Exercise:
Wie berechnet man das Trägheitsmoment einer Kugel?

Solution:
center tikzpicturescale.tdplot_main_coordslatex % coordinate system coordinate O at ; draw- colorgreen!!black -- nodeanchornorth eastx; draw- colorgreen!!black -- nodeanchornorth westy; draw- colorgreen!!black -- nodeanchorsouthz; % Punkt P tdplotsetcoordPrvecthetavecphivec % --- gefüllter Winkel phi in der xy-Ebene --- fillred!opacity. O -- . arcstart angle anglephivecradius. -- cycle; % Vektor und Hilfslinien draw-stealthcolorred O -- P nodemidway right r; drawdashed colorred O -- Pxy; drawdashed colorred P -- Pxy; drawdotted colorred Py -- Pxy; drawdotted colorred Px -- Pxy; drawdotted colorred Pz -- P; % phi-Bogen tdplotdrawarcO.phivecanchornorthvarphi % --- gefüllter Winkel theta in der gedrehten Ebene --- tdplotsetthetaplanecoordsphivec scopetdplot_rotated_coords fillblue!opacity. -- . arcstart angle anglethetavecradius. -- cycle; scope % theta-Bogen tdplotdrawarctdplot_rotated_coords.thetavec anchorsouth westvartheta node at . x r shetacosphi; node at . y r shetasinphi; node at . z r costheta; tikzpicture center Trägheitsmoment: I tilde r^ ddm tilde r^ rho ddV Aus Kugelsymmetrie: I fracI_x+I_y+I_z frac left y^+z^ ddm + dots right frac x^+y^+z^mboxdm frac r^ ddm Kugel mit Radius R und homogener Dichte rho: I frac r^ rho ddV frac r^ rho ddxddyddz frac r^ rho r^sinthetaddthetaddphiddr frac rho tcbhighmath_^pisinthetaddtheta _^piddphi _^R r^ddr fracrho leftleft-costhetarightright_^pi pi frac R^ frac rho fracpi R^ R^ frac mR^ textcolorredJacobi-Matrix für textcolorredKugelkoordinaten ddxddyddzrightarrow ddr ddtheta ddphi: det J det pmatrix fracpartial xpartial r & fracpartial xpartial theta & fracpartial xpartial phi fracpartial ypartial r & fracpartial ypartial theta & fracpartial ypartial phi fracpartial zpartial r & fracpartial zpartial theta & fracpartial zpartial phi pmatrix r^ sheta