How Do You Calculate the Depth of Oil in a Sinking Barge?

[chimmy48]

Homework Statement

A rectangular barge floats in water, ρw. When it is empty it is immersed at depth D below the surface. Oil with density, ρo, is poured into the barge until it is about to sink. Find a relationship for the depth of the oil at this point in terms of the initial depth, D, the total height of the barge H and the barge width W.

Homework Equations

P = ρgh

The Attempt at a Solution

Well, we basically have three situations here. One when its floating, the other when its immersed (the container is empty) and the last one, when oil is poured into the container until it sinks. So we have the depth of the oil that is poured into the container to figure out in terms of H, D and W.

At first, i started to think, maybe we could establish a volume relationship, but then i ran into problems, whilst doing that. So, i guess its recommended to include pressure into it, since we have density of the oil and the water. But this is all i have thought about. I really don't know how to approach it. I really really need help please. Thanks!

 

[TVP45]

Can you show some of your work? Perhaps you could start with a free body diagram identifying all the forces acting on the barge in each situation.

 

[piercas]

I would first start by identifying the variables in the problem: the density of water (ρw), the density of oil (ρo), the initial depth of the barge (D), the total height of the barge (H), and the width of the barge (W).

Next, I would use the equation for pressure, P = ρgh, to relate the pressure at the bottom of the barge (when it is empty) to the pressure at the bottom of the barge when it is filled with oil. This can be done by setting the two pressures equal to each other:

Pempty = Pfilled

ρwgh = ρogho + ρwghw

where ho is the depth of the oil at the point when the barge is about to sink and hw is the remaining depth of water in the barge.

We can then rearrange the equation to solve for ho:

ho = (ρwgh - ρwghw) / ρog

Substituting in the values for ρw, ρo, and g, we get:

ho = (ρwD - ρw(H - D)) / ρo

Simplifying, we get the final relationship:

ho = (ρwD + ρwD - ρwH) / ρo

or, ho = (2ρwD - ρwH) / ρo

This relationship gives us the depth of the oil at the point when the barge is about to sink in terms of the initial depth, the total height, and the width of the barge.