homework and exercises - Free Body Diagram of Fluid Statics Problem

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

A hollow cylinder with closed ends is $300\,\rm mm$ in diameter and $450\,\rm mm$ tall. It has a mass of $27\,\rm kg$ and a tiny hole at the bottom. It is lowered slowly into water and then released. 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.

Answer

Question a

The maximum theoretical buoyancy of the cylinder is the maximum cavity volume:

$$V = \pi R^2L$$

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'?

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

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,

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

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. $27\,\rm kg$).

Question 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.

Question c

You will need to think about the volume of water required to float the cylinders mass of $27\,\rm kg$ 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.