... or a fun exploration of volume, mass, density, floatation, global warming, and how to float in a swimming pool.

You are watching: An ice cube floats in a glass of water. as the ice melts, what happens to the water level?

by Jared Smith

Principles

Archimedes" Principles:

Any floating object displaces a volume of water equal in weight to the object"s MASS.Any submerged object displaces a volume of water equal to the object"s VOLUME.

Formula

Mass / Density = Volume

Melting ice cube

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If you place water and an ice cube in a cup so that the cup is entirely full to the brim, what happens to the level of water as the ice melts? Does it rise (overflow the cup), stay the same, or lower?

The ice cube is floating, so based on Archimedes" Principle 1 above, we know that the volume of water being displaced (moved out of the way) is equal in mass (weight) to the mass of the ice cube. So, if the ice cube has a mass of 10 grams, then the mass of the water it has displaced will be 10 grams.

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We know the density (or compactness, weight per unit) of the ice cube is less than that of the liquid water, otherwise it wouldn"t be floating. Water is one of the very few solids that is less dense than when in its liquid form. If you take a one pound bottle of water and freeze it, it will still weigh one pound, but the molecules will have spread apart a bit and it will be less dense and take up more volume or space. This is why water bottles expand in the freezer. It"s similar to a Jenga tower. When you start playing it contains a fixed number of blocks, but as you pull out blocks and place them on top, the tower becomes bigger, yet it still has the same mass/weight and number of blocks.

Fresh, liquid water has a density of 1 gram per cubic centimeter (1g = 1cm^3, every cubic centimeter liquid water will weigh 1 gram). By the formula above (Mass / Density = Volume) and basic logic, we know that 10 grams of liquid water would take up 10 cubic cm of volume (10g / 1g/cm^3 = 10cm^3).

So let"s say that our 10 gram ice cube has a density of only .92 grams per cubit centimeter. By the formula above, 10 grams of mass that has a density of .92 grams per cubic centimeter will take up about 10.9 cubic centimeters of space (10g / .92g/cm^3 = 10.9cm^3). Again, the volume of 10 grams of frozen water is more than the volume of 10 grams of its liquid counterpart.

The floating ice cube has a mass of 10 grams, so based on Archimedes" Principle 1, it is displacing 10 grams of water (which has 10cm^3 of volume). You can"t squeeze a 10.9cm^3 ice cube into a 10cm^3 space, so the rest of the ice cube (about 9% of it) will be floating above the water line.

So what happens when the ice cube melts? The ice shrinks (decreases volume) and becomes more dense. The ice density will increase from .92g/cm^3 to that of liquid water (1g/cm^3). Note that the weight will not (and cannot) change. The mass just becomes more dense and smaller - similar to putting blocks back into their original positions in our Jenga tower. We know the ice cube weighed 10 grams initially, and we know it"s density (1g/cm^3), so let"s apply the formula to determine how much volume the melted ice cube takes. The answer is 10 cubic centimeters (10g / 1g/cm^3 = 10cm^3), which is exactly the same volume as the water that was initially displaced by the ice cube.

In short, the water level will not change as the ice cube melts

Other oddities

Anchors away

Using this same logic, there are some fun analogies. Consider an aluminum boat in a swimming pool. If you put a 5 gallon bucket full of 100 pounds of lead or some other metal into the boat, the boat will get lower in the water and the additionally displaced water in the pool will cause the pool level to rise. And based on Archimedes" Principle 1 for floating objects, it would rise by the volume of water equal in weight to the 100 pound lead bucket. Water weighs 8.3 pounds per gallon, so the boat will displace an additional 12 gallons of water (12 gallons * 8.3 pounds per gallon = 100 pounds).

What would happen if you throw the bucket of lead overboard into the pool? Will the pool level increase, decrease, or stay the same?

When we toss the bucket of lead overboard, the pool level goes down 12 gallons (the volume of water no longer displaced by the weight in the boat). But when it enters the water, it will be submerged, so we now need to apply Archimedes" Principle 2 for submerged objects (it will displace a volume of water equal to the object"s volume). The water level will then go up by the volume of the lead bucket, which is 5 gallons. So, the net difference is that the pool level will go down by 7 gallons, even though the bucket is still technically in the pool.

Just remember that mass and density don"t matter for submerged objects. Volume is everything. Consider dropping a brick of clay and a brick of gold into a bucket. The gold has more mass and is more dense than the clay, yet if both bricks are the same size, both will displace the same amount of water.

A sinking ship

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Similarly, if your aluminum boat weighed 100 pounds, it would displace 100 pounds of water (12 gallons) when floating. But if the boat springs a leak and sinks, the pool level would decrease 12 gallons minus the volume of the aluminum in the boat. The boat"s volume (amount of space comprised of aluminum metal) would be much less than 12 gallons. In fact, based on the density of aluminum (.1 pounds/in^3), we can determine using our formula that the volume of our 100 pound boat will be about 1000 cubic inches (100/.1 = 1000). There are 231 cubic inches per gallon, so the boat is comprised of about 4.3 gallons of aluminum (1000/231 = 4.3) and thus displaces 4.3 gallons of water when submerged, much less than the 12 gallons that same aluminum displaced when floating. In conclusion, when our boat sinks, the pool level goes down by 7.7 gallons.

Experiment

As an experiment, fill a sink with 5 or 6 inches of water and note the water level. Next set a heavy glass down into the sink while balancing it right side up (i.e., so it doesn"t tip over and fill with water). The water level will notably rise to make room for the empty glass and you"ll note that it"s difficult to get the glass to sink while also it is upright. The heavy glass displaces a lot of water because of the heavy mass of the glass (Archimedes" Principle 1), yet it still floats because of it"s low density (don"t forget about all the air inside the glass). The glass will feel lighter to you because of the buoyancy principle (the force of the displaced water pushing up against the weight of the object displacing it). It will float perfectly in the water when the weight of the glass equals the weight of the water it is displacing.

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Now lay the glass down sideways and let it submerge in the sink. The water level will be only barely higher than the original level. It now displaces very little water because the glass has a very low volume (Archimedes" Principle 2).

A marble in ice

Back to our original scenario, what if the ice cube had a small marble embedded inside of it? When the ice melts, would the water level increase, decrease, or stay the same?

Let"s say we have the same ice cube as before (10g with a density of .92g/cm^3 and volume of 10.9cm^3) and a 1 gram marble with a density of 2g/cm^3. Using the formula above, we know the marble has to have a volume of .5 cubic centimeters (1g / 2g/cm^3 = .5cm^3). Obviously the marble would just sink if we tossed it in the glass because its density (2g/cm^3) is higher than water"s (1g/cm^3). And we know when submerged it would displace .5cm^3 of water (Archimedes" Principle 2). But when embedded in the ice cube, what happens?

Will it float?

First, we need to determine whether the ice cube will sink or float now that it has the marble in it. To do this, we need to figure out the combined density of the ice cube AND marble. We know that the ice cube has a mass of 10 grams and the marble has a mass of 1 gram, for a combined mass of 11 grams. We also know that the ice cube has a volume of 10.9cm^3 and the marble has a volume of .5cm^3, for a combined volume of 11.4cm^3. Using the formula, we can determine that the combined density is .965g/cm^3 (11g / Density = 11.4cm^3, or 11/11.4 = .965). In other words, the small marble obviously increases the combined density, but it"s still less than the density of water, so the thing will definitely float!