Why is the angular momentum of a rotating body the same about any point on its axis of rotation?

Question

The angular momentum of a rigid body is defined with respect to an origin. But when the body rotates about an axis we define it with respect to the axis; as if it would be the same about any point on the axis. But why would it be the same?

Answer 1

They are not the same. The angular momentum about an axis is the component along that axis of the angular momentum with respect to any point in the axis. It turns out that this component of the angular momentum (but not the other two) is independent of the point in the axis you take to compute the angular momentum. That this component is independent of the point in the axis used to compute the angular momentum is very easy to prove, and you should try to do so.

For example, if the axis of rotation is the $z$-axis, then the angular momentum about this axis is the $z$-component of the angular momentum about any point in the $z$-axis. In that case the $x$ and $y$ components of the angular momentum will in general depend of which point in the $z$-axis you use to compute the angular momentum, but not the $z$ component. Here you need to see the difference between $\vec L_{p}$, the angular momentum about a point $p$ in the $z$-axis, and $L_z$, the $z$ component of this vector which is independent of $p$ (provided $p$ is in the $z$-axis). This $L_z$ is the angular momentum about the $z$-axis. It is just one of the components of $\vec L_{p}$, so it is not the same as $\vec L_{p}$, which in general has also $x$ and $y$ components.