en>Trappist the monk: /* References */replace mr template with mr parameter in CS1 templates; using AWB

2014-09-25T21:41:36Z

References: replace mr template with mr parameter in CS1 templates; using AWB

← Older revision		Revision as of 23:41, 25 September 2014
Line 1:		Line 1:
	~~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:~~		Wilber Berryhill is what his wife enjoys to contact him and he completely loves this name. One of the very very best issues in the globe for him is performing ballet and he'll be starting something else alongside with it. Alaska is where he's always been residing. My working day job is a journey agent.<br><br>My site online psychic chat ([http://www.onbizin.co.kr/xe/?document_srl=320614 click the up coming webpage])

	~~:<math>m = \int\int_{}{}\rho\ (x,y)\,dx\,dy</math>, over the [[closed set]]~~.

	~~The [[center of mass]]~~ of the ~~lamina~~ is ~~at the point~~

	~~:<math> \left(\frac{M_y}{m},\frac{M_x}{m}\right) </math>~~

	~~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~~.

	~~:<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~~.

	:<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.

	~~Example 1.~~

	~~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>~~.

	:<~~math~~>~~m = \int_0^2{\int_x^{4-x}}_{}{}\,2x+3y+2\,dy\,dx~~<~~/math~~>

	~~:<math>m=\int_0^2~~ (~~2xy+\frac{3y^2}{2}+2y)\|_x^{4-x}\,dx</math>~~

	:~~<math>m=\int_0^2 -4x^2-8x+32\,dx<~~/~~math>~~

	~~:<math>m= (\frac{-4x^3}{3}-4x^2+32x)\|_0^2<~~/~~math>~~

	~~:<math>m= \frac{112}{3}<~~/~~math>~~

	~~:<math>M_y = \int_0^2{\int_x^{4-x}}{}{}x\,(2x+3y+2)\,dy\,dx<~~/~~math>~~

	~~:<math>M_y~~=~~\int_0^2 (2x^2y+\frac{3xy^2}{2}+2xy)\|_x^{4-x}\,dx</math>~~

	~~:<math>M_y=\int_0^2 -4x^3-8x^2+32x\,dx</math>~~

	~~:<math>M_y= (-x^4-\frac{8x^3}{3}+16x^2)\|_0^2</math>~~

	~~:<math>M_y= \frac{80}{3}</math>~~

	~~:<math>M_x = \int_0^2{\int_x^{4-x}}{}{}y\,(2x+3y+2)\,dy\,dx</math>~~

	~~:<math>M_x = \int_0^2 (xy^2+y^3+y^2)\|_x^{4-x}\,dx</math>~~

	~~:<math>M_x = \int_0^2 (-2x^3+4x^2-40x+80\,dx</math>~~

	~~:<math>M_x= \left.\left(\frac{-x^4}{2}+\frac{4x^3}{3}-20x^2+80x\right)\right\|_0^2</math>~~

	~~:<math>M_x= \frac{248}{3}</math>~~

	~~center of mass is at~~ the ~~point~~

	~~:<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>~~

	~~Planar laminas can be used to determine [[moments of inertia]], or center of mass.~~
	~~[[Category:Measure theory]]~~

	~~{{mathanalysis-stub}}~~

en>Helpful Pixie Bot: ISBNs (Build KG)

2012-05-11T20:15:57Z

ISBNs (Build KG)

New page

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:

:<math>m = \int\int_{}{}\rho\ (x,y)\,dx\,dy</math>, over the [[closed set]].

The [[center of mass]] of the lamina is at the point

:<math> \left(\frac{M_y}{m},\frac{M_x}{m}\right) </math>

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.

:<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.

:<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.

Example 1.

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>.

:<math>m = \int_0^2{\int_x^{4-x}}_{}{}\,2x+3y+2\,dy\,dx</math>

:<math>m=\int_0^2 (2xy+\frac{3y^2}{2}+2y)|_x^{4-x}\,dx</math>

:<math>m=\int_0^2 -4x^2-8x+32\,dx</math>

:<math>m= (\frac{-4x^3}{3}-4x^2+32x)|_0^2</math>

:<math>m= \frac{112}{3}</math>

:<math>M_y = \int_0^2{\int_x^{4-x}}{}{}x\,(2x+3y+2)\,dy\,dx</math>

:<math>M_y=\int_0^2 (2x^2y+\frac{3xy^2}{2}+2xy)|_x^{4-x}\,dx</math>

:<math>M_y=\int_0^2 -4x^3-8x^2+32x\,dx</math>

:<math>M_y= (-x^4-\frac{8x^3}{3}+16x^2)|_0^2</math>

:<math>M_y= \frac{80}{3}</math>

:<math>M_x = \int_0^2{\int_x^{4-x}}{}{}y\,(2x+3y+2)\,dy\,dx</math>

:<math>M_x = \int_0^2 (xy^2+y^3+y^2)|_x^{4-x}\,dx</math>

:<math>M_x = \int_0^2 (-2x^3+4x^2-40x+80\,dx</math>

:<math>M_x= \left.\left(\frac{-x^4}{2}+\frac{4x^3}{3}-20x^2+80x\right)\right|_0^2</math>

:<math>M_x= \frac{248}{3}</math>

center of mass is at the point

:<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>

Planar laminas can be used to determine [[moments of inertia]], or center of mass.
[[Category:Measure theory]]

{{mathanalysis-stub}}

Rotation system - Revision history

en>Trappist the monk: /* References */replace mr template with mr parameter in CS1 templates; using AWB

en>Helpful Pixie Bot: ISBNs (Build KG)