Water level of ice in cup [duplicate]

Question

Since ice is less dense than water, it floats on it, we all know this. Now suppose, if ice is floating in a glass cup, when the ice melts there should be some change at water level in the glass cup. But when I found this on the internet, it states that the water level will be same. Can someone justify/explain this?

In my way, water level will decrease because ice has greater volume than water, so when it floats, it will displace some water equal to its volume for floating. But when it melts, its volume decreases and hence water level will also decrease. Is it right or wrong what I am thinking?

Answer 1

Let's say that you have an ice cube floating on water. How much of the volume is submerged? Or, put another way: how much water is displaced?

When the ice is floating, weight and buoyant force are in equilibrium: (subscript $i$ for ice, subscript $w$ for water, subscript $d$ for displaced.)

$$\rho_{i}gV_{i}=\rho_{w}gV_{d} $$

so that

$$\frac{\rho_{i}}{\rho_{w}}V_{i}=V_{d} $$

Now, what is the volume of the ice, $V_i$? It's clearly

$$V_i=\frac{m_i}{\rho_i} $$

Put in the formula above, this gives:

$$V_{d}=\frac{m_i}{\rho_w} $$

This formula tells you that the Volume displaced by the floating ice cube is independent of its density. So when the ice cube melts, the displaced volume stays the same and the water level doesn't rise.

Edit (I sent the answer half finished by accident...):

this principle is known as the principle of flotation:

Any floating object displaces its own weight of fluid.

Since the mass of the ice cube stays the same - even if it's melted - the amount displaced stays also the same.

Answer 2

NoEigenvalue is absolutely correct, good job. If you aren't familiar with the math or don't really understand it, I suggest this little experiment to help you understand it. Put some water in a glass and mark its level. Then pour some of the water into a smaller container and freeze it. Now, put the ice back into the water and see that even though some of the ice now floats above the water, the water level returns to its original level. The mass of the water displaced by the ice is the same as the water poured out and frozen. The ice floating above the water represents the volume increase of the water due to the freezing process. I hope this helps understand the process.