A solid sphere of brass (bulk modulus of 14.0 ×10^10 N /...

A solid sphere of brass (bulk modulus of $14.0 \times 10^{10} \mathrm{~N} / \mathrm{m}^{2}$ ) with a diameter of $3.00 \mathrm{~m}$ is thrown into the ocean. By how much does the diameter of the sphere decrease as it sinks to a depth of $1.00 \mathrm{~km} ?$

Key Concepts

Bulk Modulus

Bulk modulus is a material property that quantifies how resistant a substance is to uniform compression. It is defined as the ratio of an applied pressure change to the resulting relative change in volume. In problems involving compression under pressure, this modulus allows us to relate the external pressure applied to the material with the change in its dimensions or volume.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. As an object sinks deeper into a fluid, the pressure increases linearly with depth, which can cause changes in the volume of submerged objects. This concept is crucial for determining the additional pressure applied to objects in fluids at various depths.

Volumetric Strain

Volumetric strain represents the relative change in volume of a material when subjected to external pressure, calculated as the change in volume divided by the original volume. In the context of compression, knowing the volumetric strain helps in predicting how much the volume of a body will decrease under a given pressure change.

Linear vs. Volumetric Strain Conversion

For isotropic materials, a change in volume due to compression is uniformly distributed in all directions. Therefore, the linear strain (or fractional change in a particular dimension such as the diameter) is approximately one third of the volumetric strain. This conversion is essential when a problem asks for a change in a linear dimension (like diameter) instead of the overall volume change.

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