User:Markozeta/sandbox

Area moments of inertia

Description	Figure	Area moment of inertia	Comment	Reference
a filled circular area of radius r		$I_{x}={\frac {\pi }{4}}r^{4}$ $I_{y}={\frac {\pi }{4}}r^{4}$ $I_{z}={\frac {\pi }{2}}r^{4}$
an annulus of inner radius r1 and outer radius r2		$I_{x}={\frac {\pi }{4}}\left({r_{2}}^{4}-{r_{1}}^{4}\right)$ $I_{y}={\frac {\pi }{4}}\left({r_{2}}^{4}-{r_{1}}^{4}\right)$ $I_{z}={\frac {\pi }{2}}\left({r_{2}}^{4}-{r_{1}}^{4}\right)$	For thin tubes, $r\equiv r_{1}\approx r_{2}$ and $r_{2}\equiv r_{1}+t$. We can say that $\left(r_{2}^{4}-r_{1}^{4}\right)=\left(\left(r_{1}+t\right)^{4}-r_{1}^{4}\right)=\left(4r_{1}^{3}t+6r_{1}^{2}t^{2}+4r_{1}t^{3}+t^{4}\right)$ and because $r_{1}>>t$ this bracket can be simplified to $\left(4r_{1}^{3}t+6r_{1}^{2}t^{2}+4r_{1}t^{3}+t^{4}\right)\approx 4r_{1}^{3}t$. Ultimately, for a thin tube, $I_{x}=I_{y}=\pi {r}^{3}{t}$.
a filled circular sector of angle θ in radians and radius r with respect to an axis through the centroid of the sector and the center of the circle		$I_{0}=\left(\theta -\sin \theta \right){\frac {r^{4}}{8}}$	This formula is valid for only for 0 ≤ $\theta$ ≤ $\pi$
a filled semicircle with radius r with respect to a horizontal line passing through the centroid of the area		$I_{0}=\left({\frac {\pi }{8}}-{\frac {8}{9\pi }}\right)r^{4}\approx 0.1098r^{4}$
a filled semicircle as above but with respect to an axis collinear with the base		$I={\frac {\pi r^{4}}{8}}$	This is a consequence of the parallel axis theorem and the fact that the distance between these two axes is ${\frac {4r}{3\pi }}$
a filled semicircle as above but with respect to a vertical axis through the centroid		$I_{0}={\frac {\pi r^{4}}{8}}$
a filled quarter circle with radius r entirely in the 1st quadrant of the Cartesian coordinate system		$I={\frac {\pi r^{4}}{16}}$
a filled quarter circle as above but with respect to a horizontal or vertical axis through the centroid		$I_{0}=\left({\frac {\pi }{16}}-{\frac {4}{9\pi }}\right)r^{4}\approx 0.0549r^{4}$	This is a consequence of the parallel axis theorem and the fact that the distance between these two axes is ${\frac {4r}{3\pi }}$
a filled ellipse whose radius along the x-axis is a and whose radius along the y-axis is b		$I_{x}={\frac {\pi }{4}}ab^{3}$ $I_{y}={\frac {\pi }{4}}a^{3}b$
a filled rectangular area with a base width of b and height h		$I_{x}={\frac {bh^{3}}{12}}$ $I_{y}={\frac {b^{3}h}{12}}$
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
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
a filled triangular area with a base width of b and height h with respect to an axis through the centroid		$I_{0}={\frac {bh^{3}}{36}}$
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
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.
An equal legged angle		$I_{x}=I_{y}={\frac {t(5L^{2}-5Lt+t^{2})(L^{2}-Lt+t^{2})}{12(2L-t)}}$ $I_{(}xy)={\frac {L^{2}t(L-t)^{2}}{4(t-2L)}}$ $I_{a}={\frac {t(2L^{4}-4L^{3}t+8L^{2}t^{2}-6Lt^{3}+t^{4})}{12(2L-t)}}$ $I_{b}={\frac {t(2L-t)(2L^{2}-2Lt+t^{2})}{12}}$	$I_{(}xy)$ is the often unused product of inertia, used to define inertia with a rotated axis
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 center of mass and the perpendicular distance (r) between the axes.