Moment_of_inertia_of_a_uniform_disc

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.

Description	Figure	Moment(s) of inertia	Comment
Thin cylindrical shell with open ends, of radius r and mass m		$I\; =\; m\; r^2\; \backslash ,\backslash !$	This expression assumes the shell thickness is negligible. It is a special case of the next object for r1=r2.
Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h, density ρ and mass m		$I\_z\; =\; \backslash frac\{1\}\{2\}\; m\backslash left(\{r\_1\}^2\; +\; \{r\_2\}^2\backslash right)$Classical Mechanics - Moment of inertia of a uniform hollow cylinder. LivePhysics.com. Retrieved on 2008-01-31. $=\; \backslash frac\{1\}\{2\}\; \backslash pi\backslash rho\; h\backslash left(\{r\_2\}^4\; -\; \{r\_1\}^4\backslash right)$ $I\_x\; =\; I\_y\; =\; \backslash frac\{1\}\{12\}\; m\backslash left[3\backslash left(\{r\_1\}^2\; +\; \{r\_2\}^2\backslash right)+h^2\backslash right]$ or when defining the normalized thickness tn = t/r and letting r = r2, then $I\_z\; =\; mr^2\backslash left(1-t\_n+\backslash frac\{1\}\{2\}t\_n^2\backslash right)$	—
Solid cylinder of radius r, height h and mass m		$I\_z\; =\; \backslash frac\{m\; r^2\}\{2\}\backslash ,\backslash !$ $I\_x\; =\; I\_y\; =\; \backslash frac\{1\}\{12\}\; m\backslash left(3r^2+h^2\backslash right)$	This is a special case of the previous object for r1=0.
Thin, solid disk of radius r and mass m		$I\_z\; =\; \backslash frac\{m\; r^2\}\{2\}\backslash ,\backslash !$ $I\_x\; =\; I\_y\; =\; \backslash frac\{m\; r^2\}\{4\}\backslash ,\backslash !$	This is a special case of the previous object for h=0.
Thin circular hoop of radius r and mass m		$I\_z\; =\; m\; r^2\backslash !$ $I\_x\; =\; I\_y\; =\; \backslash frac\{m\; r^2\}\{2\}\backslash ,\backslash !$	This is a special case of torus object for b=0.
Solid sphere of radius r and mass m		$I\; =\; \backslash frac\{2\; m\; r^2\}\{5\}\backslash ,\backslash !$	—
Hollow sphere of radius r and mass m		$I\; =\; \backslash frac\{2\; m\; r^2\}\{3\}\backslash ,\backslash !$	—
Oblate Spheroid of major a, minor b and mass m		$I\; =\; \backslash frac\{2\; m\; b^2\}\{3\}\backslash ,\backslash !$	—
Right circular cone with radius r, height h and mass m		$I\_z\; =\; \backslash frac\{3\}\{10\}mr^2\; \backslash ,\backslash !$ $I\_x\; =\; I\_y\; =\; \backslash frac\{3\}\{5\}m\backslash left(\backslash frac\{r^2\}\{4\}+h^2\backslash right)\; \backslash ,\backslash !$	—
Solid cuboid of height h, width w, and depth d, and mass m		$I\_h\; =\; \backslash frac\{1\}\{12\}\; m\backslash left(w^2+d^2\backslash right)$ $I\_w\; =\; \backslash frac\{1\}\{12\}\; m\backslash left(h^2+d^2\backslash right)$ $I\_d\; =\; \backslash frac\{1\}\{12\}\; m\backslash left(h^2+w^2\backslash right)$	For a similarly oriented cube with sides of length $s$, $I\_\{CM\}\; =\; \backslash frac\{m\; s^2\}\{6\}\backslash ,\backslash !$.
Thin rectangular plane of height h and of width w and mass m		$$ I_c = \frac {m(h^2 + w^2)}{12}
Thin rectangular plane of height h and of width w and mass m		$I\_e\; =\; \backslash frac\; \{m(h^2)\}\{3\}+\backslash frac\; \{m(w^2)\}\{12\}$
Rod of length L and mass m		$I\_\{\backslash mathrm\{center\}\}\; =\; \backslash frac\{m\; L^2\}\{12\}\; \backslash ,\backslash !$	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=d=0.
Rod of length L and mass m		$I\_\{\backslash mathrm\{end\}\}\; =\; \backslash frac\{m\; L^2\}\{3\}\; \backslash ,\backslash !$	This expression assumes that the rod is an infinitely thin (but rigid) wire.
Torus of tube radius a, cross-sectional radius b and mass m.		About a diameter: $\backslash frac\{1\}\{8\}\backslash left(4a^2\; +\; 5b^2\backslash right)m$ About the vertical axis: $\backslash left(a^2\; +\; \backslash frac\{3\}\{4\}b^2\backslash right)m$	—
Thin, solid, regular polygon shaped plate with vertices $\backslash vec\{P\}\_\{1\}$, $\backslash vec\{P\}\_\{2\}$, $\backslash vec\{P\}\_\{3\}$, ..., $\backslash vec\{P\}\_\{N\}$ and mass $m$.		$I=\backslash frac\{m\}\{6\}\backslash frac\{\backslash sum\_\{n=1\}^\{N\}$	\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_{n=1}^{N}	\vec{P}_{n+1}\times\vec{P}_{n}	}	—

See also

• Moment of inertia
• List of area moments of inertia
• List of moment of inertia tensors

References

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