User:BeaumontTaz/sandbox

The following is a list of second moments of area. The second moment of area or area moment of inertia has a unit of dimension L⁴, and should not be confused with the mass moment of inertia. If the piece is thin, however, the mass moment of inertia equals the area density times the area moment of inertia. Each of the following entries is with respect to an axis through the centroid of the given shape, unless otherwise specified.

Notes to Self

TO BE REPLACING THE "LIST OF SECOND MOMENT OF AREA" PAGE. ALSO, REDIRECTING THE "LIST OF ARE MOMENTS OF INERTIA" PAGE TO THE NEWLY EDITED PAGE AND CORRECTING THE "SECOND MOMENT OF AREA" ARTICLE TO REFLECT THIS CHANGE.

REPLACE REFERENCES WITH GERE!

NEED TO FIX THE O AND C MARKERS ON SEMICIRCLE THROUGH CENTROID IMAGE!

REDO ALL PICTURES TO HAVE A WHITE BACKGROUND!!!

Second moments of area

Description	Figure	Area moment of inertia^[1]	Comment
Solid circle of radius, $r$ , centered at the origin.		$I_{x}=I_{y}={\frac {\pi }{4}}r^{4}$ $J_{O}={\frac {\pi }{2}}r^{4}$
Solid circle of radius, $r$ , tangent to axes.		$I_{x}=I_{y}={\frac {5\pi }{4}}r^{4}$ $J_{O}={\frac {5\pi }{2}}r^{4}$	This is a consequence of the parallel axis theorem and the fact that the distance between the center of the circle and either axis is $r$ .
A filled annulus of inner radius $r_{i}$ and outer radius $r_{o}$ centered at the origin.		$I_{x}=I_{y}={\frac {\pi }{4}}\left({r_{o}}^{4}-{r_{i}}^{4}\right)$ $J_{O}={\frac {\pi }{2}}\left({r_{o}}^{4}-{r_{i}}^{4}\right)$	For thin circular ring, $r\approx r_{i}\approx r_{o}$ . We know that $\left(r_{o}^{2}+r_{i}^{2}\right)\left(r_{o}+r_{i}\right)\approx 4r^{3}$ and thus, $I_{x}=I_{y}=\pi {r}^{3}t$ $J_{O}=2\pi r^{3}t$ where $t\equiv$ Thickness of ring = $r_{o}-r_{i}$ $r\equiv$ = Average radius = ${\frac {r_{o}+r_{i}}{2}}$
A filled circular sector of angle $\theta$ in radians and radius $r$ with the x-axis through the centroid of the sector and the center of the circle.		$I_{x}={\frac {r^{4}}{8}}\left(\theta -\sin \theta \right)$ $I_{y}={\frac {r^{4}}{8}}\left(\theta +\sin \theta \right)$ $J_{O}={\frac {\theta r^{4}}{4}}$	Valid for $0\leq \theta \leq \pi$
A filled semicircle of radius $r$ with center at the origin lying entirely in the first and second quadrants.		$I_{x}=I_{y}={\frac {\pi r^{4}}{8}}$ $J_{O}={\frac {\pi r^{4}}{4}}$	Note that the second moments of area for a full circle are twice the second moments of area for the semicircle. A principle explained by the method of composite sections. Also note that this is a special case of the circle sector above where $\theta =\pi$ .
A filled semicircle with radius $r$ , as above, but with the centroid at the origin.	NEED TO FIX THE O AND C MARKERS ON THIS IMAGE!	$I_{x}=\left({\frac {\pi }{8}}-{\frac {8}{9\pi }}\right)r^{4}\approx 0.1098r^{4}$ $I_{y}={\frac {\pi r^{4}}{8}}$ $J_{O}=\left({\frac {\pi }{4}}-{\frac {8}{9\pi }}\right)r^{4}\approx 0.5025r^{4}$	This is a consequence of the parallel axis theorem and the fact that, in the semicircle above, the distance between the centroidal x-axis and the x-axis is $4r/3\pi$ .
A filled quarter circle of radius $r$ , lying entirely in the first quadrant.		$I_{x}=I_{y}={\frac {\pi r^{4}}{16}}$ $J_{O}={\frac {\pi r^{4}}{8}}$	Note that the second moments of area for a full circle are four times the second moments of area for the quarter circle and the second moments of area for a semicircle are twice the second moments of area for the quarter circle. A principle explained by the method of composite sections.
A filled quarter circle as above but with respect to a horizontal or vertical axis through the centroid.		$I_{x}=I_{y}=\left({\frac {\pi }{16}}-{\frac {4}{9\pi }}\right)r^{4}\approx 0.05488r^{4}$ $J_{O}=\left({\frac {\pi }{8}}-{\frac {8}{9\pi }}\right)r^{4}\approx 0.1098r^{4}$	This is a consequence of the parallel axis theorem and the fact that, in the quarter circle above, the distance between the centroidal x-axis and the x-axis is $4r/3\pi$ .
a filled ellipse whose radius along the x-axis is a and whose radius along the y-axis is b		$I_{0}={\frac {\pi }{4}}ab^{3}$
a filled rectangular area with a base width of b and height h		$I_{0}={\frac {bh^{3}}{12}}$		^[2]
a filled rectangular area as above but with respect to an axis collinear with the base		$I={\frac {bh^{3}}{3}}$	This is a result from the parallel axis theorem	^[2]
a filled rectangular area as above but with respect to an axis collinear, where r is the perpendicular distance from the centroid of the rectangle to the axis of interest		$I={\frac {bh^{3}}{12}}+bhr^{2}$	This is a result from the parallel axis theorem	^[2]
a filled triangular area with a base width of b and height h with respect to an axis through the centroid		$I_{0}={\frac {bh^{3}}{36}}$		^[3]
a filled triangular area as above but with respect to an axis collinear with the base		$I={\frac {bh^{3}}{12}}$	This is a consequence of the parallel axis theorem	^[3]
a filled regular hexagon with a side length of a		$I_{0}={\frac {5{\sqrt {3}}}{16}}a^{4}$	The result is valid for both a horizontal and a vertical axis through the centroid, and therefore is also valid for an axis with arbitrary direction that passes through the origin.
Any plane region with a known area moment of inertia for a parallel axis. (Main Article parallel axis theorem)		$I_{z}=I_{x}+Ar^{2}$	This can be used to determine the second moment of area of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's centre of mass and the perpendicular distance (r) between the axes.