Problem 971 - JEE Physics

Question

When \(100 \mathrm{~N}\) tensile force is applied to a rod of \(10^{-6} \mathrm{~m}^{2}\) cross-sectional area, its length increases by \(1 \%\) so young's modulus of material is \(\ldots \ldots \ldots \ldots\) (A) \(10^{12} \mathrm{~Pa}\) (B) \(10^{11} \mathrm{~Pa}\) (C) \(10^{10} \mathrm{~Pa}\) (D) \(10^{2} \mathrm{~Pa}\)

Step-by-Step Solution

Answer

The Young's modulus of the material is \(10^{10} \mathrm{~Pa}\).

Step 1: Find the change in length ΔL

We are given that the length increases by 1% when a tensile force is applied. So, ΔL = L0 × 0.01

Step 2: Rearrange the formula for Young's modulus

Young's modulus (Y) = (Tensile force (F) × Original length(L0)) / (Area (A) × Change in length (ΔL))

Step 3: Substitute the values into the formula

Substitute the values given in the problem: Tensile force (F) = 100 N, Area (A) = 10^{-6} m², and the change in length (ΔL) = L0 × 0.01 into the formula: Y = (100 × L0) / (10^{-6} × L0 × 0.01) You can notice that 'L0' on the numerator and denominator can be canceled out.

Step 4: Simplify the formula

After canceling out L0, we have: Y = (100) / ( 10^{-6} × 0.01) Now, let's simplify: Y = (100) / (10^{-8}) Y = 100 × 10^{8} Y = 10^{10} Pa So, the Young's modulus of the material is \(10^{10} \mathrm{~Pa}\). The correct answer is option (C).

Key Concepts

Understanding Tensile Force

When you apply a tensile force to a rod, you're essentially pulling it along its length. This force tries to stretch the material, changing its shape slightly. In our example, a tensile force of 100 Newtons is applied.
This force is crucial in understanding material behavior as it helps determine how much a material can stretch before it deforms permanently.

• Tensile Force Measurement: It's measured in units of force, like Newtons (N) in the metric system.
• Effect on Materials: Applying tensile force makes materials more elastic until they reach their limit.

Grasping how tensile force impacts materials can help you understand stress and strain, essential for structural engineering and materials science.

Cross-Sectional Area Explained

The cross-sectional area of an object like a rod is the area of the face you see when you cut through it. It's typically measured in square meters in the metric system.
In the given problem, we have a cross-sectional area of \(10^{-6} \text{ m}^{2}\). This small area shows that the rod is very thin.

• Importance in Calculations: The cross-sectional area is key in calculating stress, which is force per unit area.
• Relation to Material Properties: Larger cross-sectional areas can bear more force without deforming.

Understanding this area helps predict how much force a material can withstand before it changes shape significantly.

What is Change in Length?

When you apply a force to an object, like a tensile force on a rod, its length may change. This change in length is the difference between the original length and the new length. In our problem, the rod's length increases by 1%.
This value is crucial for determining Young's modulus, as it reflects how much stretching occurs under force.

• Percentage Change: Given as a percentage, it helps in knowing how flexible or elastic a material is.
• Role in Calculations: Integral for computing strain, which is the amount of deformation experienced by the body in the direction of the applied force.

Understanding change in length helps you evaluate material performance under load, which is vital in design and safety analysis.