Confused: Rotation About an Axis or a Point?

[DocZaius]

I had always thought that the concept of rotation required it being around an axis rather than a point, but now that I have been exposed to using vectors to represent rotation, I am a little confused.

Angular momentum is defined as the position vector cross the linear momentum vector. Suppose a particle is moving down the x axis, parallel with the x axis, at some height above the x-axis in a right hand coordinate system.

Its position vector measured from the origin of the coordinate system to its location would obviously change with time and yet it seems that its angular momentum should not. Evaluating angular momentum about a point (the origin) seems to make no sense to me.

If instead I evaluated the angular momentum of that particle about the x axis. I would realize that its position in regards to the x-axis does not change and I would not end up with this problem of the particle's position (from the origin) changing over time.

Another question: Which phrase below is the one that is the least ambiguous and most "correct"?

- The particle's angular momentum
- The particle's angular momentum about the origin
- The particle's angular momentum about the x axis

P.S. You don't have to address any specific concerns I have. The above was only meant to illustrate my obvious confusion. If you can think of ways of explaining to me how to look at rotation in terms of it around a point vs an axis, that would be fine, too. But if you do so, please bring up cases where the angular momentum is different based on what (point or axis) it is evaluated in regards to.

 

[pabloenigma]

Angular Momentum(etc) is always evaluated about a point.In case of fixed axis rotation,the angular velocity vector is constant for all points of a rigid body,and in that special case the angular momentum is same for any point on the axis considered as origin.

In the general case,it may happpen that the axis of rotation of the body varies(as in top motion).In that case the axis of rotation of the body is instantaneous.So,evaluating the angular momentum of the body about a particular axis makes no sense

Thoroughly read a standard text like Herbert Goldstien.May help.

 

[HallsofIvy]

Sounds to me like your confusion is that you are trying to do a three dimensional problem in two dimensions. An object moving in a circle, say, around the origin, (0, 0), in the xy- plane is actually rotating around the z-axis in xyz-space.

If the object is moving along the line [itex]y= y_0[/itex], with speed v, then its position vector is given by [itex]<vt, y_0, 0>[/itex]. It's linear momentum vector, assuming mass m, is [itex]<mv, 0, 0>[/itex].

The cross product of those two vectors is [itex]<0, 0, -mvy_0>[/itex]

 

[DocZaius]

Sounds to me like your confusion is that you are trying to do a three dimensional problem in two dimensions. An object moving in a circle, say, around the origin, (0, 0), in the xy- plane is actually rotating around the z-axis in xyz-space.

If the object is moving along the line [itex]y= y_0[/itex], with speed v, then its position vector is given by [itex]<vt, y_0, 0>[/itex]. It's linear momentum vector, assuming mass m, is [itex]<mv, 0, 0>[/itex].

The cross product of those two vectors is [itex]<0, 0, -mvy_0>[/itex]

Thanks. Would you say the cross product at the end of your post is "the angular momentum about the origin" ? Also, could you elaborate on your interpretation that the source of my confusion is 2D/3D? Is it inappropriate to mention rotation about an axis when in 3D?

 

[D H]

Another question: Which phrase below is the one that is the least ambiguous and most "correct"?

- The particle's angular momentum
- The particle's angular momentum about the origin
- The particle's angular momentum about the x axis

The first is wrong in that it implies angular momentum is absolute. It isn't; it is instead a frame-dependent quantity. The last is also wrong: What if the particle is moving parallel to but offset from the x-axis?

That leaves the second one, but I don't quite like that one either. IMO, it is better to talk about the particle's angular momentum with respect to rather than about the origin.

 

[zincshow]

I had always thought that the concept of rotation required it being around an axis rather than a point, but now that I have been exposed to using vectors to represent rotation, I am a little confused.

Angular momentum is defined as the position vector cross the linear momentum vector. Suppose a particle is moving down the x axis, parallel with the x axis, at some height above the x-axis in a right hand coordinate system.

Its position vector measured from the origin of the coordinate system to its location would obviously change with time and yet it seems that its angular momentum should not. Evaluating angular momentum about a point (the origin) seems to make no sense to me.

If instead I evaluated the angular momentum of that particle about the x axis. I would realize that its position in regards to the x-axis does not change and I would not end up with this problem of the particle's position (from the origin) changing over time.

Another question: Which phrase below is the one that is the least ambiguous and most "correct"?

- The particle's angular momentum
- The particle's angular momentum about the origin
- The particle's angular momentum about the x axis

P.S. You don't have to address any specific concerns I have. The above was only meant to illustrate my obvious confusion. If you can think of ways of explaining to me how to look at rotation in terms of it around a point vs an axis, that would be fine, too. But if you do so, please bring up cases where the angular momentum is different based on what (point or axis) it is evaluated in regards to.

I view it as 2 kinds of angular moment. Your #1 is the total. #2 is angular momentum due to one object rotating around another (the origin). #3 is angular momentum due to intrinsic angular momentum of the object itself, ie any rotating object like a ball or a sphere.