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List of moments of inertia

From Biocrawler, the free encyclopedia.

The following is a list of moments of inertia.

Moments of inertia

Moments of inertia have units of dimension mass × length².

Description	Moment(s) of inertia	Comment
Thin cylindrical shell with open ends, of radius $r$ and mass $m$	$I = m r^2 \,$	—
Thick cylinder with open ends, of inner radius $r 1$ , outer radius $r 2$ and mass $m$	$I = \frac{1}{2} m({r_1}^2 + {r_2}^2)$	—
Solid cylinder of radius $r$ , height $h$ and mass $m$	$I_z = \frac{1}{2} mr^2$ $I_x = I_y = \frac{1}{12} m(3r^2+h^2)$	—
Thin disk of radius $r$ and mass $m$	$I_z = \frac{1}{2} mr^2$ $I_x = I_y = \frac{1}{4} m(r^2)$	—
Solid sphere of radius $r$ and mass $m$	$I = \frac{2}{5} mr^2$	—
Hollow sphere of radius $r$ and mass $m$	$I = \frac{2}{3} mr^2$	—
Right circular cone with radius $r$ , $h$ and mass $m$	$I_z = (3/10)mr^2 \,\!$ $I_x = I_y = (3/5)m(r^2/4+h^2) \,\!$	—
Solid rectangular prism of height $h$ , width $w$ , and depth $d$ , and mass $m$	$I_h = \frac{1}{12} m(w^2+d^2)$ $I_w = \frac{1}{12} m(h^2+d^2)$ $I_d = \frac{1}{12} m(h^2+w^2)$	For a similarly oriented cube with sides of length $s$ and mass $M$ , $I_{CM} = \frac{1}{6} ms^2$ .
Rod of length $L$ and mass $m$	$I_{center} = \frac{1}{12} mL^2$	This expression is an approximation, and assumes that the mass of the rod is distributed in the form of an infinitely thin (but rigid) wire.
Rod of length $L$ and mass $m$	$I_{end} = \frac{1}{3} mL^2$	This expression is an approximation, and assumes that the mass of the rod is distributed in the form of an infinitely thin (but rigid) wire.

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Area moments of inertia

Area moments of inertia have units of dimension Length⁴. Each is with respect to a horizontal axis through the centroid of the given shape, unless otherwise specified.

Description	Area Moment(s) of inertia	Comment
a filled circular area of radius $r \,$	$I_0 = \pi r^4/4 \,$
a filled semicircle with radius $r \,$ resting atop the $x$ -axis	$I_0 = \pi r^4/8 \,$
a filled quarter circle with radius $r \,$ entirely in the upper-right quadrant of the Cartesian plane	$I_0 = \pi r^4/16 \,$
an ellipse whose radius along the $x$ -axis is $a \,$ and whose radius along the $y$ -axis is $b \,$	$I_0 = \pi ab^3/4 \,$
a filled Rectangular area with a base width of $b \,$ and height $h \,$	$I_0 = bh^3/12 \,$
an axis collinear with the base	$I = bh^3/3 \,$	This is a trivial result from the parallel axis theorem
a filled triangular area with a base width of $b \,$ and height $h$	$I_0 = bh^3/36 \,$
an axis collinear with the base	$I = bh^3/12 \,$	This is a consequence of the parallel axis theorem and the fact that the distance between these two axes is always $h/3 \,$