Problem 946 - JEE Physics

Question

What is the relationship between Young's modulus Y, Bulk modulus \(\mathrm{k}\) and modulus of rigidity \(\eta\) ? (A) \(\mathrm{Y}=[9 \eta \mathrm{k} /(\eta+3 \mathrm{k})]\) (B) \(\mathrm{Y}=[9 \mathrm{Yk} /(\mathrm{y}+3 \mathrm{k})]\) (C) \(\mathrm{Y}=[9 \eta \mathrm{k} /(3+\mathrm{k})]\) (D) \(\mathrm{Y}=[3 \eta \mathrm{k} /(9 \eta+\mathrm{k})]\)

Step-by-Step Solution

Answer

The correct relationship between Young's modulus (Y), Bulk modulus (k), and modulus of rigidity (η) is: \[Y = \frac{9ηk}{η+3k}\] Hence, the correct answer is (A).

Step 1: Understand the relation between the moduli

The relationship between Young's modulus (Y), Bulk modulus (k) and modulus of rigidity (η) can be expressed as follows: \[Y = \frac{9ηk}{3k + η}\] Step 2 - Identify the correct formula

Step 2: Identify the correct formula

Now, we will check each option, one by one and compare it to the established relationship: (A) Y = [9ηk /(η+3k)] Comparing to the correct formula, this option matches the established relationship. (B) Y=[9Yk /(y+3k)] This option contains incorrect terms and doesn't match the correct relationship. (C) Y=[9ηk /(3+k)] This option does not match the established relationship due to wrong placement of the terms in the denominator. (D) Y=[3ηk /(9η+k)] This option also has wrong coefficients and doesn't match the correct relation. So, the correct answer is (A). Y = [9ηk /(η+3k)]

Key Concepts

Young's Modulus

Young's Modulus is a fundamental concept in materials science and physics. It measures a material's ability to withstand changes in length under lengthwise tension or compression. Essentially, it describes how stiff or elastic a material is. The higher the Young's Modulus, the stiffer the material.To determine Young's Modulus mathematically, you use the formula:\[ Y = \frac{\text{Stress}}{\text{Strain}} \]Where stress is the force applied per unit area and strain is the deformation experienced by the material. Typically, it's measured in Pascals (Pa).

• Stress is the force acting per unit area.
• Strain is the relative change in shape or size.

Young's Modulus plays a critical role in engineering and construction, as it helps in selecting the right materials that ensure structures remain strong yet flexible enough to handle stressors.

Bulk Modulus

Bulk Modulus is another key measurement of a material's elasticity. Unlike Young's Modulus which focuses on stretching and compression along one dimension, Bulk Modulus considers the material's response to changes in pressure in all directions - essentially volumetric compression.The Bulk Modulus (\( k \)) is defined as:\[ k = -V \cdot \frac{\Delta P}{\Delta V} \]Where:

• \( V \) is the initial volume of the object.
• \( \Delta P \) is the change in pressure.
• \( \Delta V \) is the change in volume.

A higher Bulk Modulus indicates that a material is incompressible or less susceptible to pressure changes. In applications, this concept is vital for materials used in high-pressure environments, such as the hull of a submarine.

Modulus of Rigidity

The Modulus of Rigidity, also known as the Shear Modulus (\( \eta \)), is an essential property describing a material's response to shear stress (think twisting or shearing).The formula to find the Modulus of Rigidity is:\[ \eta = \frac{\text{Shear Stress}}{\text{Shear Strain}} \]Where:

• Shear Stress is the force causing the layers of the material to slide past each other.
• Shear Strain is the deformation perpendicularly to the force's direction.

A material with a high Modulus of Rigidity will not easily change shape when a shear force is applied.
This property is critical when dealing with materials subjected to twisting forces, as in shafts or beams bearing torque.