Question

Two strings with lengths \(l_1\) and \(l_2\), and Young's moduli \(Y_1\) and \(Y_2\) are elongated under two weights as shown. Find the ratio \(\Delta l_1 / \Delta l_2\).
Case (1): String has Young's modulus \(Y\), length \(l\), cross–sectional area \(A\) and a mass \(m\) is attached. 
Case (2): String has Young's modulus \(2Y\), length \(0.5l\), cross–sectional area \(A\) and a mass \(3m\) is attached.

• A. \(3\)
• B. \(\frac{2}{3}\)
• C. \(\frac{4}{3}\)
• D. \(\frac{1}{3}\)

Answer 1

The Correct Option is C

Concept: 
Extension of a wire under load is given by \[ \Delta l = \frac{FL}{AY} \] where \(F\) = applied force, \(L\) = length of wire, \(A\) = cross-sectional area, \(Y\) = Young's modulus. 
Step 1: Find extension of first string. Force applied: \[ F_1 = mg \] Thus, \[ \Delta l_1 = \frac{mg\,l}{AY} \] Step 2: Find extension of second string. Force applied: \[ F_2 = 3mg \] Length \(=0.5l\), Young's modulus \(=2Y\). \[ \Delta l_2 = \frac{3mg(0.5l)}{A(2Y)} \] \[ \Delta l_2 = \frac{3mgl}{4AY} \] Step 3: Find the ratio. \[ \frac{\Delta l_1}{\Delta l_2} = \frac{\frac{mgl}{AY}}{\frac{3mgl}{4AY}} \] \[ \frac{\Delta l_1}{\Delta l_2} = \frac{4}{3} \]