Question

The ratio of hydraulic stress to the corresponding strain is known as
A. Compressibility
B. Bulk modulus
C. Young’s modulus
D. Rigidity modulus

Answer 1

For finding the term which is the ratio of hydraulic stress to the corresponding strain, we need to define hydraulic stress and strain. We will consider every option, check the formula, and will determine the correct term.

Complete step by step answer:
Hydraulic stress, as the term says, is the measure of the internal force per unit area acting on the liquids. It is the restoring force acting per unit area when force is applied by fluid on a body. For example: If a rubber ball is thrown into water, force will act on that ball from all the directions due to the pressure of water. As a result of which, the ball appears to be slightly contracted. It results in a restoring force which develops in the ball and its magnitude is equal to the force applied by water but the direction is opposite. This type of stress is called hydraulic stress.
Strain is defined as the response of a system to an applied stress. When a substance or a material is loaded with a force, it produces a stress, which then causes that material to deform. Strain is defined as the amount of deformation of a material in the direction of the applied force divided by the initial length of the material.
Compressibility, also called as the coefficient of compressibility or isothermal compressibility, is a measure of the relative volume change of a fluid or solid as a response to the applied pressure or mean stress change.

$\dfrac{\Delta V}{V}\approx -c\Delta P$, $c$ is the compressibility of the substance

Bulk Modulus of a substance is a constant that describes how resistant to compression that substance is. It is defined as the ratio of the infinitesimal increase in pressure to the resulting relative decrease in the volume of substance.

$B=\dfrac{-P}{\dfrac{\Delta V}{V}}$

Young's modulus of a material is a measure of the ability of that material to withstand changes in length when under lengthwise tension or compression. It is sometimes referred to as the modulus of elasticity. Young's modulus is equal to the longitudinal stress divided by the strain.
$Y=\dfrac{Fl}{A\Delta l}$

Shear modulus or modulus of rigidity is defined as the ratio of shear stress to the shear strain. Shear stress is the force which acts is the area on which the force acts, that is, shear strain.
$G=\dfrac{\tau }{\gamma }$

The ratio of hydraulic stress to the corresponding strain is defined as the Bulk modulus of that substance.
Expression for Bulk modulus:
$B=\dfrac{-P}{\dfrac{\Delta V}{V}}$
Hence, the correct option is B.

Note:
Students should not get confused between the terms Stress and Pressure. Stress is not physically the same as pressure. In pressure, external force per unit area is considered, but in case of stress, it is the internal force per unit area. In case of liquids too the hydraulic stress is defined in the same way, the internal force per unit area acting on the liquid.