First Moments

Centroid

$M_{\mathrm{yz}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^{\mathrm{\pi }}{\int }_0^{1-\cos \left(\mathrm{\phi }\right)}{\sin }^2\left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right){\mathrm{\rho }}^3\mathit{\ⅆ}\mathrm{\rho }\mathit{\ⅆ}\mathrm{\phi }\mathit{\ⅆ}\mathrm{\θ}\=0$

$\stackrel{\&conjugate0;}{x}\=\frac{M_{\mathrm{yz}}}{V}\=\frac{0}{V}\=0$

$M_{\mathrm{xz}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^{\mathrm{\pi }}{\int }_0^{1-\cos \left(\mathrm{\phi }\right)}{\sin }^2\left(\mathrm{\phi }\right)\sin \left(\mathrm{\theta }\right){\mathrm{\rho }}^3\mathit{\ⅆ}\mathrm{\rho }\mathit{\ⅆ}\mathrm{\phi }\mathit{\ⅆ}\mathrm{\θ}\=0$

$\stackrel{\&conjugate0;}{y}\=\frac{M_{\mathrm{xz}}}{V}\=\frac{0}{V}\=0$

$M_{\mathrm{xy}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^{\mathrm{\pi }}{\int }_0^{1-\cos \left(\mathrm{\phi }\right)}\cos \left(\mathrm{\phi }\right){\mathrm{\rho }}^3\sin \left(\mathrm{\phi }\right)\mathit{\ⅆ}\mathrm{\rho }\mathit{\ⅆ}\mathrm{\phi }\mathit{\ⅆ}\mathrm{\θ}\=-\frac{32}{15} \mathrm{\π}$

$\stackrel{\&conjugate0;}{z}\=\frac{M_{\mathrm{xy}}}{V}\=\frac{-\frac{32}{15} \mathrm{\π}}{\frac{8}{3}\mathrm{\π}}\=-\frac{4}{5}$

Table 8.3.7(a)   First moments and the coordinates of the centroid

First Moments

Center of Mass

$M_{\mathrm{yz}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^{\mathrm{\pi }}{\int }_0^{1-\cos \left(\mathrm{\phi }\right)}{\sin }^2\left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right){\mathrm{\rho }}^{7\/2}\mathit{\ⅆ}\mathrm{\rho }\mathit{\ⅆ}\mathrm{\phi }\mathit{\ⅆ}\mathrm{\θ}$ = $0$

$\stackrel{\&conjugate0;}{x}\=\frac{M_{\mathrm{yz}}}{m}\=\frac{0}{m}$ = $0$

$M_{\mathrm{xz}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^{\mathrm{\pi }}{\int }_0^{1-\cos \left(\mathrm{\phi }\right)}{\sin }^2\left(\mathrm{\phi }\right)\sin \left(\mathrm{\theta }\right){\mathrm{\rho }}^{7\/2}\mathit{\ⅆ}\mathrm{\rho }\mathit{\ⅆ}\mathrm{\phi }\mathit{\ⅆ}\mathrm{\θ}$ = $0$

$\stackrel{\&conjugate0;}{y}\=\frac{M_{\mathrm{xz}}}{m}\=\frac{0}{m}$ = $0$

$M_{\mathrm{xy}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^{\mathrm{\pi }}{\int }_0^{1-\cos \left(\mathrm{\phi }\right)}\cos \left(\mathrm{\phi }\right){\mathrm{\rho }}^{7\/2}\sin \left(\mathrm{\phi }\right)\mathit{\ⅆ}\mathrm{\rho }\mathit{\ⅆ}\mathrm{\phi }\mathit{\ⅆ}\mathrm{\θ}$ = $-\frac{256}{143}\sqrt{2}\mathrm{\π}$

$\stackrel{\&conjugate0;}{z}\=\frac{M_{\mathrm{xy}}}{m}\=\frac{-\frac{256}{143}\sqrt{2}\mathrm{\π}}{\frac{128}{63}\sqrt{2}\mathrm{\π}}$ = $-\frac{126}{143}$ 

Table 8.3.7(b)   First moments and the coordinates of the center of mass

Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Center of Mass≻Spherical

Table 8.3.7(c)   Centroid computed by task template that implements the CenterOfMass command

Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Center of Mass≻Spherical

Table 8.3.7(d)   Centroid computed by task template that implements the CenterOfMass command

Initialize

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{MultivariateCalculus}\right)\:$

Apply the CenterOfMass command from the Student MultivariateCalculus package

$\mathrm{CenterOfMass}\left(1\,\mathrm{\ρ}\=0..1-\cos \left(\mathrm{\φ}\right)\,\mathrm{\φ}\=0..\mathrm{\π}\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{spherical}\left[\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right]\,\mathrm{output}\=\mathrm{integral}\right)$

$\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}{\∫}_0^{1-\cos \left(\mathrm{\φ}\right)}{\sin \left(\mathrm{\φ}\right)}^2\cos \left(\mathrm{\θ}\right){\mathrm{\ρ}}^3\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}{\∫}_0^{1-\cos \left(\mathrm{\φ}\right)}{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}}\,\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}{\∫}_0^{1-\cos \left(\mathrm{\φ}\right)}{\sin \left(\mathrm{\φ}\right)}^2\sin \left(\mathrm{\θ}\right){\mathrm{\ρ}}^3\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}{\∫}_0^{1-\cos \left(\mathrm{\φ}\right)}{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}}\,\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}{\∫}_0^{1-\cos \left(\mathrm{\φ}\right)}\cos \left(\mathrm{\φ}\right){\mathrm{\ρ}}^3\sin \left(\mathrm{\φ}\right)\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}{\∫}_0^{1-\cos \left(\mathrm{\φ}\right)}{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}}$

$C≔\left[\mathrm{CenterOfMass}\left(1\,\mathrm{\ρ}\=0..1-\cos \left(\mathrm{\φ}\right)\,\mathrm{\φ}\=0..\mathrm{\π}\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{spherical}\left[\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right]\right)\right]$

$\left[\frac{4}{5}\,\mathrm{\π}\,0\right]$

Table 8.3.7(e)   Centroid in spherical coordinates

$\mathrm{eval}\left(\left[\mathrm{\ρ} \sin \left(\mathrm{\φ}\right)\cos \left(\mathrm{\θ}\right)\,\mathrm{\ρ} \sin \left(\mathrm{\φ}\right)\sin \left(\mathrm{\θ}\right)\,\mathrm{\ρ} \cos \left(\mathrm{\φ}\right)\right]\,\mathrm{Equate}\left(\left[\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right]\,C\right)\right)$ = $\left[0\,0\,-\frac{4}{5}\right]$$$

Initialize

$\mathrm{\δ}≔\sqrt{\mathrm{\ρ}}\:$

Apply the CenterOfMass command from the Student MultivariateCalculus package

$\mathrm{CenterOfMass}\left(\mathrm{\δ}\,\mathrm{\ρ}\=0..1-\cos \left(\mathrm{\φ}\right)\,\mathrm{\φ}\=0..\mathrm{\π}\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{spherical}\left[\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right]\,\mathrm{output}\=\mathrm{integral}\right)$

$\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}{\∫}_0^{1-\cos \left(\mathrm{\φ}\right)}{\sin \left(\mathrm{\φ}\right)}^2\cos \left(\mathrm{\θ}\right){\mathrm{\ρ}}^{7\/2}\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}{\∫}_0^{1-\cos \left(\mathrm{\φ}\right)}{\mathrm{\ρ}}^{5\/2}\sin \left(\mathrm{\φ}\right)\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}}\,\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}{\∫}_0^{1-\cos \left(\mathrm{\φ}\right)}{\sin \left(\mathrm{\φ}\right)}^2\sin \left(\mathrm{\θ}\right){\mathrm{\ρ}}^{7\/2}\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}{\∫}_0^{1-\cos \left(\mathrm{\φ}\right)}{\mathrm{\ρ}}^{5\/2}\sin \left(\mathrm{\φ}\right)\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}}\,\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}{\∫}_0^{1-\cos \left(\mathrm{\φ}\right)}\cos \left(\mathrm{\φ}\right){\mathrm{\ρ}}^{7\/2}\sin \left(\mathrm{\φ}\right)\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}{\∫}_0^{1-\cos \left(\mathrm{\φ}\right)}{\mathrm{\ρ}}^{5\/2}\sin \left(\mathrm{\φ}\right)\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}}$

$\mathrm{CM}≔\mathrm{simplify}\left(\left[\mathrm{CenterOfMass}\left(\mathrm{\δ}\,\mathrm{\ρ}\=0..1-\cos \left(\mathrm{\φ}\right)\,\mathrm{\φ}\=0..\mathrm{\π}\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{spherical}\left[\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right]\right)\right]\right)$

$\left[\frac{126}{143}\,\mathrm{\π}\,0\right]$

Table 8.3.7(f)   Center of mass in cylindrical coordinates

$\mathrm{eval}\left(\left[\mathrm{\rho } \sin \left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right)\,\mathrm{\rho } \sin \left(\mathrm{\phi }\right)\sin \left(\mathrm{\theta }\right)\,\mathrm{\rho } \cos \left(\mathrm{\phi }\right)\right]\,\mathrm{Equate}\left(\left[\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right]\,\mathrm{CM}\right)\right)$

$\left[0\,0\,-\frac{126}{143}\right]$