Question

The stress-strain graph of two wires A and B is shown in the figure. If \(Y_A\) and \(Y_B\) are Young’s moduli of materials of wires A and B respectively, then

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

Hint:

The steeper the slope in the stress-strain graph, the greater the Young’s modulus. Compare angles using tangent values.

Answer 1

The Correct Option is C

Step 1: Young’s modulus is defined as the slope of the stress-strain graph: \[ Y = \dfrac{\text{Stress}}{\text{Strain}} = \tan(\theta) \] Step 2: From the graph: - Wire A makes an angle of \(30^\circ\) with the strain axis, so \(Y_A = \tan(30^\circ)\) - Wire B makes an angle of \(60^\circ\) with the strain axis, so \(Y_B = \tan(60^\circ)\) Step 3: Using trigonometric values: \[ \tan(30^\circ) = \dfrac{1}{\sqrt{3}},
\tan(60^\circ) = \sqrt{3} \] \[ \Rightarrow \dfrac{Y_B}{Y_A} = \dfrac{\sqrt{3}}{1/\sqrt{3}} = 3 \Rightarrow Y_B = 3Y_A \] % Final Answer \[ \boxed{Y_B = 3Y_A} \]