First Moments

Centroid

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

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

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

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

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

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

Table 8.3.1(a)   First moments and the coordinates of the centroid

First Moments

Center of Mass

$M_{\mathrm{yz}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^3{\int }_0^{r^2}\cos \left(\mathrm{\theta }\right)r^{4}z\sin \left(\frac{\mathrm{\theta }}{6}\right) \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $-\frac{6561}{70}$

$\stackrel{\&conjugate0;}{x}\=\frac{M_{\mathrm{yz}}}{m}\=\frac{-\frac{6561}{70}}{\frac{19683}{16}}$ = $-\frac{8}{105}$

$M_{\mathrm{xz}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^3{\int }_0^{r^2}\sin \left(\mathrm{\theta }\right)r^{4}z\sin \left(\frac{\mathrm{\theta }}{6}\right) \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $-\frac{19683}{35}\sqrt{3}$

$\stackrel{\&conjugate0;}{y}\=\frac{M_{\mathrm{xz}}}{m}\=\frac{-\frac{19683}{35}\sqrt{3}}{\frac{19683}{16}}$ = $-\frac{16}{35}\sqrt{3}$

$M_{\mathrm{xy}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^3{\int }_0^{r^2}{z}^2{r}^3\sin \left(\frac{\mathrm{\theta }}{6}\right) \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $\frac{59049}{10}$

$\stackrel{\&conjugate0;}{z}\=\frac{M_{\mathrm{xy}}}{m}\=\frac{\frac{59049}{10}}{\frac{19683}{16}}$ = $\frac{24}{5}$ 

Table 8.3.1(b)   First moments and the coordinates of the center of mass

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

Table 8.3.1(c)   Centroid computed by task template that implements the CenterOfMass command

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

Table 8.3.1(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

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

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

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

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

Table 8.3.1(e)   Centroid in cylindrical coordinates

Initialize

$\mathrm{\δ}≔z r^2\sin \left(\mathrm{\theta }\/6\right)\:$

Apply the CenterOfMass command from the Student MultivariateCalculus package

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

$\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{r^2}\cos \left(\mathrm{\θ}\right)r^{4}z\sin \left(\frac{1}{6}\mathrm{\θ}\right)\ⅆz\ⅆr\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{r^2}{r}^{3}z\sin \left(\frac{1}{6}\mathrm{\θ}\right)\ⅆz\ⅆr\ⅆ\mathrm{\θ}}\,\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{r^2}\sin \left(\mathrm{\θ}\right)r^{4}z\sin \left(\frac{1}{6}\mathrm{\θ}\right)\ⅆz\ⅆr\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{r^2}{r}^{3}z\sin \left(\frac{1}{6}\mathrm{\θ}\right)\ⅆz\ⅆr\ⅆ\mathrm{\θ}}\,\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{r^2}{z}^2{r}^3\sin \left(\frac{1}{6}\mathrm{\θ}\right)\ⅆz\ⅆr\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{r^2}{r}^{3}z\sin \left(\frac{1}{6}\mathrm{\θ}\right)\ⅆz\ⅆr\ⅆ\mathrm{\θ}}$

$\mathrm{CM}≔\left[\mathrm{CenterOfMass}\left(\mathrm{\delta }\,z\=0..r^2\,r\=0..3\,\mathrm{\theta }\=0..2 \mathrm{\pi }\,\mathrm{coordinates}\=\mathrm{cylindrical}\left[r\,\mathrm{\theta }\,z\right]\right)\right]$

$\left[\frac{8}{105}\sqrt{109}\,\arctan \left(6\sqrt{3}\right)-\mathrm{\π}\,\frac{24}{5}\right]$

Table 8.3.1(f)   Center of mass in cylindrical coordinates

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

$\left[-\frac{8}{105}\,-\frac{16}{35}\sqrt{3}\,\frac{24}{5}\right]$