Angular Momentum points in WHAT direction?

  • Thread starter Rockazella
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Angular Momentum points in WHAT direction?!?

I just don't get this whole right hand rule thing. If you have a rotating disk, how the heck can its momentum vector product point perpendicular to the disk?! There is absolutely no motion perpendicular to the disk.

I may never understand this one, but would anyone mind atleast trying to explain?
 
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Because I'm about to fall asleep at my desk, am going to do the very quick and dirty answer. You have thusly been warned.

Angular momentum is the cross product of the position vector r and the linear momentum p. If you go do the vector algebra, you'll see that the resultant vector points in the z-direction (if you're dealing with a body moving in the x-y plane), direction dependent on just what direction you're moving in said x-y plane. So, next time you want to pull out your hair thinking about the right hand rule and how it just seems so arbitrary, take solace in that you can do nicely and neatly via the linear algebra.
 

Eyesee

Re: Angular Momentum points in WHAT direction?!?

Originally posted by Rockazella
I just don't get this whole right hand rule thing. If you have a rotating disk, how the heck can its momentum vector product point perpendicular to the disk?! There is absolutely no motion perpendicular to the disk.

I may never understand this one, but would anyone mind atleast trying to explain?
I think the right hand rule used here is to keep track of the direction the disk is spinning. It's not moving in the x and y direction so how else would you describe it's motion?
 

emu

If we rotate the disk 180 degrees, only the vector which passes through the center and is perpendicular to the disk will point in the same direction.
This vector is expressed in Nm, which is not a force, but what we call a moment.
 

Hurkyl

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I just don't get this whole right hand rule thing. If you have a rotating disk, how the heck can its momentum vector product point perpendicular to the disk?! There is absolutely no motion perpendicular to the disk.
Why wouldn't it!

Like the others said, if a disk is spinning in the x-y plane, there's no motion in the x or y direction either. Which way would you have the vector point?

The only thing that makes sense is to have the vector point in some direction with circular symmetry. Why? Because if we rotate in the x-y plane a disk spinning in the x-y plane, the result is a disk spinning in the x-y plane, and the same vector should represent both situations. The only choice is to have a vector that points along the z axis!


A more thorough answer is that angular momentum is really a bivector. An ordinary vector is, heuristically, a line element that points in a 1 dimentional direction. A bivector is a plane element that points in a 2 dimentional direction. The angular momentum of the aforementioned disk should really be written as a real multiple of i^j where ^ means wedge product. (it "pastes" lower dimentional vectors together to form higher dimentional vectors; here it combines a vector pointing along x with a vector pointing along y to form a bivector pointing along the x-y plane)


However, any geometrical object has a dual representation. Instead of a generalized vector that points in the direction the object is going, we can write a generalized vector that points in the direction the object is not going. In particular, in 3-D geometry, any surface can be completely characterized by its normal lines, e.g. the dual to a bivector is its normal vector.

So, we use this dual vector to represent angular momentum because it's just an ordinary vector, which means we don't have to take a course in Grassman Algebra or Tensor Analysis to understand what it is.

Hurkyl
 

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