Solving a Buoyancy Problem: Cube of Ice in Water & Ethyl Alcohol

• eri139
In summary, a cube of ice with an edge length of 17.0 mm is floating in ice-cold water with one face parallel to the water surface. When a layer of ice-cold ethyl alcohol 5.00 mm thick is added to the surface of the water, the ice cube reaches hydrostatic equilibrium again. Using the equations FB=ρVg, W = mg, and V = lwh, it can be determined that the distance from the top of the water to the bottom face of the ice cube is 11.68 mm (or 10.31 mm if the correct ice density of 917 kg/m^3 is used).
eri139

Homework Statement

A cube of ice whose edge is 17.0 mm is floating in a glass of ice-cold water with one of its faces parallel to the water surface.

Ice-cold ethyl alcohol is gently poured onto the water surface to form a layer 5.00 mm thick above the water. When the ice cube attains hydrostatic equilibrium again, what will be the distance from the top of the water to the bottom face of the block?

FB=ρVg
W = mg
V = lwh

The Attempt at a Solution

Here is my work. I have checked and rechecked it, but for some reason it's still not correct! Please help!

mg = FBalcohol + FBwater
m = ρalcoholValcohol + ρwaterVwater
ρiceV = ρalcoholValcohol + ρwaterVwater
934 x 173 = 789 x 172 x 5 mm + 1000 x 172 x h
934 x 17 = 789 x 5 + 1000h
h = 11.933 mm

The ## h ## in your 3rd to the last equation needs to be ## L=17-5-h ## to compute the volume of water that is displaced. You basically computed ## L=11.933 ##... which is how much of the cube is immersed in the water portion. ## \\ ## Edit: I read it too quickly: I was computing the ## h ## from the surface of the liquid to the top of the block. Anyway, ## L=11.93 ## mm is immersed in water, and add ## 5 ## mm to that to get their ##h ##. ## \\ ## Edit: Nope, I read it too quickly a second time: It says the surface of the water, and not the surface of the liquid. I think you may have it right, but the person who wrote out the problem didn't read his own words carefully enough. ## \\ ## Another idea: The problem could also be the sig figs in your answer. Try putting in 11.9.

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Where did you get your ρice value? What I've seen in the literature for ice at 0°C is more like ##0.9150 gm/cm^3##.

gneill said:
Where did you get your ρice value? What I've seen in the literature for ice at 0°C is more like ##0.9150 gm/cm^3##.
Google. I typed in "ice density" and the first value that popped up was that one. Unfortunately, I've just figured it out that this density is wrong and is the reason why my answers have been thrown off. Now I've finally gotten it...glad to see it wasn't an issue with my understanding, though!

eri139 said:
Google. I typed in "ice density" and the first value that popped up was that one. Unfortunately, I've just figured it out that this density is wrong and is the reason why my answers have been thrown off. Now I've finally gotten it...glad to see it wasn't an issue with my understanding, though!
Yeah, Never believe the first entry to pop up from Google. Always check the source to see if it's reputable. Too many non-academic searchers bubble disreputable sources to the top of the hit list. It's a sad thing, but its just the way it is. Glad you worked out your issue!

@eri139, Not a trivial problem. I see nothing wrong with the wording. What was your answer? I used symbols instead of numbers & won't do the arithmetic, as usual. The answer turned out to be pretty elaborate, involving all three densities plus the alcohol layer depth plus the cube's side.

(I also did a second computation to verify my answer.)

EDIT: the anser can be simplified to comprise just the three densities and one side of the cube. With the simplification the answer I got is 11.68mm using ice density at 934 kg/m^3. But post 3 is right: should be 917 kg/m^3 in which case the answer I get is 10.31mm.

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1. How do I calculate the buoyant force on a cube of ice in water and ethyl alcohol?

To calculate the buoyant force, you will need to know the density of the fluid (water or ethyl alcohol), the volume of the cube of ice, and the density of the ice. The buoyant force can be calculated using the formula Fb = ρgV, where ρ is the density of the fluid, g is the acceleration due to gravity, and V is the volume of the displaced fluid.

2. What is the difference between buoyancy in water and ethyl alcohol?

Buoyancy is the upward force exerted on an object when it is submerged in a fluid. The main difference between buoyancy in water and ethyl alcohol is the density of the fluids. Water has a higher density than ethyl alcohol, so the buoyant force will be greater for the same volume of fluid. This means that an object will float more easily in ethyl alcohol than in water.

3. How does the size of the cube of ice affect the buoyant force?

The size of the cube of ice will affect the buoyant force because the volume of the displaced fluid will change. If the cube of ice is larger, it will displace more fluid and therefore experience a greater buoyant force. However, the density of the ice will also play a role in the buoyant force, as a larger cube of ice may have a lower density and therefore experience a weaker buoyant force.

4. Is the buoyant force the only force acting on the cube of ice?

No, the buoyant force is not the only force acting on the cube of ice. Gravity is also acting on the cube of ice, pulling it downwards. The forces are balanced when the cube of ice is at equilibrium, meaning it will neither sink nor float. However, if the buoyant force is greater than the force of gravity, the cube of ice will float, and if the force of gravity is greater, it will sink.

5. How can I determine if the cube of ice will float or sink in water and ethyl alcohol?

To determine if the cube of ice will float or sink, you will need to compare the density of the ice to the density of the fluid. If the density of the ice is lower than the density of the fluid, the cube of ice will float. If the density of the ice is higher, it will sink. This is because objects with a lower density than the fluid will experience a greater buoyant force, causing them to float, while objects with a higher density will experience a weaker buoyant force and sink.

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