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In Physics , the angular velocity is a Vector quantity (more precisely, a Pseudovector ) which specifies the Angular Speed at which an object is rotating along with the direction in which it is rotating. The SI unit of angular velocity is Radians Per Second , although it may be measured in other units such as degrees per second, degrees per hour, etc. When measured in cycles or rotations per unit time (e.g. Revolutions Per Minute ), it is often called the rotational velocity and its magnitude the Rotational Speed . Angular velocity is usually represented by the symbol Omega ('''Ω''' or '''ω'''). The direction of the angular velocity vector is perpendicular to the plane of rotation, in a direction which is usually specified by the Right Hand Rule .


THE ANGULAR VELOCITY OF A PARTICLE



Two dimensions

of the velocity vector V .]]

The angular velocity of a particle in a 2-dimensional plane is the easiest to understand. As shown in the figure on the right, if we draw a line from the origin (O) to the particle (P), then the velocity vector (\mathbf{v}) of the particle will have a component along the radius (\mathrm{v}_\parallel\, - the radial component) and a component perpendicular to the radius (\mathrm{v}_\perp - the Tangential Component ).

A radial motion produces no rotation of the particle (relative to the origin), so for purposes of finding the angular velocity the parallel (radial) component can be ignored. Therefore, the rotation is completely produced by the tangential motion (like that of a particle moving along a circumference), and the angular velocity is completely determined by the perpendicular (tangential) component.

It can be seen that the rate of change of the angular position of the particle is related to the tangential velocity by:

:\mathrm{v}_\perp=r\, rac{d\phi}{dt}








where Vi is the velocity of the particle (in the lab frame) and V is the velocity of O' (the origin of the rigid body frame). The velocity of the particle is given by:

:\mathbf{V}_i=\mathbf{V}+\boldsymbol\Omega\mathbf{r}_i

Where Ω is the Angular Velocity Tensor . If we take the dual of the angular velocity tensor, we get the angular velocity pseudovector

:\boldsymbol\omega= {Link without Title}

and the matrix multiplication is replaced by the cross product, yielding:

:\mathbf{V}_i=\mathbf{V}+\boldsymbol\omega imes\mathbf{r}_i.

It can be seen that the velocity of a point in a rigid body can be divided into two terms - the velocity of a reference point fixed in the rigid body plus the cross product term involving the angular velocity of the particle with respect to the reference point. This angular velocity is the "spin" angular velocity of the rigid body as opposed to the angular velocity of the reference point O' about the origin O.

It is an important point that the spin angular velocity of every particle in the rigid body is the same, and that the spin angular velocity is independent of the choice of the origin of the rigid body system or of the lab system. In other words, it is a physically real quantity which is a property of the rigid body, independent of one's choice of coordinate system. The angular velocity of the reference point about the origin of the lab frame will, however, depend on these choices of coordinate system. It is often convenient to choose the Center Of Mass of the rigid body as the origin of the rigid body system, since a considerable mathematical simplification occurs in the expression for the Angular Momentum of the rigid body.


SEE ALSO




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