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

Expert verified
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

01

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 \(\rho_{\text{ice}} < \rho_{\text{water}}\).
02

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: \(V_{\text{melted\_ice}} = \frac{\rho_{\text{ice}}}{\rho_{\text{water}}}V_{\text{ice}}<V_{\text{ice}}\)
03

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.

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