Question:medium
A metal string A is suspended from a rigid support and its free end is attached to a block of mass M. Second block having mass 2M is suspended at the bottom of the first block using a string B. The area of cross sections of strings A and B are same. The ratio of lengths of strings of A to B is 2 and the ratio of their Young's moduli ($Y_A / Y_B$) is 0.5. The ratio of elongations in A to B is _______.
A metal string A is suspended from a rigid support and its free end is attached to a block of mass M. Second block having mass 2M is suspended at the bottom of the first block using a string B. The area of cross sections of strings A and B are same. The ratio of lengths of strings of A to B is 2 and the ratio of their Young's moduli ($Y_A / Y_B$) is 0.5. The ratio of elongations in A to B is _______.
Updated On: Jun 6, 2026
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The Correct Option is D
Solution and Explanation
Step 1: Use elongation formula
For a wire, \[ \Delta L=\frac{TL}{AY} \] Therefore, \[ \frac{\Delta L_A}{\Delta L_B} = \frac{T_A}{T_B}\cdot \frac{L_A}{L_B}\cdot \frac{A_B}{A_A}\cdot \frac{Y_B}{Y_A} \] Step 2: Find tensions
For string $B$: \[ T_B=2Mg \] For string $A$: \[ T_A=(M+2M)g=3Mg \] Thus, \[ \frac{T_A}{T_B}=\frac{3}{2} \] Step 3: Substitute ratios
Given \[ \frac{L_A}{L_B}=2 \] \[ \frac{Y_A}{Y_B}=0.5 \Rightarrow \frac{Y_B}{Y_A}=2 \] Same cross-sectional area: \[ \frac{A_B}{A_A}=1 \] So, \[ \frac{\Delta L_A}{\Delta L_B} = \frac{3}{2}\times 2\times 1\times 2 \] \[ =6 \] Final Answer: \[ \boxed{6} \]
For a wire, \[ \Delta L=\frac{TL}{AY} \] Therefore, \[ \frac{\Delta L_A}{\Delta L_B} = \frac{T_A}{T_B}\cdot \frac{L_A}{L_B}\cdot \frac{A_B}{A_A}\cdot \frac{Y_B}{Y_A} \] Step 2: Find tensions
For string $B$: \[ T_B=2Mg \] For string $A$: \[ T_A=(M+2M)g=3Mg \] Thus, \[ \frac{T_A}{T_B}=\frac{3}{2} \] Step 3: Substitute ratios
Given \[ \frac{L_A}{L_B}=2 \] \[ \frac{Y_A}{Y_B}=0.5 \Rightarrow \frac{Y_B}{Y_A}=2 \] Same cross-sectional area: \[ \frac{A_B}{A_A}=1 \] So, \[ \frac{\Delta L_A}{\Delta L_B} = \frac{3}{2}\times 2\times 1\times 2 \] \[ =6 \] Final Answer: \[ \boxed{6} \]
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