Problem 845 - JEE Physics

Question

A stress of \(3.18 \times 10^{8} \mathrm{Nm}^{2}\) is applied to steel rod of length \(1 \mathrm{~m}\) along its length. Its young's modulus is \(2 \times 10^{11}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\) Then what is the elongation produced in the rod in \(\mathrm{mm}\) ? (A) \(3.18\) (B) \(6.36\) (C) \(5.18\) (D) \(1.59\)

Step-by-Step Solution

Answer

The elongation produced in the rod is \(1.59 \mathrm{mm}\).

Step 1: Write down Hooke's Law formula

Hooke's Law relates stress (σ), Young's modulus (E), and strain (ε) in the following way: \(σ = Eε\)

Step 2: Calculate the strain

Using Hooke's Law formula, we can find the strain (ε) produced in the steel rod: \(\epsilon = \frac{\sigma}{E}\) Plug in the given values for stress and Young's modulus: \( ε = \frac{3.18 \times 10^{8} \mathrm{Nm}^{2}}{2 \times 10^{11}\left(\mathrm{N} / \mathrm{m}^{2}\right)}\)

Step 3: Solve for strain

Calculate the strain (ε) by dividing the stress by the Young's modulus: \(ε = \frac{3.18 \times 10^{8}}{2 \times 10^{11}}\) After the calculation, we get: \(ε = 1.59 \times 10^{-3}\)

Step 4: Calculate elongation

Since strain (ε) is dimensionless and represents the ratio of elongation produced by the initial length, we can use it to find the elongation (ΔL) of the steel rod: \(ΔL = ε \times L\) Plug in the given length of the steel rod and the calculated strain: \(ΔL = (1.59 \times 10^{-3}) \times (1 \mathrm{m})\)

Step 5: Convert elongation to millimeters

Calculate the elongation (ΔL) produced in the rod in meters: \(ΔL = 1.59 \times 10^{-3} \mathrm{m}\) To convert the elongation to millimeters, multiply by 1000: \(ΔL = 1.59 \times 10^{-3} \mathrm{m} \times 1000\) Which gives us the final answer in millimeters: \(ΔL = 1.59 \mathrm{mm}\) Therefore, the elongation produced in the rod is 1.59 mm, and the correct answer is (D).

Key Concepts

Hooke's Law

Hooke's Law is a fundamental principle in physics that explains how materials respond to stress and strain. It provides a simple linear relationship among stress (force per unit area), strain (deformation per unit length), and the material's Young's modulus (a measure of stiffness). Broadly speaking, Hooke's Law can be summarized by the formula:

• \(\sigma = E \epsilon\), where \( \sigma\) is stress, \( E\) is Young's modulus, and \( \epsilon \) is strain.

This law is particularly applicable for elastic materials, which can return to their original shape after the stress is removed. When a material behaves according to Hooke’s Law, it means any strain experienced is proportionate to the stress applied, up to the material's elastic limit. Understanding Hooke’s Law is crucial for predicting how different materials deform under various forces.

Strain

Strain is a measure of how much an object deforms under stress. Essentially, it describes the extent of deformation in the object's shape or size. Strain is calculated by the formula:

• \( \epsilon = \frac{\Delta L}{L} \), where \( \Delta L \) is the change in length and \( L \) is the original length.

Strain is dimensionless because it represents a ratio of two lengths. In our exercise, the strain tells us how much the steel rod deforms when a specific stress is applied. Calculating strain is essential for engineers to understand materials' behavior under different loads to ensure they don't reach their breaking point. This analysis aids in designing structures that are safe and efficient.

Stress

Stress is defined as the internal force exerted by a material per unit area opposing an external force. It quantifies the intensity of the internal forces acting within a deformed body. Stress can be calculated using the formula:

• \( \sigma = \frac{F}{A} \), where \( F \) is the force applied and \( A \) is the cross-sectional area.

In simple terms, when you apply a force to an object, that force creates stress inside the material. In the given exercise, a stress of \(3.18 \times 10^{8} \mathrm{Nm}^{2}\) acts over the steel rod, affecting its structural integrity. Understanding stress is crucial for determining if materials can withstand proposed forces without failure, making it a key concept in material science and engineering.

Elongation

Elongation is the change in length that occurs as a material is stretched under stress. It represents how much longer a material becomes when a pulling force is applied. The relationship between elongation and strain is straightforward:

• \( \Delta L = \epsilon \times L \), where \( \Delta L \) is the elongation, \( \epsilon \) is the strain, and \( L \) is the original length.

This calculation gives us the amount, typically in meters or millimeters, by which the rod expands due to the applied stress. In our exercise, the steel rod's elongation is determined to be 1.59 mm. Elongation is crucial for assessing materials used in construction and machinery. Knowing how much a material expands helps in creating precise designs and ensures components fit together optimally.