1) Is the clockwise vs counterclockwise designation a good way to determine the direction associated with ω in an unambiquous way?
2) Can you devise a better way to assign a minus or plus sign to an angular velocity?
3) Similar consideration needs to be given to torque as a vector. Can you devise a rule to assign a minus or plus sign to a torque? Describe the rule?
Note: You should be able to answer above questions no more than 3-4 sentences.
1) You need to remember that the direction of angular velocity omega is perpendicular to the surface of rotating object (wheel), thus, the direction of the angular velocity coincides with the axis of rotation of wheel. Hence, since clockwise and counterclockwise designate the direction of rotation of wheel around its axis and not along the axis, thus, you need to indicate the direction of angular velocity by indicating the direction the angular velocity vector points along wheel axis, so, you need to use the words: up and down.
2) Since the direction of angular velocity vector is perpendicular to the surface of wheel, you can use the right hand rule to determine the positive direction. Hence, you need to wrap the right hand's fingers around wheel, in the same direction of rotation, such that the thumb will point in the positive direction of angular velocity vector.
3) The torque is given by the cross product between the radius vector and the force vector. The radius vector represents the distance between axis of rotation of an object and the point of application of force that rotates the object.
`tau = bar r X bar F`
`tau = |bar r|*|bar F|*sin theta`
The right hand rule is used to determine the direction of torque. You need to align the right hand's fingers along radius and then you need to wrap the fingers in the direction of force. The thumb will indicate the postive direction of torque.
I need to make an observation here, regarding the reponses to the points 1 and 2 which in my opinion need to be corrected. If you assume that the vector `omega` is always perpendicular to the plane of rotation (plane of vectors R and v),
`v = omega xx R` (1)
saying that the rotation is clockwise or anticlockwise is enough to determine also the direction of `omega` . This vector `omega` , always obeys the rule of the right screw: it has the direction of a right screw rotating the same way as the body rotates (clockwise, or anticlockwise). If the body rotates clockwise (to the right) the right screw enters the plane of rotation, thus omega is downwards into the plane of rotation. The figure is attached.
2) Actually this is the only way to determine the direction of the resultant vector from a cross product: it has the same direction as a right screw that rotates the first vector of the product to be parallel to the second vector of the product.
In the case of the relation (1) you just have the directions of the cross product `v ` and of one of the initial vectors `R` . This is enough to determine the direction of the angular velocity `omega` .
Of course to know the direction of vector `omega` you allways need to know the direction of its defining vectors `v`and `R` , that is you need to know the direction of rotation (clockwise or not) and the plane of the rotation.