Review Problem. A solid copper ball with a diameter of 3.00...

Review Problem. A solid copper ball with a diameter of $3.00 \mathrm{~m}$ at sea level is placed at the bottom of the ocean (at a depth of $10.0 \mathrm{~km})$. If the density of seawater is $1030 \mathrm{~kg} / \mathrm{m}^{3}$, by how much (approximately) does the diameter of the ball decrease when it reaches bottom? Take the bulk modulus of copper as $14.0 \times 10^{10} \mathrm{~N} / \mathrm{m}^{2}$.

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Key Concepts

Bulk Modulus

Bulk modulus is a material property that quantifies how resistant a substance is to uniform compression when subjected to pressure. It is defined as the ratio of the applied pressure change to the resulting fractional change in volume, and a high bulk modulus indicates that the material is not easily compressible.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It increases linearly with depth in the fluid, and this concept is crucial when considering the forces experienced by objects submerged at great depths, such as in the ocean.

Volumetric Strain

Volumetric strain refers to the change in volume of an object relative to its initial volume when it is subjected to changes in pressure. It connects to the bulk modulus through the relation ?V/V = -?P/K, where ?P is the pressure change and K is the bulk modulus, indicating the degree of compression experienced by the material.

Linear Strain in Spherical Bodies

In a sphere undergoing uniform compression, the volumetric strain is related to the linear strain (i.e., changes in dimensions like radius or diameter) such that the fractional decrease in diameter is approximately one-third of the volumetric strain. This allows the calculation of dimensional changes from the overall volume change.

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