Question

You fill a tall glass with ice and then add water to the level of the glass's rim, so some fraction of the ice floats above the rim. When the ice melts, what happens to the water level? (Neglect evaporation, and assume that the ice and water remain at $0^{\circ }\mathrm{C}$ during the melting process.) a) The water overflows the rim. b) The water level drops below the rim. c) The water level stays at the top of the rim. d) It depends on the difference in density between water and ice.

Short answer

Short Answer: When the ice melts, the water level inside the glass will stay at the top of the rim. This is because the volume of the melted ice is less than the original volume of the ice, but it still displaces the same volume of water. Therefore, the total volume of water inside the glass after the ice has melted remains the same as the original volume.

Step-by-step solution

Understanding buoyancy and the density of water and ice

Buoyancy is the upward force exerted by a fluid on an object that is immersed in it. For an object to float, the buoyant force must be equal to the weight of the object. The buoyant force is given by Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid displaced by the object. Since ice floats on water, we know that its density is less than that of water. If we take the density of ice as ${\rho }_{\text{ice}}$ and the density of water as ${\rho }_{\text{water}}$, we can see that ${\rho }_{\text{ice}}<{\rho }_{\text{water}}$.

Analyzing the behavior of water level in the glass

When the ice melts, it turns into water, so its overall mass remains constant. We need to analyze the effect of the ice melting on the water level inside the glass. Let's say the volume of the water in the glass initially is $V_{\text{water}}$, and the volume of the ice in the glass is $V_{\text{ice}}$. The total volume of the ice and water inside the glass initially is $V_{\text{total}}=V_{\text{water}}+V_{\text{ice}}$. When the ice melts, it becomes water, and its mass is conserved. Let's say the volume of the melted ice is $V_{\text{melted\_ice}}$. We can find the relationship between the volumes of melted ice and the initial ice using the densities of ice and water. Since mass is conserved, we have: ${\rho }_{\text{ice}}{V}_{\text{ice}}={\rho }_{\text{water}}{V}_{\text{melted\_ice}}$ Solving for $V_{\text{melted\_ice}}$, we get: $V_{\text{melted\_ice}}=\frac{{\rho }_{\text{ice}}}{{\rho }_{\text{water}}}{V}_{\text{ice}}<V_{\text{ice}}$

Determining what happens to the water level

As the ice melts and becomes water, the volume of the ice floating above the rim reduces. Since the ice was floating, it originally displaced a volume of water equal to its weight. When the ice melts, it will still displace the same volume of water, but the volume of the melted ice itself is less than the original volume of the ice ($V_{\text{melted\_ice}}<V_{\text{ice}}$). As a result, the total volume of water inside the glass after the ice has melted will be the same as the original volume before the ice melted. So, the water level inside the glass will not overflow, but instead will stay at the top of the rim. The answer is (c) The water level stays at the top of the rim.

Key concepts

Archimedes' principle

Archimedes' principle is a fundamental concept in fluid mechanics that explains why objects float or sink when placed in a fluid. According to this principle, any object fully or partially submerged in a fluid experiences a buoyant force. This force is equal to the weight of the fluid that the object displaces. In simple terms, if an object is lighter than the water it pushes away, it will float. If it's heavier, it will sink.

For example, when ice is placed in water, it floats because the buoyant force, which comes from the weight of the displaced water, is equal to the gravitational force pulling the ice downward. Archimedes' principle helps us to understand how different objects behave in water by considering their buoyancy and their ability to displace fluid.

Density of ice vs water

The concept of density plays a vital role in understanding why ice behaves differently from water. Density is defined as mass per unit volume. Ice has a density of about 0.92 g/cm³, which is less than the density of water at 1.0 g/cm³. This difference is crucial for understanding why ice floats.

When ice is in water, it displaces a volume of water equal to its weight, not its volume. Because ice is less dense, it means that a specific mass of ice occupies a larger volume compared to the same mass of water. This is why part of the ice cube floats above the water while the larger portion remains submerged.

This difference in density also implies that when ice melts, it will turn into a smaller volume of water. This aspect is key to predicting water level changes as ice transitions from solid to liquid.

Conservation of mass

The concept of conservation of mass is fundamental in physics. It states that mass in an isolated system remains constant, irrespective of the transformations or changes that the matter undergoes. This principle applies to the transition of ice to water.

When ice melts in a glass of water, the total mass neither increases nor decreases. What happens is that the ice, despite changing from solid to liquid, remains the same in terms of mass. The crucial observation is that the volume of the water formed from the melted ice will be less than the volume of the original ice cube, due to the density difference.

This conservation ensures that the properties of the system, like the mass of ice and water, are balanced. Therefore, the mass and the total volume displaced initially by the ice does not change when it melts.

Water level behavior

Water level behavior can be quite intriguing when dealing with melting ice. In the initial state, floating ice displaces water equal to its own weight. As the ice melts, it becomes a liquid water with the same mass but less volume than when it was solid.

This difference means that the melted ice will essentially just fill the space within the glass that was originally occupied by the displaced water. Thus, no additional water is added to cause overflow, nor is any volume lost to lower the level. The water level remains constant as the weight of the displaced water and the mass of the ice remain equal.

So, when all is said and done, the melting point scenario demonstrates that the water inside the glass remains level, sticking exactly to the rim of the glass, as initially filled. This intriguing balance highlights the importance of understanding how ice and water interact to keep the system constant.