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Calculus - Multivariate≻Integration≻Center of Mass≻Polar

Table 6.5.6(a)   Calculation of center of mass via task template

Define the density $\mathrm{\ρ}$ in polar coordinates

$\mathrm{\ρ}\=\left(1\+r^2 \left(2 {\cos }^2\left(\mathrm{\theta }\right)\+3 {\sin }^2\left(\mathrm{\theta }\right)\right)\right)\/10$$\stackrel{\text{assign}}{\to }$

Obtain $m$, the total mass in region $R$ 

${\int }_0^{2 \mathrm{\π}}{\int }_0^{1\+\cos \left(\mathrm{\θ}\right)}r\cdot \mathrm{\ρ} \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{209}{320}\mathrm{\π}$$\stackrel{\text{assign to a name}}{\to }$$m$

Obtain $M_x$, the total moments about the $x$-axis

${\int }_0^{2 \mathrm{\π}}{\int }_0^{1\+\cos \left(\mathrm{\θ}\right)}{r}^2\cdot \mathrm{\ρ} \sin \left(\mathrm{\θ}\right) \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $0$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{Mx}$$$

Obtain $M_y$, the total moments about the $y$-axis

${\int }_0^{2 \mathrm{\π}}{\int }_0^{1\+\cos \left(\mathrm{\θ}\right)}{r}^2\cdot \mathrm{\ρ} \cos \left(\mathrm{\theta }\right) \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{449}{640}\mathrm{\π}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{My}$$$

Obtain $\stackrel{\&conjugate0;}{x}\=M_y\/m$ 

$\mathrm{My}\/m$ = $\frac{449}{418}$$$

Obtain $\stackrel{\&conjugate0;}{y}\=M_x\/m$ 

$\mathrm{Mx}\/m$ = $0$$$

Table 6.5.6(b)   Calculation of the centroid from first principles

Total mass

$\mathrm{\ρ}≔\left(1\+r^2 \left(2 {\cos }^2\left(\mathrm{\θ}\right)\+3 {\sin }^2\left(\mathrm{\θ}\right)\right)\right)\/10\:$

$q≔\mathrm{Int}\left(r\cdot \mathrm{\rho }\,\left[r\=0..1\+\cos \left(\mathrm{\theta }\right)\,\mathrm{\theta }\=0..2 \mathrm{\pi }\right]\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^{1\+\cos \left(\mathrm{\θ}\right)}r\left(\frac{1}{10}\+\frac{1}{10}{r}^2\left(2{\cos \left(\mathrm{\θ}\right)}^2\+3{\sin \left(\mathrm{\θ}\right)}^2\right)\right)\ⅆr\ⅆ\mathrm{\θ}$

$m≔\mathrm{value}\left(q\right)$

$\frac{209}{320}\mathrm{\π}$

First Moments

$q≔\mathrm{Int}\left(r^2\cdot \mathrm{\ρ}\cdot \sin \left(\mathrm{\θ}\right)\,\left[r\=0..1\+\cos \left(\mathrm{\theta }\right)\,\mathrm{\theta }\=0..2 \mathrm{\pi }\right]\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^{1\+\cos \left(\mathrm{\θ}\right)}{r}^2\left(\frac{1}{10}\+\frac{1}{10}{r}^2\left(2{\cos \left(\mathrm{\θ}\right)}^2\+3{\sin \left(\mathrm{\θ}\right)}^2\right)\right)\sin \left(\mathrm{\θ}\right)\ⅆr\ⅆ\mathrm{\θ}$

$M_x≔\mathrm{value}\left(q\right)$

$0$

$q≔\mathrm{Int}\left(r^2\cdot \mathrm{\ρ}\cdot \cos \left(\mathrm{\theta }\right)\,\left[r\=0..1\+\cos \left(\mathrm{\theta }\right)\,\mathrm{\theta }\=0..2 \mathrm{\pi }\right]\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^{1\+\cos \left(\mathrm{\θ}\right)}{r}^2\left(\frac{1}{10}\+\frac{1}{10}{r}^2\left(2{\cos \left(\mathrm{\θ}\right)}^2\+3{\sin \left(\mathrm{\θ}\right)}^2\right)\right)\cos \left(\mathrm{\θ}\right)\ⅆr\ⅆ\mathrm{\θ}$

$M_y≔\mathrm{value}\left(q\right)$

$\frac{449}{640}\mathrm{\π}$

Coordinates of the center of mass $\left(\stackrel{\&conjugate0;}{x}\,\stackrel{\&conjugate0;}{y}\right)$

$\left[\frac{M_y}{m}\,\frac{M_x}{m}\right]$ = $\left[\frac{449}{418}\,0\right]$$$