It's proved in my K&K mechanics textbook that in pure rotation about an axis passing through the body it's angular momentum is $I\omega$.
What about if I want to find the angular momentum about an axis outside of it? Is it going to be the same?
In the special case of the body rotating about it's center of mass ,will it's angular momentum be the same for all axis?
My attempt to prove that for the above special case the angular momentum will be the same
Let the angular momentum about the centre of mass be $L_0$ and about another axis be $L$
Then we can write$\begin{aligned} \vec{L} &=\sum R_{i} \times m_{i} v_{i} \\ &=\sum\left(R_{0}+r_{i}\right) \times m_{i} v_{i} \\ &=\sum R_{0} \times m_{i} v_{i}+\sum r_{i} \times m_{i} v_{i} \\ &=\left(R_{0} \times \sum m_{i} v_{i}\right)+L_{0} \end{aligned}$
We know that the velocity of centre of mass is zero therefore $\sum m_{i} v_{i}=0$
$\Rightarrow \quad \vec{L}=\vec{L}_{0}$
Where am I going wrong then?