A piece of ice is floating in water in a beaker. When all of the ice has completely melted, what will happen to the level of the water in the beaker?

Introduction / Context:
This question is a classic conceptual problem in physics that tests understanding of Archimedes principle, buoyancy and the behaviour of floating bodies. It asks what happens to the water level when a piece of ice floating in water melts completely. Such questions are very popular in competitive exams because they appear simple but require clear reasoning about forces and volumes rather than guesswork.

Given Data / Assumptions:

• A beaker contains pure water.
• A piece of ice is floating freely in this water.
• The ice melts completely into water.
• We ignore effects like evaporation and assume the system is closed.
• We assume normal conditions, with the density of ice less than the density of liquid water.

Concept / Approach:
Archimedes principle states that a floating body displaces a volume of liquid whose weight is equal to the weight of the body. While the ice is still solid and floating, it displaces a certain volume of water equal in weight to the ice. When the ice melts, it turns into the same mass of water. Because the density of water is higher than that of ice, the melted water occupies exactly the same volume as the water that was displaced by the floating ice. Therefore, the total water level in the beaker does not change when the ice melts.

Step-by-Step Solution:
Step 1: While the ice is floating, it displaces water of weight equal to its own weight (Archimedes principle). Step 2: Let the mass of the ice be m and the density of water be rho. Then the volume of water displaced is m / rho. Step 3: When the ice melts, its mass remains m but it becomes liquid water with density rho, so the volume of the melted water is also m / rho. Step 4: Therefore, the volume of water formed from the melted ice is equal to the volume of water that had been displaced earlier while the ice was floating. Step 5: Since the volume before and after melting is the same, the water level in the beaker remains unchanged.

Verification / Alternative check:
We can imagine marking the initial water level on the beaker. At this stage, the floating ice displaces a volume of water, which contributes to the total level. After the ice melts, no solid remains above the surface, but the melted water exactly fills the volume that had been effectively present as displaced water earlier. Many physics textbooks illustrate this by considering numerical values and confirming that the volume calculations match. This cross-check confirms that no net change in water level occurs.

Why Other Options Are Wrong:
First rise and then fall: There is no physical mechanism for a temporary rise and subsequent fall in the water level in this simple closed system.
Fall: The level does not fall because the melted water continues to occupy the same total volume that was previously displaced.
Rise: The level does not rise because there is no additional mass of water being added; the ice simply changes its state from solid to liquid.

Common Pitfalls:
A common mistake is to think that the ice, being larger in volume, must cause the water level to drop when it melts, or that more water appears and raises the level. These confusions occur when students ignore the displacement idea and focus only on visual size. Always remember that for a floating object, the weight of displaced liquid equals the weight of the object. When the object melts into the same liquid, the melted volume exactly replaces the displaced volume, so the level remains the same.

Final Answer:
When all of the floating ice melts, the water level in the beaker will remain the same.