List of moments of inertia

The following is a list of moments of inertia. This article is about the moment of inertia of a rotating object. 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 second moment of area, also known as the area moment of inertia or second moment of inertia, is a property of a shape that is used to predict its resistance to The following moments of inertia assume constant density throughout the object.

NOTE: The axis of rotation is taken to be through the centre 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. A cylinder is one of the most basic curvilinear geometric shapes the Surface formed by the points at a fixed distance from a given Straight line, the axis 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 and mass m	$I_z = \frac{1}{2} m\left({r_1}^2 + {r_2}^2\right)$ ^[1] $I_x = I_y = \frac{1}{12} m\left[3\left({r_1}^2 + {r_2}^2\right)+h^2\right]$ 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. In Geometry, a disk (also spelled disc) is the region in a plane bounded by a Circle.
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 torus object for b=0. In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar
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. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe
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.
Oblate Spheroid of major a, minor b and mass m	$I = \frac{2 m b^2}{3}\,\!$	—
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 $s$ , $I_{CM} = \frac{m s^2}{6}\,\!$ . In Geometry and Trigonometry, a right angle is an angle of 90 degrees corresponding to a quarter turn (that is a quarter of a full circle A cone is a three-dimensional Geometric shape that tapers smoothly from a flat round base to a point called the apex or vertex In anatomy the Cuboid bone is a bone in the foot See also Hyperrectangle Oblong A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex.
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 (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=d=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.
Torus of tube radius a, cross-sectional radius b and mass m. In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar	About a diameter: $\frac{1}{8}\left(4a^2 + 5b^2\right)m$ About the vertical axis: $\left(a^2 + \frac{3}{4}b^2\right)m$	—
Thin, solid, regular polygon shaped plate with vertices $\vec{P}_{1}$ , $\vec{P}_{2}$ , $\vec{P}_{3}$ , . . . , $\vec{P}_{N}$ and mass $m$ .	$I=\frac{m}{6}\frac{\sum_{n=1}^{N}\|\|\vec{P}_{n+1}\times\vec{P}_{n}\|\|(\vec{P}^{2}_{n+1}+\vec{P}_{n+1}\cdot\vec{P}_{n}+\vec{P}_{n}^{2})}{\sum_{n=1}^{N}\|\|\vec{P}_{n+1}\times\vec{P}_{n}\|\|}$	—

References

^ Classical Mechanics - Moment of inertia of a uniform hollow cylinder. This article is about the moment of inertia of a rotating object. The following is list of area moments of inertia. The area moment of inertia or second moment of area has a unit of dimension length4 and should not be confused with This list of Moment of inertia tensors is given for Principal axes of each object LivePhysics. com. Retrieved on 2008-01-31. 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common Events 1504 - France cedes Naples to Aragon. 1606 - Gunpowder Plot: Guy Fawkes

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References