If $Y, K$ and $\eta$ are the values of Young's modulus, bulk modulus and modulus of rigidity of any material respectively. Choose the correct relation for these parameters.

Question

A 2 $\text{kg}$ mass is attached to a spring with spring constant $ k = 200, \text{N/m} $. If the mass is displaced by $ 0.1, \text{m} $, what is the potential energy stored in the spring?

Answer 1

To determine the correct relation among Young's modulus $Y$, bulk modulus $K$, and the modulus of rigidity $\eta$, let's analyze the formulae connecting these physical quantities. In material science, the relationship between these elastic moduli is crucial for understanding the mechanical properties of materials.

The fundamental equations relating Young's modulus, bulk modulus, and modulus of rigidity are derived from the general formulae of elasticity:

Y = \frac{9 K \eta}{3K + \eta}
K = \frac{Y}{3(1- 2 \nu)}, where $\nu$ is the Poisson's ratio.
\eta = \frac{Y}{2(1 + \nu)}

By manipulating these equations, the correct relation between these quantities can be derived by eliminating the Poisson's ratio:

The correct relation given in the options is:

K = \frac{Y \eta}{9 \eta - 3 Y}

Let's analyze why this is correct:

Young's Modulus ($Y$): It measures the stiffness of a solid material and describes the material's ability to withstand changes in length when under lengthwise tension or compression.
Bulk Modulus ($K$): It describes a material's response to uniform pressure applied in all directions.
Modulus of Rigidity ($\eta$): Also known as shear modulus, it describes how a material deforms under shear stress.

The given formula $K = \frac{Y \eta}{9 \eta - 3 Y}$ correctly relates these three physical constants without involving the Poisson's ratio directly in the fundamental elasticity equations, thus confirming the option as valid and correct.

Hence, the correct answer is:

K = \frac{Y \eta}{9 \eta - 3 Y} N / m ^{2}

Answer 2