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| In [[mathematics]], a '''planar lamina''' is a [[closed set]] in a plane of mass <math>m</math> and surface density <math>\rho\ (x,y)</math> such that:
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| :<math>m = \int\int_{}{}\rho\ (x,y)\,dx\,dy</math>, over the [[closed set]].
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| The [[center of mass]] of the lamina is at the point
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| :<math> \left(\frac{M_y}{m},\frac{M_x}{m}\right) </math>
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| where <math>M_y </math> moment of the entire lamin about the x-axis and <math>M_x </math> moment of the entire lamin about the y-axis.
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| :<math>M_y = \lim_{m,n \to \infty}\,\sum_{i=1}^{m}\,\sum_{j=1}^{n}\,x{_{ij}}^{*}\,\rho\ (x{_{ij}}^{*},y{_{ij}}^{*})\,\Delta\Alpha = \iint_{}{} x\, \rho\ (x,y)\,dx\,dy</math>, over the closed surface.
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| :<math>M_x = \lim_{m,n \to \infty}\,\sum_{i=1}^{m}\,\sum_{j=1}^{n}\,y{_{ij}}^{*}\,\rho\ (x{_{ij}}^{*},y{_{ij}}^{*})\,\Delta\Alpha = \iint_{}{} y\, \rho\ (x,y)\,dx\,dy</math>, over the closed surface.
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| Example 1.
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| Find the center of mass of a lamina with edges given by the lines <math>x=0</math>, <math>x=y</math> and <math>y=4-x</math>, where the density is given as <math>\rho\ (x,y)\,=2x+3y+2</math>.
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| :<math>m = \int_0^2{\int_x^{4-x}}_{}{}\,2x+3y+2\,dy\,dx</math>
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| :<math>m=\int_0^2 (2xy+\frac{3y^2}{2}+2y)|_x^{4-x}\,dx</math>
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| :<math>m=\int_0^2 -4x^2-8x+32\,dx</math> | |
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| :<math>m= (\frac{-4x^3}{3}-4x^2+32x)|_0^2</math>
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| :<math>m= \frac{112}{3}</math>
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| :<math>M_y = \int_0^2{\int_x^{4-x}}{}{}x\,(2x+3y+2)\,dy\,dx</math>
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| :<math>M_y=\int_0^2 (2x^2y+\frac{3xy^2}{2}+2xy)|_x^{4-x}\,dx</math>
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| :<math>M_y=\int_0^2 -4x^3-8x^2+32x\,dx</math>
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| :<math>M_y= (-x^4-\frac{8x^3}{3}+16x^2)|_0^2</math>
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| :<math>M_y= \frac{80}{3}</math>
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| :<math>M_x = \int_0^2{\int_x^{4-x}}{}{}y\,(2x+3y+2)\,dy\,dx</math>
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| :<math>M_x = \int_0^2 (xy^2+y^3+y^2)|_x^{4-x}\,dx</math>
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| :<math>M_x = \int_0^2 (-2x^3+4x^2-40x+80\,dx</math>
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| :<math>M_x= \left.\left(\frac{-x^4}{2}+\frac{4x^3}{3}-20x^2+80x\right)\right|_0^2</math>
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| :<math>M_x= \frac{248}{3}</math>
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| center of mass is at the point
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| :<math>\left(\frac{\frac{80}{3}}{\frac{112}{3}},\frac{\frac{248}{3}}{\frac{112}{3}}\right)=\left(\frac{5}{7},\frac{31}{14}\right)</math>
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| Planar laminas can be used to determine [[moments of inertia]], or center of mass.
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| [[Category:Measure theory]]
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| {{mathanalysis-stub}}
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