First Moments

Centroid

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

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

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

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

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

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

Table 8.3.4(a)   First moments and the coordinates of the centroid

First Moments

Center of Mass

$M_{\mathrm{yz}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^{\mathrm{\pi }}{\int }_{1\+\cos \left(\mathrm{\phi }\right)}^2{\sin }^2\left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right){\mathrm{\rho }}^5\mathit{\ⅆ}\mathrm{\rho }\mathit{\ⅆ}\mathrm{\phi }\mathit{\ⅆ}\mathrm{\θ}$ = $0$

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

$M_{\mathrm{xz}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^{\mathrm{\pi }}{\int }_{1\+\cos \left(\mathrm{\phi }\right)}^2{\sin }^2\left(\mathrm{\phi }\right)\sin \left(\mathrm{\theta }\right){\mathrm{\rho }}^5\mathit{\ⅆ}\mathrm{\rho }\mathit{\ⅆ}\mathrm{\phi }\mathit{\ⅆ}\mathrm{\θ}$ = $0$

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

$M_{\mathrm{xy}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^{\mathrm{\pi }}{\int }_{1\+\cos \left(\mathrm{\phi }\right)}^2\cos \left(\mathrm{\phi }\right){\mathrm{\rho }}^5\sin \left(\mathrm{\phi }\right)\mathit{\ⅆ}\mathrm{\rho }\mathit{\ⅆ}\mathrm{\phi }\mathit{\ⅆ}\mathrm{\θ}$ = $-\frac{32}{7} \mathrm{\π}$

$\stackrel{\&conjugate0;}{z}\=\frac{M_{\mathrm{xy}}}{m}\=\frac{-\frac{32}{7} \mathrm{\π}}{\frac{64}{3} \mathrm{\π}}$ = $-\frac{3}{14}$ 

Table 8.3.4(b)   First moments and the coordinates of the center of mass

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

Table 8.3.4(c)   Centroid computed by task template that implements the CenterOfMass command

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

Table 8.3.4(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

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

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

$C≔\left[\mathrm{CenterOfMass}\left(1\,\mathrm{\rho }\=1\+\cos \left(\mathrm{\phi }\right)..2\,\mathrm{\phi }\=0..\mathrm{\pi }\,\mathrm{\theta }\=0..2 \mathrm{\pi }\,\mathrm{coordinates}\=\mathrm{spherical}\left[\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right]\right)\right]$

$\left[\frac{4}{15}\,\mathrm{\π}\,0\right]$

Table 8.3.4(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}{15}\right]$$$

Initialize

$\mathrm{\δ}≔{\mathrm{\ρ}}^2\:$

Apply the CenterOfMass command from the Student MultivariateCalculus package

$\mathrm{CenterOfMass}\left(\mathrm{\δ}\,\mathrm{\rho }\=1\+\cos \left(\mathrm{\phi }\right)..2\,\mathrm{\phi }\=0..\mathrm{\pi }\,\mathrm{\theta }\=0..2 \mathrm{\pi }\,\mathrm{coordinates}\=\mathrm{spherical}\left[\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right]\,\mathrm{output}\=\mathrm{integral}\right)$

$\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}{\∫}_{1\+\cos \left(\mathrm{\φ}\right)}^2{\sin \left(\mathrm{\φ}\right)}^2\cos \left(\mathrm{\θ}\right){\mathrm{\ρ}}^5\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}{\∫}_{1\+\cos \left(\mathrm{\φ}\right)}^2{\mathrm{\ρ}}^4\sin \left(\mathrm{\φ}\right)\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}}\,\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}{\∫}_{1\+\cos \left(\mathrm{\φ}\right)}^2{\sin \left(\mathrm{\φ}\right)}^2\sin \left(\mathrm{\θ}\right){\mathrm{\ρ}}^5\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}{\∫}_{1\+\cos \left(\mathrm{\φ}\right)}^2{\mathrm{\ρ}}^4\sin \left(\mathrm{\φ}\right)\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}}\,\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}{\∫}_{1\+\cos \left(\mathrm{\φ}\right)}^2\cos \left(\mathrm{\φ}\right){\mathrm{\ρ}}^5\sin \left(\mathrm{\φ}\right)\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}{\∫}_{1\+\cos \left(\mathrm{\φ}\right)}^2{\mathrm{\ρ}}^4\sin \left(\mathrm{\φ}\right)\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}}$

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

$\left[\frac{3}{14}\,\mathrm{\π}\,0\right]$

Table 8.3.4(f)   Center of mass in cylindrical coordinates

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

$\left[0\,0\,-\frac{3}{14}\right]$