question_answer

A uniform wire (Young’s modulus \[2\times {{10}^{11}}{{\operatorname{Nm}}^{-2}}\]) is subjected to longitudinal tensile stress of \[5\times {{10}^{7}}{{\operatorname{Nm}}^{-2}}\]. If the overall volume change in the wire is 0.02%, the fractional decrease in the radius of the wire is close to: [JEE ONLINE 22-04-2013]

A)        \[1.0\times {{10}^{-4}}\]        

B)       \[1.5\times {{10}^{-4}}\]

C) \[0.25\times {{10}^{-4}}\]                    

D)  \[5\times {{10}^{-4}}\] Correct Answer: C Solution : [c] Given, \[y=2\times {{10}^{11}}\,N{{m}^{-2}}\]

Stress \[\left( \frac{F}{A} \right)=5\times {{10}^{7}}N{{m}^{-2}}\]

\[\Delta V=0.02%=2\times {{10}^{-4}}{{m}^{3}}\]

\[\frac{\Delta r}{r}=?\]

\[\gamma =\frac{stress}{strain}\Rightarrow strain\left( \frac{\Delta \ell }{{{\ell }_{0}}} \right)=\frac{\gamma }{stress}\]            (i)

\[\Delta V=2\pi {{\ell }_{0}}\Delta r-\pi {{r}^{2}}\Delta \ell \]                              (ii)

From eqns (i) and (ii) putting the value of

\[\Delta \ell ,{{\ell }_{0}}\]and \[\Delta V\]and solving we get

\[\frac{\Delta r}{r}=0.25\times {{10}^{-4}}\]