Statics/Geometric Properties of Solids

=Mass Moments Of Inertia Of Common Geometric Shapes=

Slender Rod
$$ I_x = 0 $$

$$ I_y = I_z = \frac {1}{12} ml^2 $$

Thin Quarter-Circular Rod
$$ I_x = I_z = mr^2 (\frac {1}{2} - \frac {4}{\pi^2}) $$

$$ I_y = mr^2 (1 - \frac {8}{\pi^2}) $$

Thin Ring
$$ I_x = I_y = \frac {1}{2}mr^2 $$

$$ I_z = mr^2 $$

Sphere
$$ I_x = I_y = I_z = \frac {2}{5} mr^2 $$

Hemisphere
$$ I_x = I_y = \frac {83}{320} mr^2 $$

$$ I_z = \frac {2}{5} mr^2 $$

Thin Circular Disk
$$ I_x = I_y = \frac {1}{4} mr^2 $$

$$ I_z = \frac {1}{2} mr^2 $$

Rectangular Prism
$$ I_x = \frac {1}{12} m \left ( b^2 + c^2 \right ) $$

$$ I_y = \frac {1}{12} m \left ( a^2 + c^2 \right ) $$

$$ I_z = \frac {1}{12} m \left ( a^2 + b^2 \right ) $$

Right Circular Cylinder
$$ I_x = I_y = \frac {1}{12} m( 3r^2 + h^2) $$

$$ I_z = \frac {1}{2} mr^2 $$

Right Half Cylinder
$$ I_x = \frac {1}{12} mh^2 + mr^2( \frac {1}{4} - \frac {16}{9\pi^2}) $$

$$ I_y = \frac {1}{12} mh^2 + \frac {1}{4}mr^2 $$

$$ I_z = mr^2( \frac {1}{2} - \frac {16}{9\pi^2}) $$

Thin Rectangular Plate
$$ I_x = \frac {1}{12} mb^2 $$

$$ I_y = \frac {1}{12} ma^2 $$

$$ I_z = \frac {1}{12} m(a^2 + b^2) $$

Right Circular Cone
$$ I_x = I_y = \frac {3}{80} m ({4}{r^2} + h^2) $$

$$ I_z = \frac {3}{10} mr^2 $$

Right Tetrahedron
$$ I_x = \frac {3}{80} m (b^2+c^2) $$

$$ I_y = \frac {3}{80} m (a^2+c^2) $$

$$ I_z = \frac {3}{80} m (a^2+b^2) $$