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The direction of angular velocity is above or below the plane. But what does it mean? I mean in linear velocity the direction of velocity is in the direction of motion of body but what does it mean that body is moving in one direction while the direction of its angular velocity is in another direction?

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3 Answers 3

The direction of angular velocity is different from that of regular velocity for (arguably) two reasons. First, it points out of the plane because of the nature of angular velocity. It signifies a rotation, as such, there is not any particular direction unit vector in every coordinate space that could represent it. In spherical or cylindrical coordinates, it would of course be easy to assign it to the $\hat\theta$ direction, but what about systems like Cartesian coordinates? Thus, to signify the direction of something that points in every direction on a plane, it is easy to specify it along the one direction we can be sure the velocity isn't pointing - normal to the plane. This is a much used convention (such as with area vectors, torque, and many others). As usual as well, we use the Right Hand Rule.

The second, and perhaps more important reason is that we always want to ensure that the angular velocity does not correspond to any true velocity that would be moving in a radial direction. However, to convert angular velocity to true velocity, it is necessary to multiply by the radius (for the most part). Therefore, the equation:

$$\vec v=\vec\omega\times\vec r$$

is used. This allows us to define it in such a way that the true velocity never has a radial component due to the angular velocity.

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The thing I find strange is when referring to a magnetic fields around a current-carrying wire, the right hand rule actually refers to the direction of the magnetic field, so I would have thought that there would be a more significant reason to use the right hand rule with angular velocity rather than just insuring "the angular velocity does not correspond to any true velocity that would be moving in a radial direction." –  Ephraim Dec 22 '13 at 22:19
@Jim: This was very intuitive and helpful. Just one thing, could you please give an intuitive understanding as to why the direction of the angular velocity can be shown by the right hand rule- I now know why it is orthogonal to the plane the object is rotating but if a disk rotates clockwise, why is $\omega$ inwards?... what's the significance of this? –  Eliza Jul 11 at 18:43
@Eliza That's not so much a physics reason as it is a sociology/anthropology reason. You see, most people on Earth are right-handed. As such, when this concept was invented, there was an ambiguity. To clear this up and make sure you get the same direction from any perspective, they chose the convention of having the right-hand-rule specify the direction. Had people been primarily left-handed, I imagine we would use the left-hand-rule for it today. –  Jim Jul 14 at 13:28
@Jim: and all this time I thought there was some complicated physics behind this :) One more thing, I am surprised as to how these conventions add up...let me make myself clearer: If I apply a force on a rotating disk and this causes the disk to rotate clockwise. If I use the RH rule, the direction of the torque is inwards (same as the direction of clockwise $\omega$ vector)... is this merely a consequence of convention which ends up making sense as such a torque will cause the disk to accelerate in the clockwise sense (and thus the two orthogonal vectors are in the same direction) –  Eliza Jul 14 at 16:43
@Eliza checkout wikipedia. As it shows, torque is related to the angular acceleration, which is the time derivative of $\omega$. So a torque that increases a clockwise rotation would point in the same direction as a clockwise rotation –  Jim Jul 14 at 17:25
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A velocity is a vector, so to describe it we want a magnitude and a direction. The magnitude bit is easy; the larger the velocity, the greater the magnitude. Like you said, for linear velocity the direction is just the direction of travel, but trying to do the same thing for an angular velocity fails - the direction of travel is changing. However, it's changing in a particular way - for a constant angular velocity (no angular acceleration), a vector pointing in the direction of travel always lies in the same plane (you could try showing this, it's not too difficult). So instead of encoding the direction of travel in the angular velocity vector, we encode the orientation of this plane... which is uniquely defined by a direction orthogonal to the plane. This explains the perhaps unintuitive definition of the angular velocity vector. The only remaining degree of freedom is whether the vector points "up" or "down" from the plane; this is usually taken to encode whether the motion is clockwise or anti-clockwise. The typical convention is that "up" is anti-clockwise (agrees with the right hand rule), but in principle there's no reason you can't reverse the definition and use the left hand rule.

As a side note, angular momentum and torque vectors are defined in a very simiar way.

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It cam be simply said that if thumb from right hand rule points outward its +ve and if it points inwards the plane of paper its -ve... direction be k

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