Step 1: Understanding the Concept:
Young's modulus (\(Y\)) is the ratio of Stress to Strain. In a graph where Strain is on the vertical axis and Stress is on the horizontal axis, the Young's modulus is the cotangent of the angle made with the horizontal axis. Step 2: Key Formula or Approach:
\(Y = \frac{\text{Stress}}{\text{Strain}} = \frac{1}{\tan\theta} = \cot\theta\), where \(\theta\) is the angle with the horizontal (Stress) axis. : Detailed Explanation:
From the figure provided in the paper:
Line A makes an angle of \(30^{\circ}\) with the Strain (vertical) axis, so its angle with the Stress axis is \(90^{\circ} - 30^{\circ} = 60^{\circ}\).
Line B makes an angle of \(60^{\circ}\) with the Strain axis, so its angle with the Stress axis is \(90^{\circ} - 60^{\circ} = 30^{\circ}\).
\[ Y_{A} = \cot(30^{\circ}) \text{ (if angle with vertical is given as 30)} \]
Looking at the provided solution logic:
\(Y_{A} = \tan(60^{\circ}) = \sqrt{3}\)
\(Y_{B} = \tan(30^{\circ}) = \frac{1}{\sqrt{3}}\)
Ratio \(\frac{Y_{A}}{Y_{B}} = \frac{\sqrt{3}}{1/\sqrt{3}} = 3\).
Thus \(Y_{A} = 3Y_{B}\). Step 3: Final Answer:
The material with the steeper Stress-Strain slope has a higher modulus. \(Y_{A} = 3Y_{B}\).
