# How to Calculate Moment of Inertia when the Axis of Rotation does not Pass Through the Center of Mass?

## Answer #1

Level 2 (suitable for students)

Answered by Alexander Fufaev

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**:

**Formula for Steiner's theorem**

Formula anchor $$ \begin{align} I ~=~ I_{\text{CM}} ~+~ \class{brown}{m} \, h^2 \end{align} $$

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.