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I have read the explanation for this in several textbooks, but I am struggling to understand it via Archimedes' principle. If someone can clarify with a diagram or something so I can understand or a clear equation explanation that would be great.

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6 Answers 6

up vote 30 down vote accepted

Good question.

Assume we have one cube of ice in a glass of water. The ice displaces some of that water, raising the height of the water by an amount we will call $h$.

Archimedes principles states that the weight of water displaced will equal the upward buoyancy force provided by that water. In this case,

$$\text{Weight of water displaced} = m_\text{water displaced}g = \rho Vg = \rho Ahg$$

where $V$ is volume of water displaced, $\rho$ is density of water, $A$ is the surface area of the glass and $g$ is acceleration due to gravity.

Therefore the upward buoyancy force acting on the ice is $\rho Ahg$.

Now the downward weight of ice is $m_\text{ice}g$.

Now because the ice is neither sinking nor floating, these must balance. That is:

$$\rho Ahg = m_\text{ice}g$$

Therefore,

$$h = \frac{m_\text{ice}}{\rho A}$$

Now when the ice melts, this height difference due to buyoancy goes to 0. But now an addition mass $m_\text{ice}$ of water has been added to the cup in the form of water. Since mass is conserved, the mass of ice that has melted has been turned into an equivalent mass of water.

The volume of such water added to the cup is thus:

$$V = \frac{m_\text{ice}}{\rho}$$

and therefore,

$$Ah = \frac{m_\text{ice}}{\rho}$$

So,

$$h = \frac{m_\text{ice}}{\rho A}$$

That is, the height the water has increased due to the melted ice is exactly the same as the height increase due to buoyancy before the ice had melted.

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6  
@MattThrower But that illustrates that ice is less dense than water, which is why ice is buoyant to begin with... But ice will only displace a volume of water equivalent to the volume of ice that is below the water level... They cancel each other out? –  David Wilkins Apr 30 at 17:49
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Aha, so they counter balance perfectly? I had no idea. That's the danger of knowing more chemistry than physics :) –  Matt Thrower May 1 at 7:58
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@Jodrell, the case for polar ice caps is different, since the icecap will melt to form fresh water, where as the surrounding ocean water is salty (differing densities). The above analysis only applies if the floating solid melts to form the same liquid initially supplying buoyancy force. –  Mew May 1 at 8:48
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@Jodrell: It's the big sheets of ice sitting on Greenland and Antartica that everyone worries about. –  Hurkyl May 1 at 8:48
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Here is an explanation that needs no calculations.

Consider the following diagram, in which part1 and part2 represent the ice.

The displaced water volume equals part2 volume and has as much mass as (part1+part2)

Now look at what happens when both part1 and part2 melt:

  1. their mass does not change, it is (part1+part2)
  2. it becomes water.

And we just said that part1+part2 mass water has part2 volume.

enter image description here

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You mention "The displaced water volume equals part2 volume and has as much mass as (part1+part2)". Why does the displaced water, which has part2's volume, have as much mass as part1 + part2? The black line that separates part1 from part2 seems to be arbitrarily drawn. –  Peter Majeed May 2 at 14:09
    
The line is not arbitrary, part2 is immersed, and therefore it is the part directly responsible for displacing the water. –  kalkanistovinko May 2 at 14:35
    
Yeah, I think I get that part. But why is the mass of the displaced water equal to the mass of part1 + part2? That water's volume is definitely the volume of part2, but I'm not intuiting why it's equal to the mass of both parts. –  Peter Majeed May 2 at 14:44
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That is Archimedes' principle in application. Keep in mind that the ice density is less than that of water, therefore, a lesser amount of water is required to balance the whole ice body. –  kalkanistovinko May 2 at 15:21
    
Right - I suppose my problem is that while I understand that principle formulaically, the part of it that relates to mass isn't as intuitive by glancing at the diagram. I kinda have to assume that principle is in effect before accepting the diagram as it is. –  Peter Majeed May 2 at 16:07

Brandon, above, gets right to the point. Frozen water displaces its own mass in the rest of the water, which means in effect it displaces an amount equal to itself. While frozen it is larger in volume, and thus less dense, because of hydrogen bonding -- that's why it floats -- and when it melts it returns to the liquid state (surprise!) at essentially the same density as the surrounding water. A given quantity of water, temporarily larger in volume but correspondingly less dense because it has frozen, returning to the liquid state will thus not raise the overall level of water (assuming here no evaporation, mosquitoes stopping in to have a sip, etc etc) --

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It does, it lowers it. Frozen water takes up more space that liquid water (you'll notice it floats -that is because there is less mass per volume). As it melts it takes less space per area thus the water level goes down.

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You're forgetting that the mass of the ice is the same mass as the water that froze to make it. Floating objects only displace their mass worth of fluid. –  Brandon Enright Apr 30 at 23:29

While the two existing answers offer some good science, on why the water level does not change, they are both slightly over optimistic. While the level will not rise, it will not remain the same either. Evaporation will occur simultaneously with melting, and the water level in the glass will actually lower slightly.

Even if the ice remains frozen the process of Sublimation will allow the ice to evaporate. You have may seen this occur in your freezer, or with snow on your porch

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You are neglecting possible condensation of water vapor from the air on the ice, which would have the opposite effect. –  Oldcat Apr 30 at 22:03
    
@Oldcat, There are several variables, container design, humidity, air temp & water temp. Evaporation occurs with all the variables, I was assuming the condensation would occur in about half the scenarios, so left it out for simplicity. You are welcome to add it in to my answer if you believe I was mistaken on the ratio of condensation scenarios. –  James Jenkins May 1 at 10:32

Consider an ice cube of 10cm, assume that the density of water and ice is 10:9. At first, the ice has 9 cm in the water, when it melts, it become water of volumn 10*10*9. Just fills the melted ice original in the walter.

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