NationMaster - Encyclopedia: List of moments of inertia

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. Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg mÂ², Former British units slug ft2) quantifies the rotational inertia of a rigid body, i. ... The definition, agreement and practical use of units of measurement have played a crucial role in human endeavour from early ages up to this day. ... The second moment of area, also known as the second moment of inertia and the area moment of inertia, is a property of a shape that is used to predict its resistance to bending and deflection. ...

See also

Description

Figure

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.

Thick cylinder with open ends, of inner radius r₁, outer radius r₂, length h and mass m

$I_z = frac{1}{2} mleft({r_2}^2 - {r_1}^2right)$
$I_x = I_y = frac{1}{12} mleft[3left({r_1}^2 + {r_2}^2right)+h^2right]$
or when defining the normalized thickness t_n = t/r and letting r = r₂,
then $I_z = mr^2left(1-t_n+frac{1}{2}t_n^2right)$

—

Solid cylinder of radius r, height h and mass m

Image:moment_of_inertia_solid_cylinder.png

$I_z = frac{m r^2}{2},!$
$I_x = I_y = frac{1}{12} mleft(3r^2+h^2right)$

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.

Solid sphere of radius r and mass m

Image:moment_of_inertia_solid_sphere.png

$I = frac{2 m r^2}{5},!$

—

Hollow sphere of radius r and mass m

$I = frac{2 m r^2}{3},!$

—

Right circular cone with radius r, height h and mass m

$I_z = frac{3}{10}mr^2 ,!$
$I_x = I_y = frac{3}{5}mleft(frac{r^2}{4}+h^2right) ,!$

—

Solid cuboid of height h, width w, and depth d, and mass m

$I_h = frac{1}{12} mleft(w^2+d^2right)$
$I_w = frac{1}{12} mleft(h^2+d^2right)$
$I_d = frac{1}{12} mleft(h^2+w^2right)$

For a similarly oriented cube with sides of length

s

, $I_{CM} = frac{m s^2}{6},!$ .

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

$I_{mathrm{end}} = frac{m L^2}{3} ,!$

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.

About a diameter: $frac{1}{8}left(4a^2 + 5b^2right)m$
About the vertical axis: $left(a^2 + frac{3}{4}b^2right)m$

—

Thin, solid, polygon shaped plate with vertices $vec{P}_{1}$ , $vec{P}_{2}$ , $vec{P}_{3}$ , ..., $vec{P}_{N}$ and mass

m

$I=frac{m}{6}frac{sum_{n=1}^{N}||vec{P}_{n+1}timesvec{P}_{n}||(vec{P}^{2}_{n+1}+vec{P}_{n+1}cdotvec{P}_{n}+vec{P}_{n}^{2})}{sum_{n=1}^{N}||vec{P}_{n+1}timesvec{P}_{n}||}$

—

Categories: Mechanics | Physics lists | Introductory physics Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg mÂ², Former British units slug ft2) quantifies the rotational inertia of a rigid body, i. ... The following is list of area moments of inertia. ... This list of moment of inertia tensors is given for principal axes of each object. ... A right circular cylinder An elliptic cylinder In mathematics, a cylinder is a quadric surface, with the following equation in Cartesian coordinates: This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b). ... Image of thin cylinder rotating about CG. Image placed in the public domain by its author, at http://www. ... MetaPost image of a cylinder, with dimensions and axes specified. ... MetaPost image of a cylinder, with dimensions and axes specified. ... In geometry, a disk is the region in a plane contained inside of a circle. ... MetaPost image of a disc, with dimensions and axes specified. ... A sphere is a perfectly symmetrical geometrical object. ... MetaPost image of a sphere, with dimensions and axes specified. ... MetaPost image of a sphere, with dimensions and axes specified. ... This article is about angles in geometry. ... This article is about the geometric object, for other uses see Cone. ... MetaPost image of a cone, with dimensions and axes specified. ... In anatomy, the cuboid bone is a bone in the foot. ... Image of solid rectangular prism rotating about center. ... Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ... Image of slender rod rotating about center. ... Image of slender rod rotating about end. ... In geometry, a torus (pl. ... homology cycles on a torus. ... Image File history File links Polygon_moment_of_inertia. ...