Problem 940 - JEE Physics

Question

For a given material the Young's modulus is \(2.4\) times that of rigidity modulus. What is its poisson's ratio? (A) \(2.4\) (B) \(1.2\) (C) \(0.4\) (D) \(0.2\)

Step-by-Step Solution

Answer

The Poisson's ratio for the given material is \(0.2\), which corresponds to the answer (D).

Step 1: Write down the given values.

We are given that Young's modulus (E) is 2.4 times the rigidity modulus (G). So, we can write this relationship as: \(E = 2.4G\)

Step 2: Substitute the given relationship into the formula.

Now we substitute the relationship between E and G into the formula relating Young's modulus, rigidity modulus, and Poisson's ratio: \(2.4G = 2G (1 + ν)\)

Step 3: Solve for Poisson's ratio.

To find the Poisson's ratio (ν), we need to solve the equation obtained in step 2. Divide both sides of the equation by 2G: \(1.2 = 1 + ν\) Now, subtract 1 from both sides of the equation: \(ν = 1.2 - 1\) So, \(ν = 0.2\) The Poisson's ratio for the given material is 0.2, which corresponds to the answer (D).

Key Concepts

Young's Modulus

Young's modulus, often denoted as \(E\), is a fundamental property in material science. It measures the stiffness of a given material. Specifically, Young's modulus is defined as the ratio of tensile stress to tensile strain. In simple terms, it is a measure of how much a material deforms under tension or compression.

The value of Young's modulus is significant because it indicates the ability of a material to resist deformation. Higher values of Young's modulus mean that the material is stiffer and less prone to deformation under applied forces. It's measured in units of Pascals (Pa) in the International System of Units (SI).

• For example, steel has a high Young's modulus, which explains why it is widely used in construction.
• Rubber, on the other hand, has a low Young's modulus, which allows it to stretch easily.

Young's modulus is crucial in engineering and construction applications, especially when selecting materials that must withstand specific loads without bending or breaking.

Rigidity Modulus

The rigidity modulus, also known as the shear modulus, indicated by \(G\), is another important material property. It measures the material's ability to resist shear deformation. That means it assesses how well a material can withstand forces that try to cause the layers of the material to slide past each other.

Just like Young's modulus, the rigidity modulus is measured in Pascals (Pa). A higher rigidity modulus suggests that a material is capable of resisting shear forces without significant deformation.

• In practical terms, it's critical in applications where torsional stress is a concern, such as in shafts and beams.
• Materials like metals and alloys usually have a high rigidity modulus, making them suitable for mechanical components.

Rigidity modulus is particularly vital in scenarios where structures are subjected to twisting or torsional loads. It helps engineers design components that are strong and stable under such conditions.

Material Science

Material science is the study of the properties and applications of materials of construction or manufacture. It includes investigating how the structure of materials affects their properties and performance. This field plays a crucial role in almost every industry, from aerospace and automotive to biomedical and electronics.

Material scientists focus on connecting the microstructural characteristics of materials to their macroscopic behavior. They employ principles from chemistry, physics, and engineering to innovate and improve the materials we use in daily life.

• One of the primary goals in material science is to develop new materials with superior characteristics for specific applications.
• Understanding material properties like Young's modulus and rigidity modulus is vital for selecting suitable materials for engineering projects.

Moreover, advances in material science contribute to sustainability by discovering new materials that are environmentally friendly, durable, and efficient.