question_answer

The length of a metal wire is \[{{\ell }_{2}}\]when the tension in it is \[{{\operatorname{T}}_{1}}\]and \[{{\ell }_{2}}\] when the tension in it is\[{{\operatorname{T}}_{2}}\]. The natural length of the wire:

A) \[\frac{{{\ell }_{1}}+{{\ell }_{2}}}{2}\]           

B) \[\frac{{{\ell }_{1}}{{\operatorname{T}}_{2}}+{{\operatorname{T}}_{1}}{{\ell }_{2}}}{2}\]

C) \[\frac{{{\ell }_{1}}{{\operatorname{T}}_{2}}-{{\operatorname{T}}_{1}}{{\ell }_{2}}}{{{\operatorname{T}}_{1}}+{{\operatorname{T}}_{2}}}\]       

D) \[\sqrt{{{\ell }_{1}}{{\ell }_{2}}}\]

Correct Answer: B

Solution :

\[\operatorname{For}{{1}^{st}}\,case{{T}_{1}}=k({{\ell }_{1}}-L)\]                  .....(1) \[\operatorname{For}{{2}^{\operatorname{nd}}}\,case{{T}_{2}}=k({{\ell }_{2}}-L)\]                 .....(2) \[\left( 1 \right)\div \left( 2 \right)\] \[{{\operatorname{T}}_{1}}\left( {{\ell }_{2}}-L \right)={{T}_{2}}{{\ell }_{1}}-{{T}_{2}}L\] \[{{\operatorname{T}}_{1}}{{\ell }_{2}}-{{\operatorname{T}}_{1}}L={{T}_{2}}{{\ell }_{1}}-{{T}_{2}}L\] \[\operatorname{L}=\frac{{{T}_{2}}{{\ell }_{1}}-{{\operatorname{T}}_{1}}{{\ell }_{2}}}{{{\operatorname{T}}_{2}}-{{\operatorname{T}}_{1}}}\]