Glenmar Sailing Association
Sailing the Chesapeake Bay

Static Righting Moment

If at the dock a sailboat is pulled over by its mast, there exists a transverse force which tries to set the boat back to where it naturally wants to be. This force is named the static righting moment. It is related to, if not identical with, the Euler force. Somewhat obvious to anyone who may have tried to pull over a boat from the top of its mast, the size of this force varies according to the angle of heel. Furthermore, every boat design is different from all the others. This is a classic example of my general rule for what makes a boat go fast. The rule is “Everything matters”, but that’s a different topic and not the subject of this post

For a particular sailboat, the force changes as the boat is tipped over. When floating upright in dead calm water, the force is zero. Likewise, when capsized, the force is again zero. In between is the problem.

From figure 1 we see that righting moment (RM) can be calculated as moment arm length times displacement force. Here, displacement force is mass (displacement in \( kg \)) times gravity. Thus the units of a righting moment are \[ kg \cdot\frac{m}{s^2}\cdot m = N \cdot m. \] Since the magnitudes of these values are essentially meaningless to everyone (not just Americans), we often see graphs with the righting arm length graphed on the y-axis versus degrees of heel on the x-axis. These graphs, which are called “GZ curves” essentially describe the yacht’s stability. Gz is the traditional yacht designer’s designation for the value that I labeled d in figure 1.

Fig 1. Righting Moment. CG is the center of gravity, CB is the center of buoyancy, and d is the distance between these two forces. In most sailing calculations, variable d is named GZ and referred to as the “righting arm”. Righting moment is the product of the moment arm length and the boats displacement force.

As an example, we have the GZ graph for a J105 in figure 2. Clearly, the righting arm is very small with a maximum deviation of only a little over an inch. We also see that beginning around 130 the graph goes into negative Gz territory. That point is called the “angle of vanishing stability” with all of the values below zero termed “inverted stability”. The good news for this J-boat is that the region is quite small and it is likely that a wave would push the boat back into an area where it would get righted.

Fig. 2. GZ graph for the J105.