Animated Solution for Physics - Properties of Solids and Liquids: 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.

Have you ever wondered how a material 'knows' how to respond when you stretch it, squeeze it, or twist it? In the world of solid mechanics, a material's personality is defined by its elastic constants. Today, we are going to explore a beautiful mathematical relationship that ties together the three fundamental pillars of elasticity: Young's Modulus ($Y$), Bulk Modulus ($K$), and the Modulus of Rigidity ($\eta$).

Before we dive into the algebra, let's build a physical intuition for what these constants actually represent. Imagine you have a block of steel.

Young's Modulus ($Y$) is the measure of stiffness when you pull or push the block in one direction. It tells you how much the material resists linear stretching or compression. If you hang a heavy weight from a steel wire, Young's Modulus dictates how much that wire will elongate.

Bulk Modulus ($K$) is all about volume. Imagine taking that same steel block and submerging it deep into the ocean. The water pressure squeezes the block from all sides simultaneously. The Bulk Modulus measures the material's resistance to this uniform volumetric compression. A high Bulk Modulus means the material is nearly incompressible.

Modulus of Rigidity ($\eta$), also known as the shear modulus, governs shape changes. If you glue the bottom of the block to a table and push the top surface sideways, you are applying a shear force. The block deforms into a parallelepiped. The Modulus of Rigidity tells us how hard it is to cause this sliding, twisting deformation.

Poisson's Ratio

At first glance, stretching, squeezing, and twisting might seem like completely independent behaviors. However, they are intimately connected by a silent partner: Poisson's Ratio ($\sigma$).

When you stretch a rubber band, it doesn't just get longer; it also gets thinner. Poisson's Ratio is the ratio of this lateral contraction to the longitudinal extension. It is the fundamental geometric property that links linear deformation to volumetric and shear deformations.

Because of Poisson's ratio, we have two foundational equations in the theory of elasticity. The first connects linear and volumetric responses:

The second connects linear and shear responses:

Our mission in this problem is to find a direct relationship between $Y$, $K$, and $\eta$. To do this, we must eliminate the 'middleman', Poisson's ratio ($\sigma$).

Let's start with our first equation and isolate $\sigma$. We have:

Dividing both sides by $3K$, we get:

Rearranging the terms to solve for $\sigma$, we find:

This is our first expression for Poisson's ratio. Now, let's turn to our second foundational equation:

Again, we want to isolate $\sigma$. Dividing by $2\eta$:

Subtracting 1 from both sides gives us our second expression:

Now for the magic. Since both expressions represent the exact same physical property ($\sigma$) of the material, they must be equal to each other. Let's set them equal:

We have successfully eliminated $\sigma$! The rest is pure algebraic manipulation. Let's expand the left side:

To make things cleaner, let's group the constant terms on the left and the terms containing $Y$ on the right. Adding 1 to both sides and adding $\frac{Y}{6K}$ to both sides:

Let's multiply the entire equation by 2 to clear the denominators slightly:

Now, let's divide the entire equation by $Y$:

This is actually a very famous form of the relationship, but let's keep going to match the specific options given in the question. The options are solved for either $Y$, $K$, or $\eta$. Let's try solving for $K$.

From our previous step:

To subtract these fractions, we need a common denominator, which is $Y\eta$:

Now, we take the reciprocal of both sides:

Finally, dividing by 3, we arrive at our destination:

Looking at our options, this perfectly matches option (c)!

While the form we just derived is correct, it is not the easiest to memorize. In physics, we love symmetry. If we go back a few steps to this equation:

And multiply the entire equation by 3, we get the Standard Form of the elastic constant relationship:

This equation is a masterpiece. It cleanly and symmetrically relates the three moduli. As a JEE aspirant, memorizing this specific form is highly recommended. If a question asks for a relation, you can quickly write down this standard form and algebraically rearrange it to match the given options, saving you the time of deriving it from scratch using Poisson's ratio.

Understanding how these constants interlock not only helps you solve numerical problems but also gives you a profound appreciation for the mathematical harmony underlying the physical world.