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Center of Mass for Planar Region in Polar Coordinates

 Description Determine $\stackrel{&conjugate0;}{r}$ and $\stackrel{&conjugate0;}{\mathrm{\theta }}$, the center of mass coordinates for a planar region in polar coordinates.

Center of Mass for Planar Region in Polar Coordinates

Density:

 > ${r}$
 ${r}$ (1)

Region:

${r}_{1}\left(\mathrm{θ}\right)$

 > ${0}$
 ${0}$ (2)

${r}_{2}\left(\mathrm{θ}\right)$

 > ${1}$
 ${1}$ (3)

$a$

 > ${0}$
 ${0}$ (4)

$b$

 > $\frac{{\mathrm{\pi }}}{{2}}$
 $\frac{{1}}{{2}}{}{\mathrm{π}}$ (5)

Moments$÷$Mass:

Inert Integral -

 >
 $\frac{{{∫}}_{{0}}^{\frac{{1}}{{2}}{}{\mathrm{π}}}{{∫}}_{{0}}^{{1}}{\mathrm{cos}}{}\left({\mathrm{θ}}\right){}{{r}}^{{3}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}}{{{∫}}_{{0}}^{\frac{{1}}{{2}}{}{\mathrm{π}}}{{∫}}_{{0}}^{{1}}{{r}}^{{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}}{,}\frac{{{∫}}_{{0}}^{\frac{{1}}{{2}}{}{\mathrm{π}}}{{∫}}_{{0}}^{{1}}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){}{{r}}^{{3}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}}{{{∫}}_{{0}}^{\frac{{1}}{{2}}{}{\mathrm{π}}}{{∫}}_{{0}}^{{1}}{{r}}^{{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}}$ (6)

Explicit values for $\stackrel{&conjugate0;}{r}$ and $\stackrel{&conjugate0;}{\mathrm{\theta }}$

 >
 $\frac{{3}}{{2}}{}\frac{\sqrt{{2}}}{{\mathrm{π}}}{,}\frac{{1}}{{4}}{}{\mathrm{π}}$ (7)

Plot:

 >

 Commands Used