I am either ashamed or relieved to admit that this thought has also crossed my mind on more than one occasion.
I’d only thought about it in terms of linear motion. The act of forcing a water droplet off a plate, say, is a little like executing the tablecloth trick. You first move the glass relatively slowly, accelerating the water droplet while keeping it stationary relative to the glass. You then rapidly pull the glass back at such a speed that the droplet simply does not have time to decelerate. Its own inertia carries it over the edge of the plate.
However, your method of drying is probably more effective, as it incorporates circular motion. Say you had a plate with an intrusive drop of water in its centre. Were you to shake it back and forth, it is likely that the drop would never be able to escape. This would certainly be the case were you to move the plate at the same speed going both back and forth. The drop would undergo some kind of oscillatory motion.
In order to guarantee the drop’s departure, the plate’s position over time should be described by something like a saw-tooth wave. As the magnitude of the friction force (surface tension?) can’t be expressed as a continuous function, I can’t think of how to be any more specific.
By swinging the plate in an arc instead, there is a (fictional) force which accelerates the drop in the same direction regardless of whether the plate is going one way or the other. This centrifugal force acts parallel to the direction of your arm.
In the case of the shaking dog, I’d be hesitant to say which of these two effects was actually contributing the most. There are two components to the fictional ‘drying force’. First, the centrifugal force contributed by the spinning of the torso and skin. Second, the whipping of the fur as it reaches the skin reaches the extent of its rotation and started to accelerate in the opposite direction. The latter could probably be approximated by the linear motion I described above.
In conclusion, no, I’ve no idea how to measure the force! Or rather, I wouldn’t want to without assumptions or a bit of rejigging with frames of reference. I’m sure you’ve picked up on something I’ve overlooked. Maybe forgetting the whipping motion ‘either end’ of the arc would make things easier – then I guess you just need ‘m’, ‘v’ and ‘r’!
I’ll write down some thoughts about your previous puzzle in another post – this one is becoming too lengthy. It’s a really interesting question, and I want to do it justice! I hope to be able to write a little about what I’ve been studying too.