A single ice cube with mass 9.70 g floats in a glass...

A single ice cube with mass 9.70 g floats in a glass completely full of 420 $\mathrm{cm}^{3}$ of water. You can ignore the water's surface tension and its variation in density with temperature (as long as it remains a liquid). (a) What volume of water does the ice cube displace? (b) When the ice cube has completely melted, has any water overflowed? If so, how much? If not, explain why this is so. (c) Suppose the water in the glass had been very salty water of density 1050 $\mathrm{kg} / \mathrm{m}^{3} .$ What volume of salt water would the $9.70-\mathrm{g}$ ice cube displace? (d) Redo part (b) for the freshwater ice cube in the salty water.

Key Concepts

Phase Change Effects on Volume and Displacement

When a substance changes phase, such as ice melting into water, its density typically changes, leading to a change in volume. This concept is critical for understanding that an ice cube, when melted, may occupy a different volume than when it was frozen, influencing whether any water overflows in a container.

Fluid Displacement

Fluid displacement refers to the phenomenon where an object immersed in a fluid pushes the fluid away to occupy space. The displaced volume is crucial in buoyancy calculations because, for floating objects, it exactly balances the object's weight, regardless of the object's material properties.

Impact of Fluid Density Variations (e.g., Salinity)

Variations in fluid density, such as in the case of salty water compared to freshwater, affect the buoyant force and the volume of fluid displaced by a floating object. A higher fluid density means that a lighter object (or one with a density that is low relative to the fluid) will displace a smaller volume of fluid to balance its weight.

Archimedes’ Principle

This principle states that a body wholly or partially submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the body. It is fundamental in understanding why objects float or sink and is directly applied when determining the volume of displaced water by a floating ice cube.

Density

Density, defined as mass per unit volume, determines how substances compare in terms of weight and volume. In the context of this problem, the densities of ice, water, and salty water dictate how much fluid is displaced by a floating object and influence whether a floating ice cube will cause water to overflow when it melts.

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