Disappearing Ice
Ice melting in water within a cup has a surprising effect on the water level.
In this article, we'll talk about some of the elementary ideas of fluid statics. Let's start with the puzzle and then we'll see why it happens. The image shows the scenario that we'll be focusing on today. Just some normal ice cubes in a glass of water. We can begin by marking the level at which the water level sits. Then we wait for the ice cubes to melt and look at the water level with respect to the marking again. What do you think happened?
For this experiment, we need to familiarize ourselves with the basic principles of fluid statics. Let's talk about Archimedes' principle. The name might sound familiar to you. He is famously known as the man who yelled "Eureka!" (means "I have it!" in greek") while running naked through the streets of Syracuse. The story goes that the King had been given a crown that was supposed to be of pure gold, but was suspicious of silver being mixed into it. He asked Archimedes to find out the truth. Archimedes realized to solve this he had to find the density of the crown and compare it to the known value of gold's density, but he had to do it without damaging the crown. One day as Archimedes sat down in his bath, he noticed water falling out of the sides. This is when he realized that if he put the crown in the water, it would displace measurable water equal to the volume of the crown. By dividing the mass of the crown by the volume, he could find the density.
So we have now introduced the principle that an object in water will displace the same volume of water as the volume of the object inside the water. Here's an example. If I drop an object of 5 cm³ into a cup of water and it is fully submerged, the water level will rise. If I drop a cube of water with volume 5 cm³, the water level will also rise. Furthermore, they will rise the exact same amount. However, in our case things are a little different because ice floats in water. This means that not all of the ice will be submerged under the water.
This is where we're going to have to introduce some math. Archimedes had a principle that stated the buoyant force exerted upwards on an object is equal to the weight of the fluid the body displaces. That means that the force that pushes up on the ice is equal to the weight of water of the volume of the object in the water. Let's look at the image to see where the calculations begin.
The calculations in the next image use Archimedes' Principle to determine that there is always 91 percent of a piece of ice left under water when it floats. This is regardless of the mass of the ice cube (as long as we assume water is incompressible). This is why if you saw an iceberg in the ocean, you would only be seeing about a tenth of the entire mass. We have to understand that the density of ice is less than that of water. This means that if we had a block of ice and a block of water of the same size, if we melted the block of ice there would only be 0.91 times the amount of water as compared to the block of water. When ice floats in water, the water level is already pushed up from its initial position. It rises such that there was an amount of water added of 0.91 times the volume of the ice cube. This is because only 91% of the ice cube is submerged, and water will be displaced equal to the volume of the object in it.
Let's say the ice cube has a volume "V". So we understand that the water level has risen as if 0.91V of water was added (91% of the volume of the ice cube is actually under water). We also know that because of differences in density, the amount of liquid water in the ice cube is also 0.91V. These are the exact same! So, as the ice cube melts, the amount of water displaced by the volume of the ice decreases at some rate. The amount of water added by the melting ice increases at the exact same rate. This may be a little hard to believe so here are the before and after photos of the experiment. As you can see, the water level stays at the exact level of the marker before and after the ice melts. I have also attached a youtube video of a time lapse of the process.
