Formula Cuboid with Axis of Rotation Through the Center Point Moment of inertia Mass Width Length
$$\class{brown}{I} ~=~ \frac{\class{brown}{m}}{12} \, \left( l^2 + w^2 \right)$$ $$\class{brown}{I} ~=~ \frac{\class{brown}{m}}{12} \, \left( l^2 + w^2 \right)$$ $$\class{brown}{m} ~=~ \frac{12I}{{\class{brown}{I}}^2 + w^2}$$ $$w ~=~ \sqrt{ \frac{12I}{\class{brown}{m}} - {\class{brown}{I}}^2 }$$ $$l ~=~ \sqrt{ \frac{12\class{brown}{I}}{\class{brown}{m}} - w^2 }$$
Moment of inertia
$$ \class{brown}{I} $$ Unit $$ \mathrm{kg} \, \mathrm{m}^2 $$ According to \( M ~=~ I \, \alpha \) (\(\alpha\): angular acceleration), the moment of inertia determines how hard it is to generate a torque \(M\) on the body. Moment of inertia \(I\) depends on the mass distribution and on the choice of the axis of rotation. Here we calculate the moment of inertia of a cuboid whose axis of rotation passes through its center.
Mass
$$ \class{brown}{m} $$ Unit $$ \mathrm{kg} $$ Homogeneously distributed total mass of the cuboid.
Width
$$ w $$ Unit $$ \mathrm{m} $$ Width of the cuboid. When the width is doubled, the moment of inertia is quadrupled.
Length
$$ l $$ Unit $$ \mathrm{m} $$ Length of the cuboid. If the length is doubled, the moment of inertia is quadrupled.