Floating Cylinder?

Question

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.

A liquid with a density of 1.06 g/cm3 is poured into the glass
The cylinder is replaced with one that has the same height and diameter, but with density of 0.89 g/cm3.
A liquid with a density of 0.76 g/cm3 is poured into the glass.
Some of the water is removed from the glass
The cylinder is replaced with one that has the same density and diameter, but with half the height.
The cylinder is replaced with one that has the same density and height, but 1.5× the diameter.

Answer 1

When the heavier fluid is poured into the glass it sinks which raises the level of the light blue water. The cylinder rises with the water at the same displacement.

When the heavier cylinder is placed into the glass, it displaces more water since it will displace water equal to it's weight. Therefore the level of the water rises and the cylinder floats lower.

When the lighter liquid is poured in it also floats on the surface of the water. It also means that the cylinder will sink some since part of what it is displacing is less dense fluid.

When some of the water is removed, the level of everything goes down. If the cylinder comes in contact with the heavier fluid at the bottom, it will float higher.

Same density and diameter, but half the height, means half the weight, therefore half as much total weight of fluid must be displaced for the thing to float at equilibrium.

Same density and height, but 1.5X the diameter. Since the volume of the cylinder is .25*pi*diameter^2*height, this cylinder has weight that is 1.5^2 heavier than the smaller cylinder, so that much more fluid must be displaced for equilibrium.

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