what is constant for uniform circular motion?
In the uniform linear motion the vector velocity v is constant, because the acceleration is zero. In the uniform circular motion, the absolute value of the vector velocity `|v|` is constant, but its direction is always changing because this time there is a nonzero acceleration `a_(cp)` that is directed towards the center of the circle representing the trajectory.
However there is one vector quantity in the circular motion that is analog to the linear velocity and that does not change. Its name is angular speed `omega` and it is defined as a vector is such a way that
`v = omega xx R`
Here, all three quantities are vectors and `xx` represent the vectorial (cross) product.
Similar to the linear speed that represents the space traveled in the time unit, the angular speed represent the angle with which the position has changed in the time unit. It is measured is `(rad)/s` .
For a body moving uniform counterclockwise in a circle the angular speed vector `omega` will be perpendicular to the plane of the circle (representing the trajectory) and directed upwards. The figure is attached.
First in circular motion the speed is always the cross product
of the angular speed with the velocity (not the scalar product as in your answer):
`v =omega xx R`
Second, for uniform circular motion, there is no such thing as variable angular speed. The angular speed `omega` is constant by definition.
The only two quantities that varies in the relation above are the direction of radius `R` which is continuously sweeping the circle surface and the velocity `v` which is continuously changing direction. In absolute value all three quantities are constant.
Since uniform motion would imply a constant velocity, then we are talking about the linear velocity being constant. Now, with circular motion, the linear velocity is:
v = `omega` *r
v = linear velocity
`omega` = angular velocity
r = radius
Therefore, either `omega` and r are constant, also, or one goes up the specific amount while the other goes down a specific amount so that their product is still constant.