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This is my first time asking a question here so I'm sorry if it's not supposed to be here. I am working on a project relating to the forces on a small dinghy I own. More specifically, I'm looking into how the shape of a sail affects the heeling moment of the boat. Anyways, one thing I haven't been able to work out is how the formula for $F_\text{LAT}$ works.

Some of the forces on boats, wikipedia "Forces on sails"

This is a diagram of the forces I'm looking at. Below is the formula I'm confused by $$ F_\text{LAT}=L\cos(α)+D\sin(α) $$ I figured out that α is the angle between drag and $F_\text{R}$, but I can't figure out which pieces of the final vector the $L\cos(α)$ and $D\sin(α)$ represent.

Edit:

$L$ represents the lift generated by the sail.

$D$ represents the drag generated by the sail.

$V_{a}$ represents the apparent wind direction

$\alpha$ represents the angle between the apparent wind, and the chord line of the sail. this is the angle at which the wind strikes the sail. enter image description here

Sorry if this is a bad question, but none of my teachers have been able to help.

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    $\begingroup$ It would help if you could define the terms in your diagram, which seem to be forces and velocities. What is $D$, $L$ etc? What angle is $\alpha$? There probably is a drag force due to the water? $\endgroup$
    – Toffomat
    Commented Mar 3, 2022 at 6:09
  • $\begingroup$ My bad. Ill add these in $\endgroup$ Commented Mar 4, 2022 at 3:34

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This is about expressing the force exerted by the wind on the sail in two different bases.

The overall force of the wind$^*$ is $F_\text{T}$ in your top diagram, and you can decompose it in two ways$^\dagger$.

  • $D$ and $L$ are the components in the "wind system", i.e. the drag force in the the direction of the wind, and the lift orthogonal to it.
  • $F_\text{LAT}$ and $F_\text{R}$ are in the "boat system", i.e. the lateral force acting sideways and the driving force pushing the boat forward.

The "wind system" and the "boat system" are rotated against each other by the angle $\alpha$ (note that $F_\text{T}$ acts along some direction that is not directly related to $\alpha$), and now the equation you're asking about is simply a statement about how the components are related.

In the boat system, the drag force has a forward component $-D \cos\alpha$ (negative, because it tends to slow the boat) and a sideways component $D\sin\alpha$. For the lift, the forward component is $L\sin\alpha$, and teh sideways component is $L\cos\alpha$. Hence, the overall lateral force is $$F_\text{LAT}=D\sin\alpha + L\cos\alpha\,,$$ and the overall forward force is $$F_\text{R}=D\cos\alpha + L\sin\alpha\,.$$


$^*$ Note that form the boat's perspective, the "appararent" wind is the relevant quantity, even though the name suggests otherwise.

$^\dagger$ Of course, you can always decompose any vector in infinitely many ways. What I mean is that there are two ways that are, in an obvious-but-not-easily-formalised way, "natural" in this situation.

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