Wikipedia:Reference desk/Archives/Mathematics/2015 January 6

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January 6

Area moment of inertia about y axis

If finding the area moment of inertia for a rectangle about the y axis does the equation become depth*breadth^3/12? 94.14.218.233 (talk) 20:27, 6 January 2015 (UTC)

It depends on the orientation of the rectangle. In general it would be given by $\iint _{R}(x^{2}+z^{2})dS$, a surface integral over the rectangle R. If you are talking about only the xy plane, I still don't have enough information because I still don't know the placement of the rectangle.--Jasper Deng (talk) 20:33, 6 January 2015 (UTC)

Yes if the rectangle is symmetric about the y-axis (the y-axis goes through the middle of the rectangle and is parallel to the depth dimension) - but you need to multiply by the mass-per-unit-area ($\scriptstyle {\rho }$). The calculation is:

${\begin{aligned}I&=\int _{0}^{\mathrm {depth} }\int _{-{\frac {\mathrm {breadth} }{2}}}^{\frac {\mathrm {breadth} }{2}}\rho x^{2}\partial x\partial y\\&=\int _{0}^{\mathrm {depth} }\left[{\frac {1}{3}}\rho x^{3}\right]_{x=-{\frac {\mathrm {breadth} }{2}}}^{x={\frac {\mathrm {breadth} }{2}}}\partial y\\&=\int _{0}^{\mathrm {depth} }{\frac {\rho }{12}}\mathrm {breadth} ^{3}\partial y\\&=\left[{\frac {\rho }{12}}\mathrm {breadth} ^{3}y\right]_{y=0}^{y=\mathrm {depth} }\\&={\frac {\rho }{12}}\mathrm {breadth} ^{3}\mathrm {depth} \end{aligned}}$

It is sometimes more convenient to express this in terms of the total mass of the rectangle

$m=\mathrm {breadth} \mathrm {depth} \rho$

So substituting

$\rho ={\frac {m}{\mathrm {breadth} \mathrm {depth} }}$

gives

${\begin{aligned}I&={\frac {\rho }{12}}\mathrm {breadth} ^{3}\mathrm {depth} \\&={\frac {m}{12}}\mathrm {breadth} ^{2}\end{aligned}}$

--catslash (talk) 22:55, 6 January 2015 (UTC)

I think by "area moment of inertia" he wants it when the density is equal to 1. Also, the density has to be constant on the rectangle, otherwise the integral has to be done differently.--Jasper Deng (talk) 23:10, 6 January 2015 (UTC)

@Catslash: Use upright or at least normal 'd' in the integral, it's not a partial derivative here:
$I=\int _{0}^{\mathrm {depth} }\int _{-{\frac {\mathrm {breadth} }{2}}}^{\frac {\mathrm {breadth} }{2}}\rho x^{2}\,\mathrm {d} x\,\mathrm {d} y$
or
$I=\int _{0}^{\mathrm {depth} }\int _{-{\frac {\mathrm {breadth} }{2}}}^{\frac {\mathrm {breadth} }{2}}\rho x^{2}\,dx\,dy$
instead of
$I=\int _{0}^{\mathrm {depth} }\int _{-{\frac {\mathrm {breadth} }{2}}}^{\frac {\mathrm {breadth} }{2}}\rho x^{2}\partial x\partial y$
Also separate words denoting values:
with a narrow space: $m=\mathrm {breadth} \,\mathrm {depth} \,\rho$
better with a normal space: $m=\mathrm {breadth} \ \mathrm {depth} \ \rho$
or with an explicit multiplication dot: $m=\mathrm {breadth} \cdot \mathrm {depth} \cdot \rho$
CiaPan (talk) 07:01, 7 January 2015 (UTC)