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I need help determining the forces, and the direction of them in this fluid statics problem.

A hollow cylinder with closed ends is 300mm in diameter and 450mm tall. It has a mass of 27kg and a tiny hole at the bottom. It is lowered slowly into water and then released. I need to calculate:

a. The pressure of the air inside the cylinder.

b. The height to which the water will rise.

c. The depth to which the cylinder will sink.

I was already given the answers to these problems for clarification but I think I'm going to start with the free body diagram (FBD) to part c.

Now I know there is a weight from the cylinder going down and a buoyant force going up because of the water and the submerged volume of the cylinder. Now does the air trapped in the cylinder also create a buoyant force or some kind of force also pushing up? And how would I calculate this force if there is one? After that, I'm sure I can get the rest of the calculations.

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Are those measures inside or outside? How deep ist that water? –  Georg Oct 24 '11 at 16:10
    
@Georg, it doesnt say in the problem. I think its just a shell so the cylinder is hollow. I just want to know the force on the free body diagram, because I already have equations for the first two parts of the problem. But I also dont want a straight forward answers. I by no means want to be spoon fed the answer –  Greg Harrington Oct 24 '11 at 18:39
    
Do you think the volume is much too small to float the 27 kgs even with the pocket of air that is created at the top? Even if it was big enough, this is a theoretical question and I am only concerned with the forces. I would think there is the weight of the cylinder acting down, the buoyant force due to the water acting up, and then a pressure force from the air acting up which is F = PA where P is the air pressure inside the cylinder and A is the cross sectional area of the cylinder. What do you guys think? –  Greg Harrington Oct 24 '11 at 21:18
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I deleted my comment, I had an error in calculation. –  Georg Oct 27 '11 at 10:12
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1 Answer 1

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Greg.

Question a.

The maximum theoretical buoyancy of the cylinder is the maximum cavity volume; $V = \pi R^{2} L$, where $R$ is the cylinders radius and $L$ is it length. I will ignore the "tiny hole at the bottom" as it is poorly defined. Assuming the air inside the cylinder is incompressible what buoyancy force does the cylinder provide, that is what can it 'float'?

$ V = \pi R^{2} L = 0.031809 \mathrm{m}^2 = 31.809 \mathrm{L}$.

From this the mass ($M$) it can float in the ideal case specified above is

$m = \rho V \approx 997.0 \times 0.031809 = 31.7 \mathrm{kg}$.

So, no we have cleared that up, you want to do the following force balance to get the equilibrium pressure inside the cylinder for the 'real' case of compressible air. If you consider the system to be the can plus the air inside, the only forces are the weight of the cylinder and the pressure between the air and the liquid. That is,

$m \mathtt{g} - \pi R^{2} P = 0$,

where $P$ is the pressure inside the cylinder at equilibrium, $\mathtt{g}$ is the gravitation constant and $m$ is the weight of the cylinder/weight you want the massless cylinder to float (i.e. 27 kg).

Questions b. you can have a go at. For b you will need to think about the pressure and volume of the air inside the cylinder initially $P_{0}$ and $V_{0}$ and how they relate to the air inside the container during its compressed state $P_{1}$ and $V_{1}$, where you have worked out the pressure $P_{1}$ above. Hint. Assume one of the ideal gas relationships to get the unknown volume $V_{1}$. Use this to get the depth...

For question c. you will need to think about the volume of water required to float the cylinders mass of 27kgs plus the water inside it. From this volume you will be able to extract the length of cylinder that needs to be submerged in order to provide sufficient buoyancy to float.

Hope this helps.

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