Effect of Density and Volume Changes on Buoyancy in a Glass of Water

[Anony-mouse]

A solid cylinder of uniform density of 0.85 g/cm3 floats in a glass of water tinted light blue by food coloring

Its circular surfaces are horizontal. What effect will the following changes, each made to the initial system, have on X, the height of the upper surface above the water? The liquids added do not mix with the water, and the cylinder never hits the bottom.

(1)A liquid with a density of 1.26 g/cm3 is poured into the glass
(2)The cylinder is replaced with one that has the same height and diameter, but with density of 0.81 g/cm3.
(3)A liquid with a density of 0.51 g/cm3 is poured into the glass.
(4)Some of the water is removed from the glass
(5)The cylinder is replaced with one that has the same density and diameter, but with twice the height.
(6)The cylinder is replaced with one that has the same density and height, but 1.5× the diameter.

The Attempt at a Solution

I thought that X would:
(1) No change
Because the dense liquid falls to the bottom of the container, and X is unaffected.
(2) Increase
Lighter Density than water, causes increase in X as less displacement
(3) Increase
The lighter fluid goes on top of both the object and water. The object displaces both water and the fluid, so the buoyant force of the fluid causes and increase in X from the water surface.
(4) No Change
Nothing is added or removed of importance, so X remains the same.
(5) Decrease
Volume increases, so water displacement increases, object sinks, and X decreases.
(6) Decrease
Volume increases, so water displacement increases, object sinks, and X decreases.

But that didn't work. I then changed it so (3) Decreases, but that was also wrong. Any help?

 

[JaWiB]

I'm not so sure about the last two.

5)We're talking about height of the top of the cylinder above the water surface right? So while the cylinder does sink, it's also twice as tall.

6)Same issue, but change the diameter by a given factor has a different effect on the volume (and therefore mass) than changing the height

 

[mgb_phys]

1, slightly unclear, if the denser liquid doesn't mix and is below the level of the bottom of the cylinder - you are correct. It's exactly the same as dropping stones in the water, the water level rises but so does the floating cylinder. If it surrounds the cylinder then the question is similair to 3.

2, correct.

3, No, the object receives an "upthrust equal to the weight of fluid displaced" if you replace the fluid with a lighter one, there is less upthrust and so it sinks. Picture, you can float wood on water but you can float steel on mercury.

4, Correct - so long as the cylinder is still floating and doesn't touch the bottom!

5, No - if it is the same material and the same shape then the PROPORTION of it above the water will be the same but the total distance above will increase. All icebergs have the same 1/7 above the surface but larger bergs stick up futher.

6, No - the diameter doesn't change the floating. Imagine you had two identical cylinders floating next to each other. Now stick them together side by side - do they change at all?

 

[Anony-mouse]

For (4), i think by height they mean the height of the cylinder (which is on its side, so it is the length).

From what I'm understanding, (1)-(4) are correct, but (5) should be no change and (6) should have an increase in X? X is the height from the water to the top of the cylinder at initial conditions.

 

[mgb_phys]

Just saw the note about the liquids not mixing and it not touching bottom - sorry to confuse you.

(5)The cylinder is replaced with one that has the same density and diameter, but with twice the height.

The new cylinder has twice the mass, so it will sink to the point where it has twice the upthrust as the previous one, that is the point where it has twice as much volume below the surface as the previous one. Suppose the original had 15% of it's height above the surface then the new block will also have 15% above the surface. But the 15% is twice as many metres as the old one.

(6)The cylinder is replaced with one that has the same density and height, but 1.5× the diameter.

Pretty much the same argument, 1.5x diameter is approx twice the total mass.
But it is easier to picture if you imagine sawing it lengthways into two half cylinders, you wouldn't expect them to float any differently if you joined them back together, just like you wouldn't expect to boats to float differently if you tied a rope between them.

Interestingly this was Galileo's argument for all weights falling at the same speed. If you threw two cannon balls off a tower at the same time. You might think that a cannon ball twice as large would fall faster. But what if the two were stuck together (ie making the weight twice as heavy), what about if they were only just touching or were tied together with a single hair? At some point it becomes obvious that them being together can't affect how they fall and so all objects must fall at the same speed.