Conceptual question on floating

[kevinnn]

Homework Statement

An ice cube with many air bubbles is places in water, does the water level rise, sink, or stay the same when it melts? Does your answer change if the ice cube is filled with many grains of sand?

Homework Equations

The Attempt at a Solution

I'm not too positive for the ice cube with the air but i believe it will stay the same. I also believe the volume will stay the same if the ice cube had many grains of sand. What I'm confused about is I don't know if the ice cubes are floating or submerged so it must not be important to the question. But I would think it would be because an object floating will displace a volume of water equal to its weight and a submerged object will displace a volume of water equal to its volume. Can someone point a finger in the right direction. Thanks.

 

[haruspex]

For the sand case, it's not clear whether you're to assume the cubes floated initially. Suppose they did. You agree that when floating the water displaced matched the cubes by weight. You can think of that as an amount displaced by the weight of the ice and another amount by the weight of the sand.
Having melted, the water portion of the cubes is displacing an amount by volume, but that's the same as the amount it would displace by weight, and its weight has not changed. Is the same true for the sand?
How does the answer change if the cubes carried so much sand that they sank right off?

 

[kevinnn]

The sands weight has not changed of course but it was floating so it should be displacing more water. If it sank to the bottom then the water level should not change because then the sand is displacing a volume of water equal to its volume. But in a sense when the sand is in the ice cube it is all ready submerged and thus already displacing water, except the water is in solid form.

 

[haruspex]

The sands weight has not changed of course but it was floating so it should be displacing more water. If it sank to the bottom then the water level should not change because then the sand is displacing a volume of water equal to its volume. But in a sense when the sand is in the ice cube it is all ready submerged and thus already displacing water, except the water is in solid form.

I'm not entirely sure what you're saying. When the sand was in the floating ice cubes, the liquid water it effectively displaced corresponded to its weight. Yes, it was in a sense submerged, but it was floating, not resting on the bottom, and that's what matters. And yes, it was displacing by volume within the ice cube, but the volume of the ice cube does not affect the level of the liquid water, so that's irrelevant.
When the ice melts, the liquid water it displaces corresponds to its volume, so it now displaces less, right?

 

[kevinnn]

Right. Now my concern is that the question did not specify whether or not the ice cubes were floating or not. The answer is the water level would go down if they were floating with the sand dissolved in it but would remain unchanged if the ice cubes were to be held down with fork or something correct.

But now what about the air?

 

[haruspex]

Right. Now my concern is that the question did not specify whether or not the ice cubes were floating or not. The answer is the water level would go down if they were floating with the sand dissolved in it but would remain unchanged if the ice cubes were to be held down with fork or something correct.

If the ice cubes had been held down with a fork then the level would be sure to go down when they melted, sand or no sand. The uncertainty is whether there was so much sand in the cubes that they sank. What do you will happen in that case?

But now what about the air?

Here's one way to think about this case. Imagine that the ice cube were tightly enclosed in a weightless, infinitely thin box. Now redistribute the air in the cube so that it's all at the top, but still held in by the box. Does that change the level at which the cubes float? Now take away the box. Does anything change?

 

[kevinnn]

If there was so much sand they sank the volume of water would go down because water expands as it freezes so the volume of water would go down as a result of that. But I don't think it will go down as a result of the sand. Because the sand is all ready submerged.
Nothing changes after the box is removed.

 

[haruspex]

If there was so much sand they sank the volume of water would go down because water expands as it freezes so the volume of water would go down as a result of that. But I don't think it will go down as a result of the sand. Because the sand is all ready submerged.
Nothing changes after the box is removed.

Yes and yes.

 

[kevinnn]

So for the air the volume of water does not go down because the air does not carry enough weight? What about a massive block of ice with a lot of air in it. When the cube melted (if the block is floating) the water level would still not change because as the ice melts and the air escaped, more air fills its place right? The water level only changes when the cube is submerged and the air escapes.

 

[Dick]

So for the air the volume of water does not go down because the air does not carry enough weight? What about a massive block of ice with a lot of air in it. When the cube melted (if the block is floating) the water level would still not change because as the ice melts and the air escaped, more air fills its place right? The water level only changes when the cube is submerged and the air escapes.

Warm up with a different question. Suppose the block is pure ice? What happens then?

 

[kevinnn]

Nothing. The water level will stay the same because the volume of water displaced will be equal to the volume of the solid block of ice.

 

[Dick]

Nothing. The water level will stay the same because the volume of water displaced will be equal to the volume of the solid block of ice.

No, the volume of water displaced will have a volume whose weight is equal to the weight of the block. Not the volume of the block.

 

[kevinnn]

I'm sorry that is what I meant to put. I understand that concept but it is when you get to the air and the sand. For the sand the water level will not fall until the sand has fallen out of the ice cube and under the surface of the water right?

 

[Dick]

I'm sorry that is what I meant to put. I understand that concept but it is when you get to the air and the sand. For the sand the water level will not fall until the sand has fallen out of the ice cube and under the surface of the water right?

Imagine a block of sand enclosed in a block of ice. As long as the ice is melting the weight lost by the block is exactly compensated by the weight of the displaced volume gained by the water, right? So the level doesn't change. What happens when the ice finally melts?

 

[kevinnn]

When the ice finally melts the sand will fall to the bottom and then no longer displace a volume of water equal to its weight but to its volume. So the water level should go down.

 

[Dick]

When the ice finally melts the sand will fall to the bottom and then no longer displace a volume of water equal to its weight but to its volume. So the water level should go down.

That sounds exactly right.

 

[kevinnn]

OK thanks. So now what about the air?

 

[Dick]

OK thanks. So now what about the air?

Apply the same reasoning to that case.

 

[kevinnn]

So in that case as the ice melts the air will escape and the water volume will go down. The weight of the air makes a noticeable difference?

 

[Dick]

So in that case as the ice melts the air will escape and the water volume will go down. The weight of the air makes a noticeable difference?

Air is a thousand times less dense than water. Ignore that in a first approximation. Apply your logic from the sand case to this case.

 

[haruspex]

So in that case as the ice melts the air will escape and the water volume will go down. The weight of the air makes a noticeable difference?

The escape of the air will only make a difference to the extent that the bubbles were at a pressure other than atmospheric. To see this, consider my infinitely-thin box thought experiment in post 6.

 

[kevinnn]

So the water level should not change because the ice is displacing a volume of water equal to its weight, but that is the same as the volume it will occupy when melted. Correct?

 

[haruspex]

So the water level should not change because the ice is displacing a volume of water equal to its weight, but that is the same as the volume it will occupy when melted. Correct?

Yes.

 

[kevinnn]

Ok cool. Thanks.