Problem 884 - JEE Physics

Question

In steel the young's modulus and the strain at the breaking point are \(2 \times 10^{11}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\) and \(0.15\) respectively the stress at the breaking point for steel is therefore........... (A) \(1.33 \times 10^{11}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\) (B) \(1.33 \times 10^{12}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\) (C) \(7.5 \times 10^{-13}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\) (D) \(3 \times 10^{10}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\)

Step-by-Step Solution

Answer

The stress at the breaking point for steel is \(3 \times 10^{10}\mathrm{~N/m^2}\).

Step 1: Identify the given values and formula

We are given: - Young's Modulus (E) = \(2 \times 10^{11} \mathrm{~N/m^2}\) - Strain at the breaking point (ε) = 0.15 We will use the formula: σ = E ε

Step 2: Calculate the stress at the breaking point

Plug the given values into the formula: σ = \((2 \times 10^{11})(0.15)\)

Step 3: Solve for the stress at the breaking point

Multiplying the given values, we get: σ = \(3 \times 10^{10}\mathrm{~N/m^2}\)

Step 4: Match the calculated stress with the given options

We look for the option that matches our calculated stress: (A) \(1.33 \times 10^{11}\mathrm{~N/m^2}\) (Incorrect) (B) \(1.33 \times 10^{12}\mathrm{~N/m^2}\) (Incorrect) (C) \(7.5 \times 10^{-13}\mathrm{~N/m^2}\) (Incorrect) **(D) \(3 \times 10^{10}\mathrm{~N/m^2}\) (Correct)** Thus, the stress at the breaking point for steel is \(3 \times 10^{10}\mathrm{~N/m^2}\), and the correct answer is (D).

Key Concepts

Stress-Strain Relationship

The stress-strain relationship is a fundamental concept in the field of material science. It describes how a material deforms, specifically how it responds to stress. Stress is the internal force exerted by a material per unit area, while strain is the measure of deformation or elongation that occurs in response to the applied stress.

When a material is subjected to stress, it responds by stretching or compressing. For most materials, there is a linear relationship between stress and strain at the initial stage. This is typically represented by the equation: \[ \sigma = E \varepsilon \] where \( \sigma \) is the stress, \( E \) is Young’s modulus, and \( \varepsilon \) is strain.

This relationship holds true only up to a certain limit known as the proportional limit. Beyond this, the material may enter the plastic deformation stage, where the changes become permanent. However, within the linear elastic region, the material will return to its original form upon the removal of stress. Understanding this relationship is crucial in predicting how materials will behave under different conditions.

Breaking Point

The breaking point of a material is the stress level at which it becomes incapable of withstanding further stress and fails or fractures. It's a critical value because it informs engineers and scientists about the maximum load a material can withstand before it fractures.

At the breaking point, the structure of the material gives way, causing a complete separation into two or more parts. This is particularly important in engineering and construction, where safety and reliability must be ensured.

For example, in the problem involving steel, the breaking point corresponded to a stress of \(3 \times 10^{10}\, \mathrm{N/m^2}\). This information can help in determining the suitable applications for steel, ensuring that it is used in a manner that doesn't exceed its breaking capacity. Recognizing this point helps in preventing catastrophic failures by allowing one to design structures safely within material capacities.

Material Science

Material science is a field that explores the properties of materials and how they can be manipulated for various uses. It is an interdisciplinary field that combines elements of physics, chemistry, and engineering to understand and improve the materials that make up our world.

A significant aspect of material science is understanding how materials react under various conditions. This includes studying properties like Young's modulus, tensile strength, and ductility. Young's modulus, for example, provides a measure of the stiffness of a material, indicating how much it will stretch under stress. Steel, with its high Young's modulus, indicates that it is very rigid and can withstand significant stress before deforming.

The study of breaking points also falls within material science, as it seeks to identify the maximum stress materials can handle before failure. This knowledge informs the development of new materials and contributes to improving their performance in applications ranging from construction to electronics.

Calculating Stress

Calculating stress in materials involves determining the force applied over the area on which the force acts. Stress is defined mathematically as: \[ \sigma = \frac{F}{A} \] where \( \sigma \) is the stress, \( F \) is the force applied, and \( A \) is the cross-sectional area the force acts upon.

In the example of steel, we used Young’s modulus and strain to calculate stress using the formula \( \sigma = E \varepsilon \). This formula equates the stress to the product of Young’s modulus \( E \) and the strain \( \varepsilon \). It's particularly useful when you know the material properties and the deformation that occurs.

Understanding how to calculate stress ensures that structures can be safely designed with appropriate material thicknesses, preventing failure under load. It is a crucial procedure in fields like engineering and architecture to ensure the safety and reliability of structures and machinery.