A material has density ρand bulk modulus k. The increase in the density of the material when it

step 1 As we know that k is equal to k is equal to minus v d p by dv So I can write the value of dv by

A material has density ρand bulk modulus k. The increase in...

A material has density $\rho$ and bulk modulus $k$. The increase in the density of the material when it is subjected to an external pressure $P$ from all sides is
(a) $\frac{P}{\rho k}$
(b) $\frac{k}{\rho P}$
(c) $\frac{\rho P}{k}$
(d) $\frac{\rho k}{P}$

Key Concepts

Bulk Modulus

The bulk modulus (k) is a fundamental property of materials that quantifies their resistance to uniform compression. It is defined as the ratio of an applied pressure change to the resulting fractional change in volume, expressed as k = -V (?P/?V). A high bulk modulus indicates that a material requires a large pressure change to achieve a given decrease in volume, reflecting its stiffness under compression.

Volumetric Strain

Volumetric strain is the measure of the relative change in volume of a material when subjected to an external pressure. It is given by the change in volume divided by the original volume (?V/V), and it directly relates to internal phenomena such as density change. In cases of uniform compression, a decrease in volume results in a corresponding increase in density.

Density

Density is a physical property defined as mass per unit volume. When a material is compressed by an external pressure, its volume decreases while its mass remains constant, leading to an increase in density. The relationship between density change and volumetric strain is given by ??/? = -?V/V, linking mechanical compression with material density.

Material Compressibility

Material compressibility is the inverse of the bulk modulus and represents how much a material can be compressed under an applied pressure. A more compressible material has a lower bulk modulus and exhibits a greater change in volume—and therefore density—for a given pressure change. This concept is essential for predicting material behavior under various loading conditions.

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