First Moments

Centroid

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

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

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

$\stackrel{\&conjugate0;}{y}\=\frac{M_{\mathrm{xz}}}{V}\=\frac{2 \mathrm{\π}}{\frac{16}{3}}\=\frac{3 \mathrm{\π}}{8}$

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

$\stackrel{\&conjugate0;}{z}\=\frac{M_{\mathrm{xy}}}{V}\=\frac{\mathrm{\π}}{\frac{16}{3}}$ = $\frac{3 \mathrm{\π}}{16}$

Table 8.3.2(a)   First moments and the coordinates of the centroid

First Moments

Center of Mass

$M_{\mathrm{yz}}$ = ${\int }_0^{\mathrm{\pi }}{\int }_0^2{\int }_0^{r\sin \left(\mathrm{\theta }\right)}\cos \left(\mathrm{\theta }\right)r^3{z}^2\cos \left(\frac{\mathrm{\theta }}{3}\right) \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $\frac{10368}{35035}$

$\stackrel{\&conjugate0;}{x}\=\frac{M_{\mathrm{yz}}}{m}\=\frac{\frac{10368}{35035}}{\frac{81}{20}}$ = $\frac{512}{7007}$

$M_{\mathrm{xz}}$ = ${\int }_0^{\mathrm{\pi }}{\int }_0^2{\int }_0^{r\sin \left(\mathrm{\theta }\right)}{r}^3\sin \left(\mathrm{\theta }\right)z^2\cos \left(\frac{\mathrm{\theta }}{3}\right) \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $\frac{124416}{35035}\sqrt{3}$

$\stackrel{\&conjugate0;}{y}\=\frac{M_{\mathrm{xz}}}{m}\=\frac{\frac{124416}{35035}\sqrt{3}}{\frac{81}{20}}$ = $\frac{6144}{7007}\sqrt{3}$

$M_{\mathrm{xy}}$ = ${\int }_0^{\mathrm{\pi }}{\int }_0^2{\int }_0^{r\sin \left(\mathrm{\theta }\right)}{z}^3{r}^2\cos \left(\frac{\mathrm{\theta }}{3}\right) \mathrm{dz} \mathrm{dr} d\mathrm{\θ}$ = $\frac{93312}{35035}\sqrt{3}$

$\stackrel{\&conjugate0;}{z}\=\frac{M_{\mathrm{xy}}}{m}\=\frac{\frac{93312}{35035}\sqrt{3}}{\frac{81}{20}}$ = $\frac{4608}{7007}\sqrt{3}$

Table 8.3.2(b)   First moments and the coordinates of the center of mass

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

Table 8.3.2(c)   Centroid computed by task template that implements the CenterOfMass command

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

Table 8.3.2(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

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

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

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

$\left[\frac{3}{8}\mathrm{\π}\,\frac{1}{2}\mathrm{\π}\,\frac{3}{16}\mathrm{\π}\right]$

Table 8.3.2(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{3}{8}\mathrm{\π}\,\frac{3}{16}\mathrm{\π}\right]$$$$$

Initialize

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

Apply the CenterOfMass command from the Student MultivariateCalculus package

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

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

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

$\left[\frac{512}{7007}\sqrt{433}\,\arctan \left(12\sqrt{3}\right)\,\frac{4608}{7007}\sqrt{3}\right]$

Table 8.3.2(f)   Center of mass in cylindrical coordinates

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

$\left[\frac{512}{7007}\,\frac{6144}{7007}\sqrt{3}\,\frac{4608}{7007}\sqrt{3}\right]$

$\mathrm{evalf}\left(\mathrm{cm}\right)$ = $\left[0.07306978736\,1.518727011\,1.139045258\right]$$$