Problem 902 - JEE Physics

Question

A steel wire of \(1 \mathrm{~m}\) long and \(1 \mathrm{~mm}^{2}\) cross sectional area \(1 \mathrm{~s}\) hung from rigid end when weight of \(1 \mathrm{~kg}\) is hung from it then change in length will be........... [ \(\left.\mathrm{Y}=2 \times 10^{11}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\right]\) (A) \(0.5 \mathrm{~mm}\) (B) \(0.25 \mathrm{~mm}\) (C) \(0.05 \mathrm{~mm}\) (D) \(5 \mathrm{~mm}\)

Step-by-Step Solution

Answer

The change in length of the steel wire is approximately \(0.05\,\text{mm}\).

Step 1: List the given information

We are given the following information - Original length of wire \(L = 1\,\text{m}\) - Cross-sectional area of wire \(A = 1\,\text{mm}^2 = 1 \times 10^{-6}\,\text{m}^2\) - Weight of \(1\,\text{kg}\) (force applied) \(F = 1\,\text{kg} \times 9.81\,\text{m/s}^2 = 9.81\,\text{N}\) - Young's modulus \(Y = 2 \times 10^{11}\,\text{N/m}^2\) Now, we will plug the given information into the formula to calculate the change in length.

Step 2: Calculate the change in length using the formula

We will use the formula \(\Delta L = \frac{FL}{AY}\) to calculate the change in length. \[\Delta L = \frac{9.81\,\text{N} \times 1\,\text{m}}{1 \times 10^{-6}\,\text{m}^2 \times 2\times 10^{11}\,\text{N/m}^2}\]

Step 3: Simplify and find the change in length

Now, simplify the equation: \[\Delta L = \frac{9.81}{2 \times 10^5} \approx 4.905 \times 10^{-5}\,\text{m}\] To convert the change in length into millimeters, multiply by 1000: \[\Delta L \approx 4.905 \times 10^{-5}\,\text{m} \times 1000 = 0.04905\,\text{mm}\] Looking at the given options, the closest answer is (C) \(0.05\,\text{mm}\)

Key Concepts

Elasticity

Elasticity is a fundamental property of materials that describes how they deform and return to their original shape when forces are applied and then removed. It's like when you stretch a rubber band and then let go—it snaps back to its original form. Materials like steel, used in the problem, are considered elastic as they can withstand some deformation without permanent changes.

When a material is deformed, it experiences both stress and strain. This allows us to measure its elastic properties. Not all materials have the same elasticity. This is why rubber is more flexible, and steel is more rigid. Elasticity is essential in designing structures and tools, ensuring they can handle daily stress without breaking or permanently deforming.

Young's Modulus is the measure of a material's elasticity, defining the relationship between stress (force per unit area) and strain (proportional deformation). It essentially tells us how "stiff" or "flexible" a material is. In the given exercise, Young's Modulus is crucial for calculating the change in length of the steel wire when weight is applied.

Stress and Strain

Stress and strain are key concepts in understanding how materials deform under force. **Stress** is the force applied per unit area, and it's what causes the material to change shape. In our steel wire example, stress is calculated using the weight of the object (gravity acting on mass) divided by the cross-sectional area of the wire.

On the other hand, **strain** is the deformation or change in dimension experienced by the material relative to its initial length. It is a ratio and hence, dimensionless. Strain measures how much a material stretches or compresses compared to its original size.

• **Stress**: \( ext{Stress} = rac{ ext{Force}}{ ext{Area}}\)
• **Strain**: \( ext{Strain} = rac{ ext{Change in length}}{ ext{Original length}}\)

These are closely connected, as stress applied to a material will cause strain. Young's Modulus connects these two by showing how a particular material will react to a certain amount of stress.

Hooke's Law

Hooke's Law is a principle of physics that relates the force needed to extend or compress a spring to the distance it is stretched or compressed. Though initially associated with springs, it applies broadly to elastic materials. This law states that the strain in a solid is proportional to the applied stress, up to the elastic limit of that solid.

In mathematical terms, Hooke's Law is expressed as:
\[ F = k imes ext{strain} \]
where \( k \) is a constant value unique to each material. The relatie between Hooke's Law and Young's Modulus is that while Hooke's Law deals with forces and displacements directly, Young's Modulus considers stress and strain, a broader application conceptually.

In our problem, Hooke's Law helps explain why the steel wire returns to its original length after the weight is removed. It provides the foundational principles for calculating the deformation seen in materials under stress, making it a key principle in understanding mechanical properties in physics.