Onglet sphérique

En coordonnées sphériques

${\displaystyle x_{G}={\frac {\int _{-\alpha /2}^{\alpha /2}\int _{0}^{\pi }\int _{0}^{r}\rho \sin \theta \cos \varphi \cdot \rho ^{2}\sin \theta \,\mathrm {d} \rho \,\mathrm {d} \theta \,\mathrm {d} \varphi }{\int _{-\alpha /2}^{\alpha /2}\int _{0}^{\pi }\int _{0}^{r}\rho ^{2}\sin \theta \,\mathrm {d} \rho \,\mathrm {d} \theta \,\mathrm {d} \varphi }}={\frac {M_{x}}{V}}}$

On sait déjà que ${\displaystyle V={\frac {2}{3}}\alpha r^{3}}$. Il reste à calculer ${\displaystyle Mx}$.

${\displaystyle Mx=\int _{-\alpha /2}^{\alpha /2}\cos \varphi \,\mathrm {d} \varphi \cdot \int _{0}^{\pi }\sin ^{2}\theta \,\mathrm {d} \theta \cdot \int _{0}^{r}\rho ^{3}\,\mathrm {d} \rho }$

${\displaystyle Mx={\Big [}\sin \varphi {\Big ]}_{-\alpha /2}^{\alpha /2}\cdot \left[{\frac {\theta }{2}}-{\frac {\sin(2\theta )}{4}}\right]_{0}^{\pi }\cdot \left[{\frac {\rho ^{4}}{4}}\right]_{0}^{r}}$

${\displaystyle Mx=2\sin(\alpha /2)\cdot {\frac {\pi }{2}}\cdot {\frac {r^{4}}{4}}={\frac {1}{4}}\pi \sin(\alpha /2)r^{4}}$

Donc

${\displaystyle x_{G}={\frac {{\frac {1}{4}}\pi \sin(\alpha /2)r^{4}}{{\frac {2}{3}}\alpha r^{3}}}={\frac {3}{4}}{\frac {\pi \sin(\alpha /2)r}{2\alpha }}}$