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Below is an animation that shows the balance of forces: actually it is technically the balance of a concept called “moments.” First off then, we’d better explain moments. Simply explained: if you hold your hand out straight and someone puts a pound weight in your hand, that is harder to hold than if they put the same pound on your elbow. Even easier—if they put the pound weight right by your shoulder. It’s the same pound weight, but it was the “moment” that was straining your muscles, not the weight. Moment (nothing to do with a moment in time), then, is weight multiplied by distance.
- What tends to tip the boat over is the moment of the wind force high up in the sails.
- What tends to right the boat back is the keel weight and the distance it is heeled over away from the vertical center line.
Balance of Moments Animation
So now watch the animation 10 times over or so and watch each dynamic as it is happening. Then refer to the text below, which will further explain.
Use the green “incr. wind” button.
The force on the sails is the pressure multiplied by the area on the sails on which the wind is acting. The pressure is proportional to the velocity of the wind squared. Why? It just is! It’s one of those formulas that make up the universe.
From above, the moment is the force multiplied by the height of the position where the wind is considered to singularly act. This position is called the center of the pressure (COP). The center of pressure is the position on the sail whereby if we replaced all the wind all over the sail with an equal force at some position that would be the position called the center of pressure. On a right triangle sail, the point is 1/3 of the way up the mast starting at the boom.
So the tipping moment is proportional to area, height, and wind speed squared.
So what really happens is this: the wind tips the boat over a bit and this shifts the keel weight off the centerline a bit. The boat will continue to heel over until the tipping moment by the wind is equal to the moment from the keel being off center. At this point, the boat will stop heeling over further and the moments are balanced.
Now the wind picks up again, and again the boat heels over farther and the keel does some righting. But also notice that the area of the sail presented to the wind has reduced and the height to the COP has also reduced. So as the boat heels: area and height decrease on the tipping side of the equation.
At all times, for the boat to not continue to heel farther, the moments of tipping and righting have to be balanced.
This equation must balance: keel weight x distance = area X height x velocity²
(Note to the puritanical: ok, ok, well not quite. There is the factor of ½ density of air but that is constant. We’re trying to make it simple here.)
The only dynamic input to the system is the wind. Everything else in the equation is just working to balance the velocity squared.
So the heel angle of the boat is purely a mathematical balance of the wind force on the rig versus the keel weight off-center. Durh! While you knew that, perhaps you had not seen it in equation form.
Now go back and run the animation some more. Notice that the two moments are always in balance.
To further extrapolate: when the boat heels way way way over, there is almost no sail area presented to the wind and the height (h) has reduced also. Additionally, the keel distance off-center has moved way out, which is acting to pull the keel down (boat upright) again.
So next time you’re out there and the boat heels way way way over, don’t worry; every little thing is going to be all right. You’ve got mathematical equations working in your favor. Area and height are reducing and keel distance off center is increasing.
Best you check the keel bolts every now and again, however! Yup, the keel dropping off would be a problem!
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