First Moments

Centroid

$M_{\mathrm{yz}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^1{\int }_{3r^2}^{4-r^2}\cos \left(\mathrm{\theta }\right)r^2 \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = 0

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

$M_{\mathrm{xz}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^1{\int }_{3r^2}^{4-r^2}\sin \left(\mathrm{\theta }\right)r^2 \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $0$ 

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

$M_{\mathrm{xy}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^1{\int }_{3r^2}^{4-r^2}zr \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $\frac{14}{3}\mathrm{\π}$ 

$\stackrel{\&conjugate0;}{z}\=\frac{M_{\mathrm{xy}}}{V}\=\frac{\frac{14}{3} \mathrm{\π}}{2 \mathrm{\π}}$ = $\frac{7}{3}$ 

Table 8.3.8(a)   First moments and the coordinates of the centroid

First Moments

Center of Mass

$M_{\mathrm{yz}}$ = ${\int }_0^{2\mathrm{\pi }}{\int }_0^1{\int }_{3r^2}^{4-r^2}\cos \left(\mathrm{\theta }\right)r^{3}z\cos \left(\frac{\mathrm{\theta }}{6}\right) \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $-\frac{\sqrt{3}}{14}$

$\stackrel{\&conjugate0;}{x}\=\frac{M_{\mathrm{yz}}}{m}\=\frac{-\frac{\sqrt{3}}{14}}{\frac{136}{35}\sqrt{3}}$ = $-\frac{67}{3256}$

$M_{\mathrm{xz}}$ = ${\int }_0^{2\mathrm{\pi }}{\int }_0^1{\int }_{3r^2}^{4-r^2}\sin \left(\mathrm{\theta }\right)r^{3}z\cos \left(\frac{\mathrm{\θ}}{6}\right) \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $\frac{3}{7}$

$\stackrel{\&conjugate0;}{y}\=\frac{M_{\mathrm{xz}}}{m}\=\frac{\frac{7}{3}}{\frac{136}{35}\sqrt{3}}$ = $\frac{67}{1628}\sqrt{3}$

$M_{\mathrm{xy}}$ = ${\int }_0^{2\mathrm{\pi }}{\int }_0^1{\int }_{3r^2}^{4-r^2}{z}^2{r}^2\cos \left(\frac{\mathrm{\theta }}{6}\right) \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $\frac{3256}{315}\sqrt{3}$

$\stackrel{\&conjugate0;}{z}\=\frac{M_{\mathrm{xy}}}{m}\=\frac{\frac{3256}{315}\sqrt{3}}{\frac{136}{35}\sqrt{3}}$ = $\frac{8402}{3663}$

Table 8.3.8(b)   First moments and the coordinates of the center of mass

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

Table 8.3.8(c)   Centroid computed by task template that implements the CenterOfMass command

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

Table 8.3.8(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

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

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

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

$\left[0\,0\,\frac{37}{21}\right]$

Table 8.3.8(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{37}{21}\right]$$$

Initialize

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

Apply the CenterOfMass command from the Student MultivariateCalculus package

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

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

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

$\left[\frac{67}{3256}\sqrt{13}\,-\arctan \left(2\sqrt{3}\right)\+\mathrm{\π}\,\frac{8402}{3663}\right]$

Table 8.3.8(f)   Center of mass in cylindrical coordinates

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

$\left[-\frac{67}{3256}\,\frac{67}{1628}\sqrt{3}\,\frac{8402}{3663}\right]$

$\mathrm{evalf}\left(\mathrm{cm}\right)$ = $\left[-0.02057739558\,0.07128218926\,2.293748294\right]$$$