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I've been looking at some buoyancy problems and one continues to vex me, how to calculate the new fluid height within a container after a floating solid is added to it.

I understand that for a completely submerged solid Archimedes comes in to play, volume of the solid equals volume of the displaced fluid. Volume of the fluid divided by the surface area of the fluid in the container added to the original fluid height gives me the new fluid height... h_{with object}= (V_{solid}/A_{liquid})+h_{start}...easy.

Searching around I find that a floating solid displaces it's own weight in fluid. This sounds great, however, I also find that the percent of the solid submerged is equal to the difference between the fluid density and the solid density. So I understand that and object with average density of 0.4g/cm^{3}will have 40% of it's body submerged in water.

But, what I don't get is that in the case of the water surface area in the container being very close to the surface area of the solid, only one of the above can be true. Either it displaces it's weight in fluid and sinks below the surface, or it remains partially submerged and it doesn't replace it's own weight in fluid.

So I assume there must be a formula to combine the two to work out how high the the fluid will be. Could someone please tell me what it is, or at least tell me what I've misunderstood?

I've attached a picture to help demonstrate what I'm on about.

Cheers

Amo

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# Calculate new water height in container after adding floating object

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