First Moments

Centroid

$M_{\mathrm{yz}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^3{\int }_0^{9-r^2}\cos \left(\mathrm{\theta }\right)r^2 \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = 0

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

$M_{\mathrm{xz}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^3{\int }_0^{0-r^2}\sin \left(\mathrm{\theta }\right)r^2 \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $0$

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

$M_{\mathrm{xy}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^3{\int }_0^{9-r^2}zr \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $\frac{243}{2}\mathrm{\π}$

$\stackrel{\&conjugate0;}{z}\=\frac{M_{\mathrm{xy}}}{V}\=\frac{\frac{243}{2}\mathrm{\π}}{\frac{81}{2} \mathrm{\π}}$ = $3$ 

Table 8.3.6(a)   First moments and the coordinates of the centroid

First Moments

Center of Mass

$M_{\mathrm{yz}}$ = ${\int }_0^{2\mathrm{\pi }}{\int }_0^3{\int }_0^{9-r^2}\cos \left(\mathrm{\theta }\right)r^{2}z \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $0$ 

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

$M_{\mathrm{xz}}$ = ${\int }_0^{2\mathrm{\pi }}{\int }_0^3{\int }_0^{9-r^2}\sin \left(\mathrm{\theta }\right)r^{2}z \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $0$ 

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

$M_{\mathrm{xy}}$ = ${\int }_0^{2\mathrm{\pi }}{\int }_0^3{\int }_0^{9-r^2}{z}^{2}r \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $\frac{2187}{4}\mathrm{\π}$

$\stackrel{\&conjugate0;}{z}\=\frac{M_{\mathrm{xy}}}{m}\=\frac{\frac{2187}{4}\mathrm{\π}}{\frac{243}{2}\mathrm{\π}}$ = $\frac{9}{2}$

Table 8.3.6(b)   First moments and the coordinates of the center of mass

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

Table 8.3.6(c)   Centroid computed by task template that implements the CenterOfMass command

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

Table 8.3.6(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

$\mathrm{CenterOfMass}\left(1\,z\=0..9-r^2\,r\=0..3\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{cylindrical}\left[r\,\mathrm{\θ}\,z\right]\,\mathrm{output}\=\mathrm{integral}\right)$

$\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{-r^2\+9}\cos \left(\mathrm{\θ}\right)r^2\ⅆz\ⅆr\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{-r^2\+9}r\ⅆz\ⅆr\ⅆ\mathrm{\θ}}\,\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{-r^2\+9}\sin \left(\mathrm{\θ}\right)r^2\ⅆz\ⅆr\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{-r^2\+9}r\ⅆz\ⅆr\ⅆ\mathrm{\θ}}\,\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{-r^2\+9}zr\ⅆz\ⅆr\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{-r^2\+9}r\ⅆz\ⅆr\ⅆ\mathrm{\θ}}$

$C≔\left[\mathrm{CenterOfMass}\left(1\,z\=0..9- r^2\,r\=0..3\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{cylindrical}\left[r\,\mathrm{\theta }\,z\right]\right)\right]$

$\left[0\,0\,3\right]$

Table 8.3.6(e)   Centroid in cylindrical coordinates

Initialize

$\mathrm{\δ}≔z\:$

Apply the CenterOfMass command from the Student MultivariateCalculus package

$\mathrm{CenterOfMass}\left(\mathrm{\δ}\,z\=0..9-r^2\,r\=0..3\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{cylindrical}\left[r\,\mathrm{\theta }\,z\right]\,\mathrm{output}\=\mathrm{integral}\right)$

$\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{-r^2\+9}{r}^2\cos \left(\mathrm{\θ}\right)z\ⅆz\ⅆr\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{-r^2\+9}zr\ⅆz\ⅆr\ⅆ\mathrm{\θ}}\,\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{-r^2\+9}{r}^2\sin \left(\mathrm{\θ}\right)z\ⅆz\ⅆr\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{-r^2\+9}zr\ⅆz\ⅆr\ⅆ\mathrm{\θ}}\,\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{-r^2\+9}{z}^{2}r\ⅆz\ⅆr\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{-r^2\+9}zr\ⅆz\ⅆr\ⅆ\mathrm{\θ}}$

$\mathrm{CM}≔\mathrm{simplify}\left(\left[\mathrm{CenterOfMass}\left(\mathrm{\δ}\,z\=0..9-r^2\,r\=0..3\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{cylindrical}\left[r\,\mathrm{\theta }\,z\right]\right)\right]\right)$

$\left[0\,0\,\frac{9}{2}\right]$

Table 8.3.6(f)   Center of mass in cylindrical coordinates

$\mathrm{cm}≔\mathrm{eval}\left(\left[r \cos \left(\mathrm{\theta }\right)\,r \sin \left(\mathrm{\theta }\right)\,z\right]\,\mathrm{Equate}\left(\left[r\,\mathrm{\theta }\,z\right]\,\mathrm{CM}\right)\right)$

$\left[0\,0\,\frac{9}{2}\right]$