First Moments

Centroid

$M_{\mathrm{yz}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^2{\int }_0^{4-r^2}\cos \left(\mathrm{\theta }\right)r^2 \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = 0

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

$M_{\mathrm{xz}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^2{\int }_0^{4-r^2}\sin \left(\mathrm{\theta }\right)r^2 \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$

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

$M_{\mathrm{xy}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^2{\int }_0^{4-r^2}zr \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $\frac{32}{3}\mathrm{\π}$ 

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

Table 8.3.11(a)   First moments and the coordinates of the centroid

First Moments

Center of Mass

$M_{\mathrm{yz}}$ = ${\int }_0^{2\mathrm{\pi }}{\int }_0^2{\int }_0^{4-r^2}\cos \left(\mathrm{\theta }\right)r^3{z}^2\sin \left(\frac{\mathrm{\θ}}{3}\right) \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $-\frac{24}{5}$

$\stackrel{\&conjugate0;}{x}\=\frac{M_{\mathrm{yz}}}{m}\=\frac{-\frac{24}{5}}{\frac{4096}{105}}$ = $-\frac{63}{512}$

$M_{\mathrm{xz}}$ = ${\int }_0^{2\mathrm{\pi }}{\int }_0^2{\int }_0^{4-r^2}\sin \left(\mathrm{\theta }\right)r^3{z}^2\sin \left(\frac{\mathrm{\θ}}{3}\right) \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $-\frac{24}{5}\sqrt{3}$

$\stackrel{\&conjugate0;}{y}\=\frac{M_{\mathrm{xz}}}{m}\=\frac{-\frac{24}{5}\sqrt{3}}{\frac{4096}{105}}$ = $-\frac{63}{512}\sqrt{3}$

$M_{\mathrm{xy}}$ = ${\int }_0^{2\mathrm{\pi }}{\int }_0^2{\int }_0^{4-r^2}{z}^3{r}^2\sin \left(\frac{\mathrm{\θ}}{3}\right) \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $\frac{32768}{385}$

$\stackrel{\&conjugate0;}{z}\=\frac{M_{\mathrm{xy}}}{m}\=\frac{\frac{32768}{385}}{\frac{4096}{105}}$ = $\frac{24}{11}$

Table 8.3.11(b)   First moments and the coordinates of the center of mass

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

Table 8.3.11(c)   Centroid computed by task template that implements the CenterOfMass command

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

Table 8.3.11(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

$\mathrm{CenterOfMass}\left(1\,z\=0..4-r^2\,r\=0..2\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{cylindrical}\left[r\,\mathrm{\θ}\,z\right]\,\mathrm{output}\=\mathrm{integral}\right)$

$\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^2{\∫}_0^{-r^2\+4}\cos \left(\mathrm{\θ}\right)r^2\ⅆz\ⅆr\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^2{\∫}_0^{-r^2\+4}r\ⅆz\ⅆr\ⅆ\mathrm{\θ}}\,\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^2{\∫}_0^{-r^2\+4}\sin \left(\mathrm{\θ}\right)r^2\ⅆz\ⅆr\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^2{\∫}_0^{-r^2\+4}r\ⅆz\ⅆr\ⅆ\mathrm{\θ}}\,\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^2{\∫}_0^{-r^2\+4}zr\ⅆz\ⅆr\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^2{\∫}_0^{-r^2\+4}r\ⅆz\ⅆr\ⅆ\mathrm{\θ}}$

$C≔\left[\mathrm{CenterOfMass}\left(1\,z\=0..4-r^2\,r\=0..2\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{cylindrical}\left[r\,\mathrm{\theta }\,z\right]\right)\right]$

$\left[0\,0\,\frac{4}{3}\right]$

Table 8.3.11(e)   Centroid in cylindrical coordinates

$\mathrm{eval}\left(\left[r \cos \left(\mathrm{\theta }\right)\,r \sin \left(\mathrm{\theta }\right)\,z\right]\,\mathrm{Equate}\left(\left[r\,\mathrm{\theta }\,z\right]\,C\right)\right)$ = $\left[0\,0\,\frac{4}{3}\right]$$$

Initialize

$\mathrm{\δ}≔z^{2}r \sin \left(\mathrm{\theta }\/3\right)\:$

Apply the CenterOfMass command from the Student MultivariateCalculus package

$\mathrm{CenterOfMass}\left(\mathrm{\δ}\,z\=0..4-r^2\,r\=0..2\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{cylindrical}\left[r\,\mathrm{\theta }\,z\right]\,\mathrm{output}\=\mathrm{integral}\right)$

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

$\mathrm{CM}≔\left[\mathrm{CenterOfMass}\left(\mathrm{\δ}\,z\=0..4-r^2\,r\=0..2\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{cylindrical}\left[r\,\mathrm{\theta }\,z\right]\right)\right]$

$\left[\frac{63}{256}\,-\frac{2}{3}\mathrm{\π}\,\frac{24}{11}\right]$

Table 8.3.11(f)   Center of mass in cylindrical coordinates

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

$\left[-\frac{63}{512}\,-\frac{63}{512}\sqrt{3}\,\frac{24}{11}\right]$

$\mathrm{evalf}\left(\mathrm{cm}\right)$ = $\left[-0.1230468750\,-0.2131234393\,2.181818182\right]$$$