Chemistry - List of moments of inertia

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.

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 $h/3 \,$

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