Question

The stress versus strain graphs for wires of two materials A and B are as shown. If \(Y_A\) and \(Y_B\) are the Young's moduli of the materials, then:

• A. \( Y_B = 2Y_A \)
• B. \( Y_A = Y_B \)
• C. \( Y_B = 3Y_A \)
• D. \( Y_A = 3Y_B \)

Hint:

If graph is strain vs stress, then slope \(=\frac{1}{Y}\). Higher slope $\Rightarrow$ Lower Young’s modulus.

Answer 1

The Correct Option is C

Concept: \[ Y = \frac{\text{Stress}}{\text{Strain}} \] Hence, slope of stress vs strain graph: \[ \text{slope} = \frac{\text{Strain}}{\text{Stress}} = \frac{1}{Y} \] So, slope is inversely proportional to Young’s modulus.

Step 1:From the graph:}
•Line A makes \(60^\circ\)
•Line B makes \(30^\circ\)

Step 2:Slopes:} \[ \text{slope}_A = \tan 60^\circ = \sqrt{3}, \quad \text{slope}_B = \tan 30^\circ = \frac{1}{\sqrt{3}} \]

Step 3:Ratio of slopes:} \[ \frac{\text{slope}_A}{\text{slope}_B} = \frac{\sqrt{3}}{1/\sqrt{3}} = 3 \]

Step 4:Using inverse relation:} \[ \frac{1/Y_A}{1/Y_B} = 3 \Rightarrow \frac{Y_B}{Y_A} = 3 \] \[ \Rightarrow Y_B = 3Y_A \]