \frac{m s^2}{20}\,\!

I_{hollow} = \frac{m s^2}{12}\,\!

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}{5}\,\!
Sphere (hollow) 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).

Moment of inertia hollow sphere.svgI = \frac{2 m r^2}{3}\,\!  [1]
Ball (solid) 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). Also, it can be taken to be made up of infinitesimally thin, hollow spheres, where the radius differs from 0 to r.

Moment of inertia solid sphere.svgI = \frac{2 m r^2}{5}\,\!  [1]
Sphere (shell) of radius r2, with centered spherical cavity of radius r1 and mass m.

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.

Spherical 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]
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.

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

Torus cycles.pngAbout a diameter: \frac{1}{8}\left(4a^2 + 5b^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 m.

For a similarly oriented cube with sides of length s, I_{CM} = \frac{m s^2}{6}\,\!.

Moment 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)
Solid cuboid of height D, width W, and length L, and mass m with the longest diagonal as the axis.

For a cube with sides s, I = \frac{m s^2}{6}\,\!.

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)}
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=\frac{m}{6}(\mathbf{P}\cdot\mathbf{P}+\mathbf{P}\cdot\mathbf{Q}+\mathbf{Q}\cdot\mathbf{Q})
Plane polygon with vertices P1, P2, P3, ..., PN 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}\|\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}\|}
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 origina stands for side length.I=\frac{ma^2}{24}[1 + 3\cot^2(\tfrac{\pi}{n})]  [5]
Infinite disk with mass normally distributed on two axes around the axis of rotation with mass-density as a function of x and y:
\rho(x,y) = \tfrac{m}{2\pi ab}\, e^{-((x/a)^2+(y/b)^2)/2}\,,
Gaussian 2D.pngI = m (a^2+b^2) \,\!

List of 3D inertia tensors[edit]

This list of moment of inertia tensors is given for principal axes of each object.

To obtain the scalar moments of inertia I above, the tensor moment of inertia I is projected along some axis defined by a unit vector n according to the formula:

\mathbf{n}\cdot\mathbf{I}\cdot\mathbf{n}\equiv n_i I_{ij} n_j\,,

where the dots indicate tensor contraction and we have used the Einstein summation convention. In the above table, n would be the unit Cartesian basis ex, ey, ez to obtain Ix, Iy, Iz respectively.

DescriptionFigureMoment of inertia tensor
Solid sphere of radius r and mass mMoment of inertia solid sphere.svg I = \begin{bmatrix}   \frac{2}{5} m r^2 & 0 & 0 \\   0 & \frac{2}{5} m r^2 & 0 \\    0 & 0 & \frac{2}{5} m r^2 \end{bmatrix}
Hollow sphere of radius r and mass mMoment of inertia hollow sphere.svg

 I = \begin{bmatrix}   \frac{2}{3} m r^2  & 0 & 0 \\   0 & \frac{2}{3} m r^2 & 0 \\    0 & 0 & \frac{2}{3} m r^2 \end{bmatrix}

Solid ellipsoid of semi-axes a, b, c and mass mEllipsoide.png I = \begin{bmatrix}   \frac{1}{5} m (b^2+c^2) & 0 & 0 \\   0 & \frac{1}{5} m (a^2+c^2) & 0 \\    0 & 0 & \frac{1}{5} m (a^2+b^2) \end{bmatrix}
Right circular cone with radius r, height h and mass m, about the apexMoment of inertia cone.svg I = \begin{bmatrix}   \frac{3}{5} m h^2 + \frac{3}{20} m r^2  & 0 & 0 \\   0 & \frac{3}{5} m h^2 + \frac{3}{20} m r^2 & 0 \\    0 & 0 & \frac{3}{10} m r^2 \end{bmatrix}
Solid cuboid of width w, height h, depth d, and mass m
180x
 I = \begin{bmatrix}   \frac{1}{12} m (h^2 + d^2) & 0 & 0 \\   0 & \frac{1}{12} m (w^2 + d^2) & 0 \\    0 & 0 & \frac{1}{12} m (w^2 + h^2) \end{bmatrix}
Slender rod along y-axis of length l and mass m about end
Moment of inertia rod end.svg

 I = \begin{bmatrix}   \frac{1}{3} m l^2  & 0 & 0 \\   0 & 0 & 0 \\   0 & 0 & \frac{1}{3} m l^2  \end{bmatrix}

Slender rod along y-axis of length l and mass m about center
Moment of inertia rod center.svg

 I = \begin{bmatrix}   \frac{1}{12} m l^2  & 0 & 0 \\   0 & 0 & 0 \\   0 & 0 & \frac{1}{12} m l^2 \end{bmatrix}

Solid cylinder of radius r, height h and mass mMoment of inertia solid cylinder.svg

 I = \begin{bmatrix}   \frac{1}{12} m (3r^2+h^2)  & 0 & 0 \\   0 & \frac{1}{12} m (3r^2+h^2) & 0 \\    0 & 0 & \frac{1}{2} m r^2\end{bmatrix}

Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass mMoment of inertia thick cylinder h.svg

 I = \begin{bmatrix}   \frac{1}{12} m (3({r_1}^2 + {r_2}^2)+h^2)  & 0 & 0 \\   0 & \frac{1}{12} m (3({r_1}^2 + {r_2}^2)+h^2) & 0 \\    0 & 0 & \frac{1}{2} m ({r_1}^2 + {r_2}^2)\end{bmatrix}

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. 
  5. ^ Karel Rektorys (1994). Survey of Applicable Mathematics, second ed., Vol. II. Kluwer Academic Publisher. p. 942. ISBN 0-7923-0681-3. 

External links[edit]