# eli5 what is the meaning of the cross product of angular velocity and angular momentum…

28 views
0

This term shows up in euler’s equations of rigid body dynamics… what is the meaning of this term… why are angular momentum and velocity not collinear

In: 0 Consider a spinning wheel. Different points on that wheel will be moving in different directions. If angular velocity were co-linear with velocity, which of the points should the one whose velocity is co-linear with angular velocity? Should the angular velocity change as that point moves in a circle, or should the selected point change?

All of these problems can be fixed by noticing that while all the velocities in the wheel are going in different directions, they are all perpendicular to one particular axis. As long as we all agree that the direction of ‘angular velocity’ is defined by the axis rather than by the linear velocity of a point, everything works out in the end. https://physics.stackexchange.com/questions/487842/angular-momentum-and-angular-velocity

For single point mass they are colinear. However, in general it depends on the shape (or rather mass distribution) of the object. Let’s start with a more formal mathematical reason and then give an example.

For a rigid body, linear momentum is given by mass × velocity, where velocity and momentum are vectors and mass is a scalar. Since mass is a scalar, velocity and momentum are trivially always colinear.

*Angular momentum*, on the other hand, is defined as L = I 𝜔, where I is the moment of inertia and 𝜔 is angular momentum. L and 𝜔 are once again vector quantities, but unlike mass, moment of inertia is no longer scalar! Rather, it is a rank 2 tensor, which you can think of as a more complicated “proportionality” represented by a 3 by 3 matrix.

Moment of inertia is a fundamentally more complicated object than mass, which is reflected in the more complicated relationship between angular velocity and angular momentum.

Now to see physically *why* this should be the case, think about a “wobbly” rotating object, [like this one](https://i.stack.imgur.com/0rkLC.png). What is the direction of angular velocity? It is, as usual, normal to the plane of rotation, and therefore in the z-direction.

What about the angular momentum? This isn’t quite as simple. Angular momentum is by definition orthogonal to the plane of the position vector and the linear momentum, which *is not*, in this case, flat to the xy plane. Angular momentum is angular inertia times angular velocity: L = I ω.

But angular inertia isn’t a scalar, it’s a matrix, so in general L and ω won’t be parallel.