# Example where angular momentum and angular velocity are not parallel

I am unable to visualize any case where angular momentum and angular velocity of an object are not parallel.

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Optical vortex? – Waqar Ahmad Oct 2 '13 at 6:44
Can you explain what is optical vortex and how it can be an example where angular momentum and angular velocity are not paralle – Saurabh Shringarpure Oct 2 '13 at 9:58
Related – Waqar Ahmad Oct 2 '13 at 13:44
@WaqarAhmad that's actually the opposite question – user6972 Oct 3 '13 at 1:23

## 1 Answer

In the basic discussion of angular momentum where something is rotating around a fixed symmetrical axis

$\vec{L}=\vec{r}\times\vec{p}$

reduces to

$\vec{L}=I*\vec{\omega}$

Like in this animation where each vector is colored appropriately:

However angular velocity and angular momentum can have different directions in two cases: If the axis of rotation is not symmetric or the axis of rotation is moving.

Here's an example:

You can see that $\vec{L}=\vec{r}\times\vec{p}$ is not the same direction as $\vec{\omega}$ nor would the simplification $\vec{L}=I*\vec{\omega}$ be correct.

The position vector $\vec{r}$ is the vector between the reference point and the mass (note these problems are ignoring the mass of the rod), only in simple rotational cases like the first case is it perpendicular to $\vec{\omega}$. In a system of masses for example these vectors to the masses about a reference point can be complex. It is much easier to take the reference point as the center of mass. In each case $\vec{r}$ is the positional vector between your reference point and the mass and their composite angular momentums will superimpose (add) together.

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In the second illustration direction of angular velocity and linear momentum are misleading. In calculating angular momentum, L ,don't we take r as a vector perpendicular to our reference axis? – Saurabh Shringarpure Oct 2 '13 at 9:55
Corect me if I am mistaken. – Saurabh Shringarpure Oct 2 '13 at 10:22
Imagine this, the dumb bell is rotating about a stick. The angular momentum is parallel to the stick. But the stick itself is inclined w.r.t another, stationary, reference axis and is rotating about that axis. The reason for using r parallel to the arms and not the stationary axis is is because r is perpendicular to the axis about which the masses are rotating. But that axis itself is rotating w.r.t another.The angular momentum vector is rotating with respect to the stationary axis. – iTunnels Oct 2 '13 at 10:57
@WaqarAhmad It is a misconception that angular momentum is always about the same axis as angular velocity. Sometime this may not be possible, in these cases the angular momentum component along the axis of rotation is the product of angular velocity and moment of inertia about the given axis of rotation. – user6972 Oct 2 '13 at 21:53
I found an answer in this paper – Saurabh Shringarpure Oct 3 '13 at 10:18

## protected by Qmechanic♦Nov 17 '15 at 9:40

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