A glass having a diameter of 50 mm is filled with water to the level shown. If an ice cube with

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A glass having a diameter of 50 mm is filled with water to...

A glass having a diameter of $50 \mathrm{~mm}$ is filled with water to the level shown. If an ice cube with $25-\mathrm{mm}$ sides is placed into the glass, determine the new height $h$ of the water surface. Take $\rho_{w}=1000 \mathrm{~kg} / \mathrm{m}^{3}$ and $\rho_{i c e}=920 \mathrm{~kg} / \mathrm{m}^{3} .$ What will the water level $h$ be when the ice cube completely melts?

A glass having a diameter of $50 \mathrm{~mm}$ is filled with water to the level shown. If an ice cube with $25-\mathrm{mm}$ sides is placed into the glass, determine the new height $h$ of the water surface. Take $\rho_{w}=1000 \mathrm{~kg} / \mathrm{m}^{3}$ and $\rho_{i c e}=920 \mathrm{~kg} / \mathrm{m}^{3} .$ What
will the water level $h$ be when the ice cube completely melts?

Key Concepts

Volume Calculation Using Geometry

Problems like these require calculating the volumes of various shapes involved, such as cubes and cylinders, using their geometric formulas. Determining the water level after displacement or melting involves computing the volume change and then relating that volume to the cross-sectional area of the container.

Displacement of Floating Objects and Volume Change

When an object floats, the volume of fluid it displaces is exactly the amount needed to counterbalance its weight. In the context of melting, the floating ice cube initially displaces a larger volume of water than the volume it occupies when melted due to differences in density. Understanding this relationship is crucial to determine the water level before and after melting.

Archimedes' Principle

This principle states that a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces. It is the foundational concept for understanding why objects float or sink in a fluid. In problems involving floating ice, it explains how the displaced volume of water is directly related to the weight of the ice cube, regardless of its volume.

Density and Phase Change

Density serves as a measure of mass per unit volume and is key to determining buoyancy. Since ice is less dense than water, it floats and its melting results in a volume change; the melted water occupies less volume than the ice did as a floating object. This concept is essential when calculating the resultant water level after the ice cube has melted.

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