question_answer

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

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

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

C) (c)\[\frac{{{\ell }_{1}}{{T}_{2}}-{{\ell }_{2}}{{T}_{1}}}{{{T}_{2}}-{{T}_{1}}}\]

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

Correct Answer: C

Solution :

[c] If \[\ell \] is the original length of wire, then change in length of first wire, \[\Delta {{\ell }_{1}}=({{\ell }_{1}}-\ell )\] Change in length of second wire, \[\Delta {{\ell }_{2}}=({{\ell }_{2}}-\ell )\] Now,  \[Y=\frac{{{T}_{1}}}{A}\times \frac{\ell }{\Delta {{\ell }_{1}}}=\frac{{{T}_{2}}}{A}\times \frac{\ell }{\Delta {{\ell }_{2}}}\] or\[\frac{{{T}_{1}}}{\Delta {{\ell }_{1}}}=\frac{{{T}_{2}}}{\Delta {{\ell }_{2}}}\,or\,\frac{{{T}_{1}}}{{{\ell }_{1}}-\ell }=\frac{{{T}_{2}}}{{{\ell }_{2}}-\ell }\] or \[{{T}_{1}}{{\ell }_{2}}-{{T}_{1}}\ell ={{T}_{2}}{{\ell }_{1}}-\ell {{T}_{2}}\] or \[\ell =\frac{{{T}_{2}}{{\ell }_{1}}-{{T}_{1}}{{\ell }_{2}}}{{{T}_{2}}-{{T}_{1}}}\]