Question

Two wires \(A\) and \(B\) made of different materials have lengths \(6.0\) cm and \(5.4\) cm, and areas of cross-sections \(3.0\times10^{-5}\,\text{m}^2\) and \(4.5\times10^{-5}\,\text{m}^2\), respectively. They are stretched by the same magnitude under the same load. If the ratio of Young’s modulus of \(A\) to that of \(B\) is \(x:3\), find the value of \(x\).

• A. \(5\)
• B. \(4\)
• C. \(2\)
• D. \(1\)

Hint:

When load and extension are same, compare Young’s modulus directly using \( Y\propto \frac{L}{A} \).

Answer 1

The Correct Option is B

Concept: Young’s modulus is defined as: \[ Y=\frac{\text{Stress}}{\text{Strain}} =\frac{FL}{A\Delta L} \] For the same load \(F\) and same extension \(\Delta L\), Young’s modulus is proportional to: \[ Y\propto \frac{L}{A} \]
Step 1: Write the ratio of Young’s moduli \[ \frac{Y_A}{Y_B} =\frac{L_A A_B}{L_B A_A} \]
Step 2: Substitute given values \[ L_A=6.0\text{ cm},\quad L_B=5.4\text{ cm} \] \[ A_A=3.0\times10^{-5},\quad A_B=4.5\times10^{-5} \] \[ \frac{Y_A}{Y_B} =\frac{6.0\times4.5}{5.4\times3.0} =\frac{27}{16.2} =\frac{5}{3} \]
Step 3: Compare with given ratio Given: \[ \frac{Y_A}{Y_B}=\frac{x}{3} \] \[ \Rightarrow x=4 \]