Why Polytarp sails do not suite larger boats.

Problem:

Suppose a sail consists of a 45-degree right triangle of height H. Suppose the equations of motion suggest that the force on that tear depends only on the speed of the wind, the mass density of the wind, and the size of the sail. What formula might describe the tearing force on the sail?

Disclaimer:

I promised the class are “real-life” problem. This is indeed such a real-life problem. Unfortunately, I cannot be sure that this solution is correct. In fact, I know it is a flawed solution. This is often the situation when one attempts to apply physics to complex situations. In this “solution” I have issues with turbulence and the fact that a properly sewn polytarp sail can be less prone to tearing if we strengthen the sides. Nevertheless, I believe this solution to the problem has some merit. I once had a physics professor who compared physics with modern art. His words were something like this:

“It’s like modern art… a nose here, a shape that looks like an ear there, maybe an eye or two. And somehow it looks like a face.”

Solution:

I always like to frame the problem in my own writing:

Here, k is dimensionless. The dimensions (or units) of S, rho, v, and H can be found from the following table:

Parameter	Symbol	Units: L=length T=time M=mass
Tearing force	S	ML/T^2
Wind speed	V	L/T
Sail height	H	L
Air’s mass density	ρ	M/L^3

Using this table, we write:

Where M is mass, L is length, and T is time. The equation circled in red becomes:

Inspecting this equation tells us the power on the M and on the T, since they appear in only one place.

The power of H is still an unknown. Let us label it as “x” and solve for x:

To get the equation for S, we return to the original variables:

Therefore our formula for the tearing force on a sail is:

I have taken the liberty (in blue) of introducing a new constant of proportionality “k-tilda”. This will facilitate the next step, which is to see how the tearing force depends on the size of the sail. In other words, the equation (in blue) with k-tilda tells us how the tearing force depends on size, provided the speed of the air remains unchanged.

Examples of how you can use this equation:

• If the height of a sail doubles, and the wind stays the same, then the tearing force on a rip should quadruple (increase by a factor of 4).
• Suppose you do an experiment on a piece of polytarp and conclude that it tears when the tearing force exceeds 5000 newtons. Suppose you build a 20 ft high sail and measure the tearing force to be 3000 newtons in a wind of 25 miles per hour.
• What would be the tearing force at the same wind speed on a sail with a height of 40 feet? Would the sail tear or not?

The sail would tear!

Another Disclaimer: I HAVE NO IDEA WHAT THE TEARING FORCE IS ON AN ACTUAL POLYTARP SAIL. THIS IS JUST AN EXAMPLE THAT SHOWS THE NUMBERS MIGHT COME OUT.