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Above Left: City Lights - reefed under light winds because these are novice sailors and because the nylon sail had stretched. Right: Two successful models, waterproof and constructed from discarded manila folders.
This is perhaps my best example of how boatbuilding can combine the usefulness of science education with the excitement of original research. First, I tasked a number of students with finding a method for making waterproof toy boats “on the cheap”. Most were college students, but a high school sophomore, his mother, and younger sister also attended the evening boatbuilding effort that met one evening per week.
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"every wrong attempt discarded is often a step
forward" ... Thomas Edison
The first attempt used those ubiquitous overhead transparencies that come with textbooks. That failure is shown just in front of the coffee cup. One success, sitting one the soda can, was a manila folder coated with paraffin wax. This idea was introduced by James, an elementary education major with a strong interest in science. A student with initials TRR proposed using foam sheets. But that was too expensive for my tastes. A number of failures involved coating cardboard or manila folder paper with various glues. We wanted very much to make a boat using school glue, but she always became waterlogged and sank in a few hours.
For me the most useful way to construct a toy boat appropriate for research is to cut panels out of manila folder paper and coat her with waterproof polyurethane glue. Though not appropriate for middle school children, polyurethane was the primary glue used to make City Lights, the (real) boat shown at the top of this page.
If you inspect the boat in the lower right corner of the first image, you will see the manila/polyurethane boat, along with fishing weights and a straw. The fishing weights along one side were used to vary the applied torque on the boat, causing her to tilt to one side. This was suggested by the mother of the two pre-college children as she was watching us struggle with a different scheme that used pennies we found in the lab. The straw was her son’s idea as a way to measure the angle of tilt. (I think we ended up using uncooked spaghetti.)
There was a reason I wanted all this done. After reading various confusing and sometimes even conflicting explanations for how water ballast stabilizes a sailboat, I decided to try my hand at calculation. It is somewhat counterintuitive that water, which is neutrally buoyant, could stabilize a sailboat. Though beyond the scope of all my collaborators, the calculation shown below should be understandable by any physics major. It predicts a linear relationship between applied torque and angle of tilt, up to the point where the bottom corner leaves the water. Hence the line in the graph terminates at this critical angle.
The experimental data were taken by an adult student, Debra, who had more manual dexterity than possessed by the pre-college students. An excellent college algebra student, she lacked the background to fully understand the theory. But we did have a lengthy discussion about whether it was significant that her data did not seem to cross the origin. I think I convinced Debra not to worry about it, and that the graph is an excellent match to the data. Was I right? We need more investigation, perhaps with computer models, to be sure.
The is a good place to stop, unless you have had a year of calculus and found it easy! There are plenty of easy projects to do from here. Contact me if you are interested.
Let b = b(x) be the beam of a Bolger Box (Advanced Sharpie), where x is measured along the boat. Only two forces act on the boat: gravity and the buoyant force. At small angle of list, the buoyant force points upward. By the definition of pressure, P, and the equation for static equilibrium,
Where g is the acceleration of gravity, h is the depth, and the Greek rho is the mass density of water. For fresh water at 39F, where water has its highest density, in English Units (pounds per cubic feet):
At small angle of tilt, the boat rotates about a point on the waterline, at the center of the boat:
This axis of rotation ensures that the total area of this underwater section remains constant. Close inspection of the graph indicates that the rotation slightly increases the area under the waterline. Hence the rotation is accompanied by a slight lifting of the boat, which becomes important only at large angle, I believe.
To calculate the force, we integrate the pressure, dF = P1 dy dx, where y is
the transverse direction, where P1 is the change in pressure associated with
the angle of list. At small angle, the change in depth is y times the angle
of list. We want the torque, so we add an extra factor of y and integrate:
,
where
.
Let b equal the width of the boat where the bottom comes up above the water. We want to consider the range from a very traditional sharpie with the bow scraping into the water, b=0, to a full-blown barge with b=B. To reasonable approximation, the width of the boat is parabolic in x. The parabola that yields a width of B at x=0, and a width of b at x=L/2 is:
.
Hence, the torque due to buoyancy is:
where for most purposes, we may use the following table of approximate values for the function, f:
f(0) = 0.46; - - - f(0.5)=0.63; - - - f(1)=1.00
This yields some insight into how shape influences stiffness. The three numerical values of f shown above address the issue of using a blunt bow to enhance stiffness. A double-ender has 46% of the stiffness of a barge. Another way to stiffen her is to widen the beam, B. Changing the beam from 6 to 7 feet results increases stiffness by a factor of (7/6)3= 1.59. If the boat is asymmetrical fore and aft, then one averages the two values of f(b/B), not b. To get a “handy” formula for most advanced sharpies, take b/B = 0.458.
where torque T is measured in foot-pounds, and both waterline length L and beam B are measured in feet. The factor 57.3 converts degrees into radians.
After checking the accuracy of these equations, we can continue and investigate larger angles using the following line of reasoning: As the chine lifts out of the waterline, the pivot point begins to approach the other chine. A proper Bolger box has equal bottom and side curvatures. Hence at 45 degrees, the buoyant force is directed along a vertical line that intersects the chine. The lever arm is formed by this vertical line and a vertical line that passes through the center-of-gravity.:
It also seems to me that we might obtain useful intuition from this calculation. For example, the center-of-gravity is above as a cross, and the dotted ellipse shows a range of locations the designer may contemplate moving it. The change in the restoring torque is about plus or minus 25% as one moves the center-of-gravity to each end of the ellipse. It would be interesting to see if simple formulas based on this approximate model work well on actual Bolger Boxes.
I suspect that the small angle model, and the model valid near 45 degrees can be combined to get a pretty good picture stability, as shown in the figure below. The straight dotted line is the small angle approximation, which terminates when the chine reaches the waterline. The dotted curve represents the situation near 45 degrees. If we are lucky, the two models will give us what we need to know, for all practical purposes. Of course, we are not always lucky, and that is the joy of science.
I haven’t worked out the details on the formula for the restoring torque near 45 degrees, but I think it looks like this:
where R is the distance from the center-of-mass to the pivot point, which is located somewhere inside the boat, near the submerged chine. Theta-naught (?0) is some reference angle associated with the height of the center-of-gravity above the bottom. If the center-of-gravity is so high that it is directly above the submerged chine at theta equal to 45 degrees, then theta-naught is also 45 degrees and there is no gravitational restoring torque.
In handy units, the restoring torque (in foot-pounds) looks like it might be near:
They say a Bolger Box loses stability a chine lifts out of the water. Take for example the AS-29, which I believe weighs about 1000#. Compare the buoyant torque at 0.1 radians (5.7 degrees):
Compare this with my estimate for the gravitational torque near 45 degrees, making the assumption that my placement of the center-of-mass is correct.
There is much more to be done here. Projects at ALL levels of difficulty are available, from finding other ways to build toy boats to doing calculus and/or computer analysis of stability.