List of moments of inertia: Information from Answers.com

The following is a list of moments of inertia. Mass moments of inertia have units of dimension mass × length². It is the rotational analogue to mass. It should not be confused with the second moment of area (area moment of inertia), which is used in bending calculations. The following moments of inertia assume constant density throughout the object.

NOTE: The axis of rotation is taken to be through the center of mass, unless otherwise specified.

Description	Moment(s) of inertia	Comment
Thin cylindrical shell with open ends, of radius r and mass m	$I = m r^2 \,\!$	This expression assumes the shell thickness is negligible. It is a special case of the next object 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.
Thick-walled cylindrical tube with open ends, of inner radius r₁, outer radius r₂, length h and mass m	$I_z = \frac{1}{2} m\left({r_1}^2 + {r_2}^2\right)$ ^[1] $I_x = I_y = \frac{1}{12} m\left[3\left({r_2}^2 + {r_1}^2\right)+h^2\right]$ or when defining the normalized thickness t_n = 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 $I_z = \frac{1}{2} \pi\rho h\left({r_2}^4 - {r_1}^4\right)$
Solid cylinder of radius r, height h and mass m	$I_z = \frac{m r^2}{2}\,\!$ $I_x = I_y = \frac{1}{12} m\left(3r^2+h^2\right)$	This is a special case of the previous object for r₁=0.
Thin, solid disk of radius r and mass m	$I_z = \frac{m r^2}{2}\,\!$ $I_x = I_y = \frac{m r^2}{4}\,\!$	This is a special case of the previous object for h=0.
Thin circular hoop of radius r and mass m	$I_z = m r^2\!$ $I_x = I_y = \frac{m r^2}{2}\,\!$	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.
Solid sphere of radius r and mass m	$I = \frac{2 m r^2}{5}\,\!$	A sphere can be taken to be made up of a stack of infinitesimal thin, solid discs, where the radius differs from 0 to r.
Hollow sphere of radius r and mass m	$I = \frac{2 m r^2}{3}\,\!$	Similar to the solid sphere, only this time considering a stack of infinitesimal thin, circular hoops.
Ellipsoid of semiaxes a, b, and c with axis of rotation a and mass m	$I = \frac{m (b^2+c^2)}{5}\,\!$	—
Right circular cone with radius r, height h and mass m	$I_z = \frac{3}{10}mr^2 \,\!$ $I_x = I_y = \frac{3}{5}m\left(\frac{r^2}{4}+h^2\right) \,\!$	—
Solid cuboid of height h, width w, and depth d, and mass m	$I_h = \frac{1}{12} m\left(w^2+d^2\right)$ $I_w = \frac{1}{12} m\left(h^2+d^2\right)$ $I_d = \frac{1}{12} m\left(h^2+w^2\right)$	For a similarly oriented cube with sides of length $s$ , $I_{CM} = \frac{m s^2}{6}\,\!$ .
Thin rectangular plate of height h and of width w and mass m	$I_c = \frac {m(h^2 + w^2)}{12}\,\!$	—
Thin rectangular plate of height h and of width w and mass m (Axis of rotation at the end of the plate)	$I_e = \frac {m h^2}{3}+\frac {m w^2}{12}\,\!$	—
Rod of length L and mass m	$I_{\mathrm{center}} = \frac{m L^2}{12} \,\!$	This expression assumes that the rod is an infinitely thin (but rigid) wire. This is a special case of the previous object for w = L and h = 0.
Rod of length L and mass m (Axis of rotation at the end of the rod)	$I_{\mathrm{end}} = \frac{m L^2}{3} \,\!$	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: h = L and w = 0.
Torus of tube radius a, cross-sectional radius b and mass m.	About a diameter: $\frac{1}{8}\left(4a^2 + 5b^2\right)m$ About the vertical axis: $\left(a^2 + \frac{3}{4}b^2\right)m$	—
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.	$I=\frac{m}{6}\frac{\sum\limits_{n=1}^{N}\\|\vec{P}_{n+1}\times\vec{P}_{n}\\|(\vec{P}^{2}_{n+1}+\vec{P}_{n+1}\cdot\vec{P}_{n}+\vec{P}_{n}^{2})}{\sum\limits_{n=1}^{N}\\|\vec{P}_{n+1}\times\vec{P}_{n}\\|}$	—
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 : $ρ(x, y)$ is the mass-density as a function of x and y.)	$I = m (a^2+b^2) \,\!$

References

^ Classical Mechanics - Moment of inertia of a uniform hollow cylinder. LivePhysics.com. Retrieved on 2008-01-31.

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References