First Moments

Centroid

$M_{\mathrm{yz}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^3{\int }_0^{5-r\sin \left(\mathrm{\theta }\right)}\cos \left(\mathrm{\theta }\right)r^2 \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = 0

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

$M_{\mathrm{xz}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^3{\int }_0^{5-r\sin \left(\mathrm{\theta }\right)}{r}^2\sin \left(\mathrm{\theta }\right) \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $-\frac{81}{4}\mathrm{\π}$

$\stackrel{\&conjugate0;}{y}\=\frac{M_{\mathrm{xz}}}{V}\=\frac{-\frac{81}{4}\mathrm{\π}}{45 \mathrm{\π}}\=-\frac{9}{20}$

$M_{\mathrm{xy}}\={\int }_0^{2\mathrm{\pi }}{\int }_0^3{\int }_0^{5-r\sin \left(\mathrm{\theta }\right)}zr \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $\frac{981}{8}\mathrm{\π}$

$\stackrel{\&conjugate0;}{z}\=\frac{M_{\mathrm{xy}}}{V}\=\frac{\frac{981}{8}\mathrm{\π}}{45 \mathrm{\π}}\=\frac{109}{40}$

Table 8.3.5(a)   First moments and the coordinates of the centroid

First Moments

Center of Mass

$M_{\mathrm{yz}}$ = ${\int }_0^{2\mathrm{\pi }}{\int }_0^3{\int }_0^{5-r\sin \left(\mathrm{\theta }\right)}\cos \left(\mathrm{\theta }\right)r^2\left(r^2\+z\right)\sin \left(\frac{\mathrm{\theta }}{4}\right) \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $-\frac{992958}{25025}$

$\stackrel{\&conjugate0;}{x}\=\frac{M_{\mathrm{yz}}}{m}\=\frac{-\frac{992958}{25025}}{\frac{131322}{175}}$ = $-\frac{165493}{3129841}$

$M_{\mathrm{xz}}$ = ${\int }_0^{2\mathrm{\pi }}{\int }_0^3{\int }_0^{5-r\sin \left(\mathrm{\theta }\right)}{r}^2\sin \left(\mathrm{\theta }\right)\left(r^2\+z\right)\sin \left(\frac{\mathrm{\theta }}{4}\right) \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $-\frac{21250536}{25025}$

$\stackrel{\&conjugate0;}{y}\=\frac{M_{\mathrm{xz}}}{m}\=\frac{-\frac{21250536}{25025}}{\frac{131322}{175}}$ = $-\frac{3541756}{3129841}$

$M_{\mathrm{xy}}$ = ${\int }_0^{2\mathrm{\pi }}{\int }_0^3{\int }_0^{5-r\sin \left(\mathrm{\theta }\right)}zr\left(r^2\+z\right)\sin \left(\frac{\mathrm{\θ}}{4}\right) \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $\frac{130252293}{50050}$

$\stackrel{\&conjugate0;}{z}\=\frac{M_{\mathrm{xy}}}{m}\=\frac{\frac{130252293}{50050}}{\frac{131322}{175}}$ = $\frac{43417431}{12519364}$

Table 8.3.5(b)   First moments and the coordinates of the center of mass

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

Table 8.3.5(c)   Centroid computed by task template that implements the CenterOfMass command

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

Table 8.3.5(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

$\mathrm{CenterOfMass}\left(1\,z\=0..5- r \sin \left(\mathrm{\θ}\right)\,r\=0..3\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{cylindrical}\left[r\,\mathrm{\θ}\,z\right]\,\mathrm{output}\=\mathrm{integral}\right)$

$\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{5-r\sin \left(\mathrm{\θ}\right)}\cos \left(\mathrm{\θ}\right)r^2\ⅆz\ⅆr\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{5-r\sin \left(\mathrm{\θ}\right)}r\ⅆz\ⅆr\ⅆ\mathrm{\θ}}\,\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{5-r\sin \left(\mathrm{\θ}\right)}{r}^2\sin \left(\mathrm{\θ}\right)\ⅆz\ⅆr\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{5-r\sin \left(\mathrm{\θ}\right)}r\ⅆz\ⅆr\ⅆ\mathrm{\θ}}\,\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{5-r\sin \left(\mathrm{\θ}\right)}zr\ⅆz\ⅆr\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{5-r\sin \left(\mathrm{\θ}\right)}r\ⅆz\ⅆr\ⅆ\mathrm{\θ}}$

$C≔\left[\mathrm{CenterOfMass}\left(1\,z\=0..5- r \sin \left(\mathrm{\θ}\right)\,r\=0..3\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{cylindrical}\left[r\,\mathrm{\theta }\,z\right]\right)\right]$

$\left[\frac{9}{20}\,-\frac{1}{2}\mathrm{\π}\,\frac{109}{40}\right]$

Table 8.3.5(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\,-\frac{9}{20}\,\frac{109}{40}\right]$$$

Initialize

$\mathrm{\δ}≔\left(r^2\+z\right) \sin \left(\mathrm{\theta }\/4\right)\:$

Apply the CenterOfMass command from the Student MultivariateCalculus package

$\mathrm{CenterOfMass}\left(\mathrm{\δ}\,z\=0..5- r \sin \left(\mathrm{\θ}\right)\,r\=0..3\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{cylindrical}\left[r\,\mathrm{\theta }\,z\right]\,\mathrm{output}\=\mathrm{integral}\right)$

$\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{5-r\sin \left(\mathrm{\θ}\right)}\cos \left(\mathrm{\θ}\right)r^2\left(r^2\+z\right)\sin \left(\frac{1}{4}\mathrm{\θ}\right)\ⅆz\ⅆr\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{5-r\sin \left(\mathrm{\θ}\right)}r\left(r^2\+z\right)\sin \left(\frac{1}{4}\mathrm{\θ}\right)\ⅆz\ⅆr\ⅆ\mathrm{\θ}}\,\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{5-r\sin \left(\mathrm{\θ}\right)}{r}^2\sin \left(\mathrm{\θ}\right)\left(r^2\+z\right)\sin \left(\frac{1}{4}\mathrm{\θ}\right)\ⅆz\ⅆr\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{5-r\sin \left(\mathrm{\θ}\right)}r\left(r^2\+z\right)\sin \left(\frac{1}{4}\mathrm{\θ}\right)\ⅆz\ⅆr\ⅆ\mathrm{\θ}}\,\frac{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{5-r\sin \left(\mathrm{\θ}\right)}zr\left(r^2\+z\right)\sin \left(\frac{1}{4}\mathrm{\θ}\right)\ⅆz\ⅆr\ⅆ\mathrm{\θ}}{{\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{5-r\sin \left(\mathrm{\θ}\right)}r\left(r^2\+z\right)\sin \left(\frac{1}{4}\mathrm{\θ}\right)\ⅆz\ⅆr\ⅆ\mathrm{\θ}}$

$\mathrm{CM}≔\mathrm{simplify}\left(\left[\mathrm{CenterOfMass}\left(\mathrm{\δ}\,z\=0..5- r \sin \left(\mathrm{\θ}\right)\,r\=0..3\,\mathrm{\θ}\=0..2 \mathrm{\π}\,\mathrm{coordinates}\=\mathrm{cylindrical}\left[r\,\mathrm{\theta }\,z\right]\right)\right]\right)$

$\left[\frac{1}{1593089069}\sqrt{3257016970918738385}\,\arctan \left(\frac{3541756}{165493}\right)-\mathrm{\π}\,\frac{43417431}{12519364}\right]$

Table 8.3.5(f)   Center of mass in cylindrical coordinates

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

$\left[-\frac{165493}{3129841}\,-\frac{3541756}{3129841}\,\frac{43417431}{12519364}\right]$

$\mathrm{evalf}\left(\mathrm{cm}\right)$ = $\left[-0.05287584896\,-1.131608922\,3.468022098\right]$$$