Druid, your conclusions are still faulty. There has been absolutely nothing to prove that water ballast works only for wide flat hulls, or that the righting moment decreases at heel angles greater than 45 degrees.
If your first FBD is of the ballast tank only, then it is incorrectly labeled. You have the weight of the ballast labeled Fg, and in your 2nd FBD, you labeled the weight of the entire boat and ballast as Fg. For clarity, lets relabel the ballast weight in the 1st FBD as Wb. You have labeled the opposing force as Fh, and said that is the bouyancy of the ballast. As you later pointed out, the purpose of a Free Body Diagram is to separate something and examine the forces acting on it. If we separate the ballast from the boat, there are only two external forces acting on it: 1) Weight (Wb) due to gravity, and 2) the supporting force (Fh) from the hull which contains the ballast. Therefore Fh = Wb. Note that the water surrounding the boat doesn’t act on the ballast- it acts on the hull. That is an important difference. Fb is the force of bouyancy on the hull, and is equal and opposite to the weight of the hull and everything on and in it, including the ballast. Fb = Fg, but neither is equal to Fh or Wb. Fh is a part of Fb, and Wb is a part of Fg. There is no moment of any kind in the first FBD because both forces act through the center of gravity of the ballast.
If the second FBD was viewed with the boat in a vertical position, there would be a zero righting moment because both Fb and Fg act along the same vertical line, and the moment arm is zero. In the arbitrary position shown, there is a righting moment, because there is a right angle distance between the lines of action of Fb and Fg because, as the boat heels, the center of bouyancy shifts to the right. This moment is equal to Fb x D as you show, but remember that Fb = Fg = the weight of the boat and everything in it, including the ballast- NOT JUST THE BOAT WEIGHT! All of this remains valid no matter what kind of hull form the boat has. If the boat is sitting vertically, Fb and Fg are both centered, so there is zero moment. Fg always acts through the center of gravity, which has a fixed position regardless of the orientation of the boat; as long as the boat and its attachments (including ballast) are symmetrical, the cg is on the centerline. The center of bouyancy, however, is dependent on the orientation of the boat (what part of the hull is in the water and supporting Fg) and will move away from the centerline in the direction of heel. How far it moves will depend on many factors, including the hull shape, angle of heel, and beam. It is true that, everything else being the same, the greater the beam, the more the center of bouyancy will be displaced, and the greater the moment arm will be. However, that does not support your conclusion that in order for water-ballast to work, the hull has to be wide and flat. By the way, one of the criticisms I’ve heard about Macs is that they are too narrow….
Your statement "Note also that when things get REALLY interesting (over 45 degrees), the moving-Fb-style righting moment begins to get LESS (not zero), while the more classical keel-based righting moment gets stonger." is also not proved. First, every sailboat, with or without water ballast, with or without a "classical" keel, will shift its center of bouyancy when heeling. Second, there is nothing magical about 45 degrees. Your 2nd FBD appears to be drawn at about 40 degrees of heel, and shows the center of bouyancy lower than the center of gravity. These positions were drawn arbitrarily. Without a lot of specific information (weights, hull dimensions, etc.) about a specific boat, you cannot say exactly where these points will be. All you can say is that the center of gravity is on the centerline of the boat, fairly low; and that the center of bouyancy, when the boat is vertical, is on the centerline at some point above the center of gravity (otherwise the boat would not sit vertically). We can say that the center of bouyancy is displaced horizontally when the boat heels; we cannot say how much, or if it also moves vertically. You can say that if the boat heels enough for the center of bouyancy and the center of gravity to be on the same horizontal line, then 1) that is the position where the greatest righting moment is produced (D is at its maximum), and 2) that any further heel will reduce D and therefore the righting moment. Again, without a lot of information about a specific boat, we cannot say at what angle of heel that will be. And again, these statements apply to all sailboats.
Free body diagrams can be very useful, but we have to be careful not to read more into them than is there. -Paul
If your first FBD is of the ballast tank only, then it is incorrectly labeled. You have the weight of the ballast labeled Fg, and in your 2nd FBD, you labeled the weight of the entire boat and ballast as Fg. For clarity, lets relabel the ballast weight in the 1st FBD as Wb. You have labeled the opposing force as Fh, and said that is the bouyancy of the ballast. As you later pointed out, the purpose of a Free Body Diagram is to separate something and examine the forces acting on it. If we separate the ballast from the boat, there are only two external forces acting on it: 1) Weight (Wb) due to gravity, and 2) the supporting force (Fh) from the hull which contains the ballast. Therefore Fh = Wb. Note that the water surrounding the boat doesn’t act on the ballast- it acts on the hull. That is an important difference. Fb is the force of bouyancy on the hull, and is equal and opposite to the weight of the hull and everything on and in it, including the ballast. Fb = Fg, but neither is equal to Fh or Wb. Fh is a part of Fb, and Wb is a part of Fg. There is no moment of any kind in the first FBD because both forces act through the center of gravity of the ballast.
If the second FBD was viewed with the boat in a vertical position, there would be a zero righting moment because both Fb and Fg act along the same vertical line, and the moment arm is zero. In the arbitrary position shown, there is a righting moment, because there is a right angle distance between the lines of action of Fb and Fg because, as the boat heels, the center of bouyancy shifts to the right. This moment is equal to Fb x D as you show, but remember that Fb = Fg = the weight of the boat and everything in it, including the ballast- NOT JUST THE BOAT WEIGHT! All of this remains valid no matter what kind of hull form the boat has. If the boat is sitting vertically, Fb and Fg are both centered, so there is zero moment. Fg always acts through the center of gravity, which has a fixed position regardless of the orientation of the boat; as long as the boat and its attachments (including ballast) are symmetrical, the cg is on the centerline. The center of bouyancy, however, is dependent on the orientation of the boat (what part of the hull is in the water and supporting Fg) and will move away from the centerline in the direction of heel. How far it moves will depend on many factors, including the hull shape, angle of heel, and beam. It is true that, everything else being the same, the greater the beam, the more the center of bouyancy will be displaced, and the greater the moment arm will be. However, that does not support your conclusion that in order for water-ballast to work, the hull has to be wide and flat. By the way, one of the criticisms I’ve heard about Macs is that they are too narrow….
Your statement "Note also that when things get REALLY interesting (over 45 degrees), the moving-Fb-style righting moment begins to get LESS (not zero), while the more classical keel-based righting moment gets stonger." is also not proved. First, every sailboat, with or without water ballast, with or without a "classical" keel, will shift its center of bouyancy when heeling. Second, there is nothing magical about 45 degrees. Your 2nd FBD appears to be drawn at about 40 degrees of heel, and shows the center of bouyancy lower than the center of gravity. These positions were drawn arbitrarily. Without a lot of specific information (weights, hull dimensions, etc.) about a specific boat, you cannot say exactly where these points will be. All you can say is that the center of gravity is on the centerline of the boat, fairly low; and that the center of bouyancy, when the boat is vertical, is on the centerline at some point above the center of gravity (otherwise the boat would not sit vertically). We can say that the center of bouyancy is displaced horizontally when the boat heels; we cannot say how much, or if it also moves vertically. You can say that if the boat heels enough for the center of bouyancy and the center of gravity to be on the same horizontal line, then 1) that is the position where the greatest righting moment is produced (D is at its maximum), and 2) that any further heel will reduce D and therefore the righting moment. Again, without a lot of information about a specific boat, we cannot say at what angle of heel that will be. And again, these statements apply to all sailboats.
Free body diagrams can be very useful, but we have to be careful not to read more into them than is there. -Paul
. Most traditional sail boats have a maximum righting arm around 60º of heel (give or take). It doesn't get stronger after 60º - it get's weaker until it vanishes at about 120-130º of heel. My Mac-M may even do a bit better than that with it's high freeboard. The M's COB at 90º heel is a very long way from the VCG compared to a traditional vessel - due to the ultra-high topsides.
