My name is Alexander FufaeV and here I write about:
Steiner's Theorem
Important Formula
$$I ~=~ I_{\text{CM}} ~+~ \class{brown}{m} \, h^2$$
$$I ~=~ I_{\text{CM}} ~+~ \class{brown}{m} \, h^2$$
$$I_{\text{CM}} ~=~ I ~-~ \class{brown}{m} \, h^2$$
$$h ~=~ \sqrt{ \frac{I ~-~ I_{\text{CM}}}{ \class{brown}{m} } }$$
$$\class{brown}{m} ~=~ \frac{I ~-~ I_{\text{CM}}}{ h^2 }$$
What do the formula symbols mean?
Moment of inertia
$$ \class{brown}{I} $$ Unit $$ \mathrm{kg} \, \mathrm{m}^2 $$
Moment of inertia of a rotating body (e.g. a cylinder) whose axis of rotation has been shifted parallel to the axis of rotation through the center of mass. With the Steiner's theorem, there is no need to calculate a complicated integral for the new axis of rotation.
Moment of inertia through CM
$$ I_{\text{CM}} $$ Unit $$ \mathrm{kg} \, \mathrm{m}^2 $$
Moment of inertia of the rotating body whose axis of rotation passes through the center of mass of the body.
Distance
$$ h $$ Unit $$ \mathrm{m} $$
Distance of the new axis of rotation from the axis of rotation through the center of mass
Mass
$$ \class{brown}{m} $$ Unit $$ \mathrm{kg} $$
Total mass of the rotating body.
The moment of inertia \( I \) always refers to a specific axis of rotation. If the axis of rotation is changed, the moment of inertia of the rotating body also changes. It becomes larger, if you shift the rotation axis parallel by the distance \( h \) from the center of mass axis. The moment of inertia through the center of mass axis is \( I_\text{CM} \).
You can calculate the new moment of inertia with the following Steiner's theorem:
$$ \begin{align} I ~=~ I_{\text{CM}} ~+~ \class{brown}{m} \, h^2 \end{align} $$
So, using Steiner's theorem, you can easily calculate the moment of inertia for a new axis of rotation without having to calculate a complicated integral.