Trägheitsmoment einer Pyramide
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The following quantities appear in the problem:
Länge \(\ell\) / Masse \(m\) / Volumen \(V\) / Fläche \(A\) / Höhe \(h\) / Trägheitsmoment \(J, \Theta, I\) / Radius \(r\) / Dichte \(\varrho\) /
The following formulas must be used to solve the exercise:
\(\varrho = \dfrac{m}{V} \quad \) \(V = \dfrac13 A h \quad \) \(J = \int r^2 \, \text{d}m \quad \) \(J = \dfrac{1}{10}ma^2 \quad \)
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Exercise:
Berechne das Trägheitsmoment einer Pyramide der Höhe h mit quadratischer Grundfläche Seitenlänge a bezüglich einer Achse durch die Spitze und den Mittelpunkt des Bodens Fusspunkt.
Solution:
I rho _^h mboxdz _-fracazh^fracazh mboxdy _-fracazh^fracazh x^+y^mboxdx rho _^h mboxdz _-fracazh^fracazh mboxdy left fracx^+y^xright_-fracazh^fracazh rho _^h mboxdz _-fracazh^fracazh mboxdy leftfracfraca^z^h^+fracazhy^right rho _^h mboxdz leftfracfraca^z^h^y+fracfracazhy^right_-fracazh^fracazh rho _^h mboxdz leftfracfraca^z^h^+fracfraca^z^h^right rho _^h fracfraca^z^h^ mboxdz fraca^rhoh^ _^h z^ mboxdz fraca^ rho h^ frac h^ fraca^ rho frac h frac frac a^ h rho a^ fracma^ Volumen der Pyramide: VfracGh center tikzpicturescale.latex draw- colorgreen!!black --- noderight y; draw- colorgreen!!black --- nodeleft x; draw- colorgreen!!black -.-- nodeabove z; drawcolorgreen!!black dashed ---; drawcolorgreen!!black dashed ---; drawthick colorred!!black ------------cycle; drawthick colorred!!black ----; drawthick colorred!!black ---; drawthick colorred!!black --; drawthick colorred!!black ---; tikzpicture center
Berechne das Trägheitsmoment einer Pyramide der Höhe h mit quadratischer Grundfläche Seitenlänge a bezüglich einer Achse durch die Spitze und den Mittelpunkt des Bodens Fusspunkt.
Solution:
I rho _^h mboxdz _-fracazh^fracazh mboxdy _-fracazh^fracazh x^+y^mboxdx rho _^h mboxdz _-fracazh^fracazh mboxdy left fracx^+y^xright_-fracazh^fracazh rho _^h mboxdz _-fracazh^fracazh mboxdy leftfracfraca^z^h^+fracazhy^right rho _^h mboxdz leftfracfraca^z^h^y+fracfracazhy^right_-fracazh^fracazh rho _^h mboxdz leftfracfraca^z^h^+fracfraca^z^h^right rho _^h fracfraca^z^h^ mboxdz fraca^rhoh^ _^h z^ mboxdz fraca^ rho h^ frac h^ fraca^ rho frac h frac frac a^ h rho a^ fracma^ Volumen der Pyramide: VfracGh center tikzpicturescale.latex draw- colorgreen!!black --- noderight y; draw- colorgreen!!black --- nodeleft x; draw- colorgreen!!black -.-- nodeabove z; drawcolorgreen!!black dashed ---; drawcolorgreen!!black dashed ---; drawthick colorred!!black ------------cycle; drawthick colorred!!black ----; drawthick colorred!!black ---; drawthick colorred!!black --; drawthick colorred!!black ---; tikzpicture center