Understanding addition of angular velocity I'm trying to understand the concept of angular velocity. I read this paragraph on Wikipedia, which asserts that if a point $p$ has angular velocity $u$ within a coordinate frame $F_1$ which itself has angular velocity $v$ within some other frame $F_2$, then the point has angular velocity $u+v$ with respect to the second frame.
I want to set aside the specific formula $u+v$ and just focus on the abstract form $f(u, v)$ - in other words, just the fact that the angular velocity of $p$ with respect to $F_2$ can be deduced from the angular velocities of $p$ with respect to $F_1$ and of $F_1$ with respect to $F_2$. That seems like it ought to be impossible to me. In the diagram below, $C$ has the same angular velocity $u$ with respect to $B$ in both figures and $B$ has the same angular velocity $v$ with respect to $A$ in both figures, yet in the first figure $C$ has angular velocity $u+v$ with respect to $A$, whereas in the second figure $C$ has angular velocity $u-v$, zero if we set $u=v$.

What am I failing to understand?
 A: You are asking about the algebra of angular velocities, especially as it relates to the area of kinematics.
Consider the tea-cup ride at Disney. You have a platter (orange) rotating about A with angular speed $\Omega$. At some radius $R$ away form the center a 2nd object (green) is pinned at point B with relative speed $\dot \theta$. This is to denote that $\theta$ is an orientation angle. On this object there are two attached points, C and D, that at some instance in time are located as shown below:



*

*What is the rotational velocity of A? 
Every particle attached to the platter moves with translational velocity except the center of rotation. Nevertheless we can state that the body rotates with $\Omega$ and the translational speed of each particles is given by $$ v= R\, \Omega $$ where $R$ is the radial distance from the center of rotation. You can say that B rotates about A, but in reality the entire body is rotating about A. A more accurate description would be that B orbits A.

*What is the rotational velocity of B?
This depends on which body is B belong to. The bottom of the pin connecting the two bodies rotates with $\Omega$ and the top of the pin with $$\omega = \Omega + \dot\theta$$ since $\dot \theta$ is the relative motion between the two bodies.

*What is the translational velocity of points A, B, C and D.


*

*$v_A = 0$ 
since it is on a fixed point

*$v_B = R\,\Omega$ 
as we saw before

*$v_C = v_B + \omega \, r = \Omega \,R + (\Omega + \dot \theta) \, r = \Omega ( R+r) + \dot \theta \, r $
To be interpreted as the velocity of the platter at C plus the relative velocity of the second body.

*$v_D = v_B - \omega \, r = \Omega \,R - (\Omega + \dot \theta) \, r = \Omega ( R-r) - \dot \theta \, r $
To be interpreted as the velocity of the platter at C minus the relative velocity of the second body.
As you can see, relative translational velocities switch signs between C and D. This is because the position of the point is either radially out or radially in from the reference point B.
On the contrary rotational velocity of the second body is only added the motion of the platter, because the location of the pin at B bears no significance in the rotational kinematics.
