Classical mechanics, angular momentum and velocity not parallel, why?

[karnten07]

Homework Statement

A single particle of mass m is revolving at steady speed in a circle of radius a about the z axis at a height h above the origin. In this situation the particle has angular velocity w = wz

1 Show that the angular momentum L is not parallel to w

2 If a second particle revolves in the same circle at the opposite end of a diameter, show that the combined angular momentumof the two particles rotating rigiidly is parallel to w.

Homework Equations

The Attempt at a Solution

For the first question, i have found a webpage that talks about this situation and describes the problem of w and L not being parallel. But I am finding it hard to extract what information to show that this is true. Here is the link: http://sites.isel.ipl.pt/fisica/pedagogia/ajp75(2007)53.pdf

I am concentrating on the first question first so any help would be greatly appreciated.

 

[simon1987]

To answer part (1), consider this:

In rotational motion, angular momentum $$\stackrel{\rightarrow}{L} = \stackrel{\rightarrow}{r} \times\stackrel{\rightarrow}{P}$$, where P is linear momentum.

 

[karnten07]

To answer part (1), consider this:

In rotational motion, angular momentum $$\stackrel{\rightarrow}{L} = \stackrel{\rightarrow}{r} \times\stackrel{\rightarrow}{P}$$, where P is linear momentum.

Does this show that the components of L are in the x and y axes as well as z. Therefore it isn't parallel because of this?

 

[karnten07]

Does this show that the components of L are in the x and y axes as well as z. Therefore it isn't parallel because of this?

I think i have the answer to the first part by showing that the moment of inertia in this case has components that are perpendicular to the z axis therefore L and w arent parallel. Any help on the second part would be much appreciated.

 

[simon1987]

I think i have the answer to the first part by showing that the moment of inertia in this case has components that are perpendicular to the z axis therefore L and w arent parallel.

Have you drawn a picture? It is incredibly helpful when considering multiple different vectors. Consider: what would the cross product be of z and P if z were parallel to P?

As for part (2), remember that "combined" is just the sum of the two angular momenta, and that each angular momentum is a separate vector. Again, draw a picture.