Problem 8 - Master Resource Book in JEE Main Physics

Question

Young modulus of steel is \(1.9 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}\). When expressed in CGS units of dyne \(/ \mathrm{cm}^{2}\), it will be equal to \(\left(1 \mathrm{~N}=10^{5}\right.\) dyne, \(1 \mathrm{~m}^{2}=10^{4} \mathrm{~cm}^{2}\) ) (a) \(1.9 \times 10^{10}\) (b) \(1.9 \times 10^{11}\) (c) \(1.9 \times 10^{12}\) (d) \(1.9 \times 10^{13}\)

Step-by-Step Solution

Answer

The Young's modulus in dyne/cm² is \(1.9 \times 10^{12}\), so the correct answer is (c).

Step 1: Identify Given Values

You are given the Young's modulus of steel as \(1.9 \times 10^{11} \ \mathrm{N} / \mathrm{m}^{2}\), and we need to convert this value into CGS units of dyne/\(\mathrm{cm}^{2}\). The conversion factors are \(1 \ \mathrm{N} = 10^{5} \ \mathrm{dyne}\) and \(1 \ \mathrm{m}^{2} = 10^{4} \ \mathrm{cm}^{2}\).

Step 2: Convert Newtons to Dynes

Convert the force unit from Newtons to dynes using the conversion factor. Multiply the Young's modulus value in Newtons by the conversion factor: \[1.9 \times 10^{11} \ \mathrm{N} = 1.9 \times 10^{11} \times 10^{5} \ \mathrm{dynes}\]This gives:\[1.9 \times 10^{16} \ \mathrm{dynes} / \mathrm{m}^{2}\]

Step 3: Convert Square Meters to Square Centimeters

Convert the area unit from square meters to square centimeters using the conversion factor. Since each square meter is equal to \(10^{4}\) square centimeters, replace \(\mathrm{m}^{2}\) with the equivalent in \(\mathrm{cm}^{2}\):\[1.9 \times 10^{16} \ \mathrm{dynes} / \mathrm{m}^{2} = \frac{1.9 \times 10^{16} \ \mathrm{dynes}}{10^{4} \ \mathrm{cm}^{2}}\]This gives:\[1.9 \times 10^{12} \ \mathrm{dynes} / \mathrm{cm}^{2}\]

Step 4: Match with Given Options

Compare the final result with the options provided:- (a) \(1.9 \times 10^{10}\)- (b) \(1.9 \times 10^{11}\)- (c) \(1.9 \times 10^{12}\)- (d) \(1.9 \times 10^{13}\)The value \(1.9 \times 10^{12}\) matches option (c), which is the correct answer.

Key Concepts

Unit Conversion Explained

Unit conversion is a vital skill, especially when dealing with scientific measurements. It involves changing the units of a given quantity without altering its value. When converting measurements, like units of Young's modulus, you'll need a conversion factor. A conversion factor is a number used to transform a measurement from one unit to another. In the example, we converted \(\text{Newtons} \, (\mathrm{N})\) to \(\text{dynes} \, (\mathrm{dyn/cm^2})\) and square meters to square centimeters.

• The force conversion: Since \(1 \, \mathrm{N} = 10^5 \, \mathrm{dyn}\), increase the number of dynes compared to Newtons by multiplying by \(10^5\).
• The area conversion: Since \(1 \, \mathrm{m^2} = 10^4 \, \mathrm{cm^2}\), decrease units of area by dividing the larger unit (meters) by \(10^4\).

Together these create a new measurement reflecting the same physical quantity but in desired CGS (centimeter-gram-second) units.

Understanding Material Properties

Materials exhibit unique characteristics that govern their behavior in applications. Young's modulus, a predominant property, measures a material's ability to withstand changes in length when under lengthwise tension or compression. It is a significant 'stiffness' metric for solids.

The larger the value of Young's modulus, the less the material will deform under stress. To grasp material properties:

• Young's modulus is a core aspect, reflecting the elastic properties of a material.
• It is calculated by the ratio of stress (force per unit area) over strain (deformation per unit length). \( E = \frac{\text{stress}}{\text{strain}} \)
• Application varies from refining construction materials to optimizing mechanical components.

Recognizing these traits is essential for mechanical engineering and materials science.

Mechanics Concepts in Context

Mechanics is a branch of physics concerned with motion and the forces that cause motion. Young's modulus plays a critical role in mechanics because it informs how structures and materials handle loads. Mechanics relies heavily upon consistent unit usage and understanding core mechanical principles.

• Force and area conversions are essential when transitioning between different measurement systems.
• In mechanics, correctly interpreting and converting Young's modulus ensures practical application of formulas and calculations.
• Understanding these calculations aids in preventing structural failures and ensuring safety in design.

By integrating unit conversions with a solid grasp of mechanics, engineers and scientists can predict and manage how materials will behave under stress effectively.