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Density is a fundamental concept in understanding buoyancy and weight. It tells us how much mass is packed into a given volume. In our rock specimen scenario, the density of the solid portion is given as \(5.0 \times 10^3\ kg/m^3\). This means every cubic meter of solid in the rock has a mass of 5000 kilograms.

When we deal with buoyancy and objects submerged in fluids, knowing the density is crucial. It allows us to determine potential buoyancy forces and help identify if parts of an object might be hollow or not. For the geologist, the known density assists in calculating the solid volume fraction of the specimen.

Density is measured as mass per unit volume, represented by

• \(\rho = \frac{m}{V}\)

where \(\rho\) is density, \(m\) is mass, and \(V\) is volume.

Volume measures how much space an object occupies. When dealing with buoyancy, both the actual solid volume and the apparent volume of the object come into play. In our case, the apparent volume \(V_a\) includes any hollow space in the rock. The solid volume \(V_s\) is just the volume of the material itself, without voids.

The relationship between these volumes is vital for finding the fraction of the specimen's volume that is actually solid. The geologist uses the weight measurements in air and in water, along with known densities, to quantify these volumes.

• \(V_s\) represents just the solid portion.
• \(V_a\) accounts for the entire space taken by the sample, including any hollow sections.

Understanding these differences in volume is key to solving problems in buoyancy and material composition.

For objects floating or submerged, knowing both apparent and solid volumes helps determine how and how much they are buoyed by the fluid.

Weight displacement is at the core of the buoyancy concept. It is based on Archimedes' principle, which states that when an object is submerged in a fluid, it experiences a buoyant force equal to the weight of the fluid displaced. In our exercise, this principle is used to solve for the fraction of the specimen’s volume that is solid.

By weighing the specimen in air and then in water, the geologist observes a change in weight due to the buoyant force. This change indicates the volume of water displaced by the specimen, which relates to the specimen's volume.

Formulas let us quantify this:

• \( W_a - W_w = \rho_w \cdot g \cdot V_a \)

Where \(W_a\) is the weight in air, \(W_w\) is the weight in water, \(\rho_w\) is the density of water, and \(V_a\) is the apparent volume.

This is crucial in determining how much of that volume corresponds to the solid part.

A geologist is a scientist who studies the solid Earth, including the materials it is composed of and the processes that affect it. In this exercise, the geologist uses principles of physics and geoscience to evaluate the rock specimen. She employs tools like density, volume calculations, and buoyancy to solve for the composition of a rock.

Her approach reflects the typical work that geologists perform:

• Assessing rock formations for their mineral content.
• Pursuing theories of Earth's formation and structural mechanics.
• Using scientific principles to solve practical geological problems.

In our situation, the geologist successfully determines that a portion of the rock is hollow by leveraging her knowledge of these foundational concepts.