Calculating the angle of roll in a rolly anchorage

We've been parked in a very rolly anchorage for the past day. Some anchorages, like this one, are fairly open to the ocean swells and if you're there at the wrong time can be so rolly as to be dangerous. While it's not quite that bad here right now, it is pretty uncomfortable.  This started a discussion around the breakfast table of estimating the angle of roll. The boys started out with pretty unrealistic ideas like 90 or 100 degrees. So then we talked about ways to estimate the roll angle to give a close approximation. Austin's final estimate was that our larger rolls were about 12 degrees. We finished with a little measuring experiment where we used the mast post, the floor and a weighted string to measure how far the boat was healing on some of the larger rolls.

To do this we made the assumption that the weighted string, taped to the top of the mast post (inside the boat) would approximate vertical. We also assumed that the mast post and floor were perpendicular. We then measured how far the weighted string moved from the mast post on a large roll. With these three pieces of information (length of side A, top of mast post to floor, length of side b, mast post to mark on floor and the 90 degree angle between sides A and B) we set out to draw a scale model of our triangle.

Side A: 183.63cm
Side B: 30.05.cm
Angle x: 90deg

We decided that reducing the scale by a ratio of 10:1 would allow us to draw the triangle on Austin's drawing paper. This was also an easy calculation as we just needed to move the decimal one point to the left. This gave us new dimensions for the scale triangle of:

Side A: 18.363cm
Side B: 3.05cm
Angle x: 90deg

We drew these two sides on the paper then added side C by joining points y and x and used a charting compass (same as a compass in a geometry kit with extra features to aid in charting bearings and courses on navigational charts) to measure the angle at point y. The result was 13 degrees. Since we felt that there were probably a few extreme rolls that were a bit larger than the ones we measured we estimate that, in the extreme cases, we are experiencing rolls of up to 15 degrees to a side (Austin worked out that it would be 26 to 30 degrees through the whole arc).

Now imagine cooking breakfast and dinner and pouring drinks while your kitchen floor (walls, table & counters) were all rolling around 30 degrees side to side. Not easy! No messes so far.