So I am trying to find the depth of immersion of this particular boat model, which has curvilinear edges (subparabolic edges) on each side. The figure of the pontoon is shown below(adsbygoogle = window.adsbygoogle || []).push({});

So now when I place this pontoon in water, the height of the water that rises, called the depth of immersion is to be found. I already know how to calculate the displacement volume, but I have no idea how to find the depth of immersion. Below are the steps taken to find the displacement volume

1. Volume of body

I assumed the whole body to be a rectangle and calculated the area of the rectangle first:B*h

And then, since the curves are subparabolic, it is defined by the equation:

y =(h/b^2)*x^2

Integrating the above equation, gives the area under the curve to be:

bh/3

So subtracting : 2*(bh/3) from (B*h) will give the cross sectional area which

C.S.A =(B*h)-(2*(bh/3))

Multiplying this C.S.A with the length of the pontoon (say L) will give the volume

V = (C.S.A)*L =[(B*h)-(2*(bh/3))] * L

So now that volume of the body is known, the following steps are taken to find the displacement volume

2. Density of body

Density = mass of body / Volume

Note: The mass was found by weighing the pontoon on a weighing machine

3. Specific weight of body

Specific weight of body = density of body * g

Note: g = 9.81 m/s^2

4. Weight of body

Weight of body = Specific weight of body * Vbody

5. Weight of water displaced

Weight of displaced water = Specific weight of water * Displaced volume

Note: Specific weight of water = 9810 N/m^3

6. Archimedes principle

According to Archimedes Principle the weight of the body acting downwards is equal to the weight of displaced acting upwards, so

Wbody = 9810 * Vdisp

7. Displaced Volume

And so the displaced volume of water is given by

Vdisp = Wbody/9180

Now another equation for the displacement volume is:

Vdisp = [(B'*h')-(2*(b'h'/3))] * L

h' is the depth of immersion

and B' is the top width at this displaced volume

The problem is that there are two unknowns in the above equation and I just can't figure out how to find the depth of immersion here. If anyone has any idea please please let me know.

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# How to find the depth of immersion of this boat model?

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