5. Parallel Axis Theorem

Objects don't always rotate around the most convenient axis

The theorem states that

$$I_p = I_{cm} + Md^2$$

where

$I_p$ is the moment of inertia around the rotation axis,

$I_{cm}$ is the moment of inertia around the centre of mass

Example

Rotation around the end of a slender rod

using the Parallel Axis Theorem

$$I_p = \dfrac{1}{12}ML^2 + M(\dfrac{L}{2})^2$$

 

$$I_p = \dfrac{1}{3}ML^2$$