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Center of Mass for 3D Region in Spherical Coordinates

 Description Determine $\stackrel{&conjugate0;}{\mathrm{\rho }},\stackrel{&conjugate0;}{\mathrm{φ}}$, and $\stackrel{&conjugate0;}{\mathrm{\theta }}$, the center of mass coordinates for a 3D region in spherical coordinates.

Center of Mass for 3D Region in Spherical Coordinates

($\mathrm{φ}$ is the colatitude, measured down from the $z$-axis)

Density:

 > ${\mathrm{\rho }}$
 ${\mathrm{ρ}}$ (1)

Region: $\left\{{\mathrm{\rho }}_{1}\left(\mathrm{φ},\mathrm{θ}\right)\le \mathrm{ρ}\le {\mathrm{\rho }}_{2}\left(\mathrm{φ},\mathrm{θ}\right),{\mathrm{φ}}_{1}\left(\mathrm{θ}\right)\le \mathrm{φ}\le {\mathrm{φ}}_{2}\left(\mathrm{θ}\right),a\le \mathrm{θ}\le b\right\}$

${\mathrm{\rho }}_{1}\left(\mathrm{φ},\mathrm{θ}\right)$

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

${\mathrm{\rho }}_{2}\left(\mathrm{φ},{\mathrm{\theta }}_{}\right)$

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

${\mathrm{φ}}_{1}\left(\mathrm{θ}\right)$

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

${\mathrm{φ}}_{2}\left(\mathrm{θ}\right)$

 > $\frac{{\mathrm{\pi }}}{{6}}$
 $\frac{{1}}{{6}}{}{\mathrm{π}}$ (5)

$a$

 > ${0}$
 ${0}$ (6)

$b$

 >
 ${2}{}{\mathrm{π}}$ (7)

Moments ÷ Mass:

Inert Integral -

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{CenterOfMass}\right]\left(,{\mathrm{ρ}}=..,{\mathrm{φ}}=..,{\mathrm{θ}}=..,\mathrm{coordinates}=\mathrm{spherical}\left[{\mathrm{ρ}}{,}{\mathrm{φ}}{,}{\mathrm{\theta }}\right],\mathrm{output}=\mathrm{integral}\right)$
 $\frac{{{∫}}_{{0}}^{{2}{}{\mathrm{π}}}{{∫}}_{{0}}^{\frac{{1}}{{6}}{}{\mathrm{π}}}{{∫}}_{{0}}^{{1}}{{\mathrm{sin}}{}\left({\mathrm{φ}}\right)}^{{2}}{}{\mathrm{cos}}{}\left({\mathrm{θ}}\right){}{{\mathrm{ρ}}}^{{4}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{ρ}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{φ}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}}{{{∫}}_{{0}}^{{2}{}{\mathrm{π}}}{{∫}}_{{0}}^{\frac{{1}}{{6}}{}{\mathrm{π}}}{{∫}}_{{0}}^{{1}}{{\mathrm{ρ}}}^{{3}}{}{\mathrm{sin}}{}\left({\mathrm{φ}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{ρ}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{φ}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}}{,}\frac{{{∫}}_{{0}}^{{2}{}{\mathrm{π}}}{{∫}}_{{0}}^{\frac{{1}}{{6}}{}{\mathrm{π}}}{{∫}}_{{0}}^{{1}}{{\mathrm{sin}}{}\left({\mathrm{φ}}\right)}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){}{{\mathrm{ρ}}}^{{4}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{ρ}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{φ}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}}{{{∫}}_{{0}}^{{2}{}{\mathrm{π}}}{{∫}}_{{0}}^{\frac{{1}}{{6}}{}{\mathrm{π}}}{{∫}}_{{0}}^{{1}}{{\mathrm{ρ}}}^{{3}}{}{\mathrm{sin}}{}\left({\mathrm{φ}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{ρ}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{φ}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}}{,}\frac{{{∫}}_{{0}}^{{2}{}{\mathrm{π}}}{{∫}}_{{0}}^{\frac{{1}}{{6}}{}{\mathrm{π}}}{{∫}}_{{0}}^{{1}}{\mathrm{cos}}{}\left({\mathrm{φ}}\right){}{{\mathrm{ρ}}}^{{4}}{}{\mathrm{sin}}{}\left({\mathrm{φ}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{ρ}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{φ}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}}{{{∫}}_{{0}}^{{2}{}{\mathrm{π}}}{{∫}}_{{0}}^{\frac{{1}}{{6}}{}{\mathrm{π}}}{{∫}}_{{0}}^{{1}}{{\mathrm{ρ}}}^{{3}}{}{\mathrm{sin}}{}\left({\mathrm{φ}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{ρ}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{φ}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}}$ (8)

Explicit values for $\stackrel{&conjugate0;}{\mathrm{\rho }},\stackrel{&conjugate0;}{\mathrm{φ}}$, and $\stackrel{&conjugate0;}{\mathrm{\theta }}$, the center of mass given in spherical coordinates:

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{CenterOfMass}\right]\left(,{\mathrm{ρ}}=..,{\mathrm{φ}}=..,{\mathrm{θ}}=..,\mathrm{coordinates}=\mathrm{spherical}\left[{\mathrm{ρ}}{,}{\mathrm{φ}}{,}{\mathrm{\theta }}\right]\right)$
 $\frac{{1}}{{20}}{}\frac{{\mathrm{π}}}{\frac{{1}}{{2}}{}{\mathrm{π}}{-}\frac{{1}}{{4}}{}\sqrt{{3}}{}{\mathrm{π}}}{,}{0}{,}{0}$ (9)

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