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Nice discussion. It's getting well beyond my level of expertise, but I do follow it for the most part. I just wanted to mention two concepts that I understand somewhat differently, in the hopes that others will comment and help me to makes sense of this puzzle.
First, for purposes of this question let's compare a shoal draft long cruising fin keel to a deep foil with a bulb, both generating the same righting moment on the same hull. Robert said: "But because the distance term in the polar moment of inertia is squared and the distance term in righting moment is linear the amount of energy to flip a shoal draft boat that has the same righting moment as a deep draft boat is less."
I have understood that the larger surface area of the long fin keel would be more resistant to capsize due to the volume and mass of water that must be forced aside as the keel pivots around the boat's longitudinal axis. In other words, the increased wetted surface area whose friction is a liability in light air sailing becomes an asset in heavy going by offering increased resistance to roll as compared to the low wetted surface area foil. Thoughts?
Second, Robert also said: "...you really need a higher righting moment on the shoal draft boat to get a similar feel as the deep draft boat. Look at an ice skater in a spin to see the effect of changing the polar moment of inertia. Bringing in the arms increases the speed of rotation." Intuitively, the need for a higher righting moment in the shoal draft boat seems to make sense -- all else equal. But I could use a bit more help in understanding the "polar moment of inertia".
I guess I am having difficulty with the ice skater analogy. My understanding of the phenomena exhibited by the ice skater is that it demonstrates a physical law known as "conservation of angular momentum" which essentially states (forgive me, I am pulling this out of the cobwebs of high school physics) that an object moving around an axis will continue to move in the same direction at the same speed unless and until acted upon by other forces (such as friction). It's the same physical law that is responsible for the Coriolis Effect, from which we get our trade winds (with the sun as the energy source).
In the case of the skater, bringing in the arms from a distance that is further from the axis of rotation to a distance that is closer to the axis of rotation requires that the RPM of the skater increase in order to "conserve" the amount of momentum the skater already had. The skater still has the same amount of angular momentum, and the arms are moving at the same speed as they were, only now the rpms are more rapid because a shorter distance is being travelled by the arms. The opposite would happen if the skater extended her arms.
But that only works when the same object increases or decreases its distance from the axis of rotation. I'm not sure you can make an analogy when dealing with two different objects. In other words, an increase or decrease in the speed of the roll between the shoal draft and deep draft boat cannot be attributed to conservation of angular momentum (the ice skater phenomena) unless one or the other changes its own draft. Or can it?
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