Formula Sphere (axis of rotation through the center) Moment of inertia Mass Radius
$$\class{brown}{I} ~=~ \frac{2}{5} \, \class{brown}{m} \, r^2$$ $$\class{brown}{I} ~=~ \frac{2}{5} \, \class{brown}{m} \, r^2$$ $$\class{brown}{m} ~=~ \frac{5\class{brown}{I}}{2r^2}$$ $$r ~=~ \sqrt{ \frac{5\class{brown}{I}}{2\class{brown}{m}} }$$
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 the moment of inertia of a homogeneous rotating sphere is calculated, whose axis of rotation passes through the center.
Mass
$$ \class{brown}{m} $$ Unit $$ \mathrm{kg} $$ Total mass of the sphere that is homogeneously distributed within the sphere. The greater the mass, the greater the moment of inertia of the sphere.
Radius
$$ r $$ Unit $$ \mathrm{m} $$ Radius of the sphere. If the radius is doubled, the moment of inertia is quadrupled.