question_answer

Two wires of same length and material but of different radii are suspended from a rigid support. Both carry the same load. Will the stress, strain and extension in them be same or different?

Answer:

                Let \[{{r}_{1}}\] and \[{{r}_{2}}\] be the radii of the two wires. (i) \[Stress=\frac{F}{A}=\frac{F}{\pi {{r}^{2}}}.\]For same load \[F,\frac{{{(stress)}_{1}}}{{{(stress)}_{2}}}=\frac{r_{2}^{2}}{r_{1}^{2}}\] (ii) Strain, \[\frac{\Delta l}{l}=\frac{F}{A\Upsilon }=\frac{F}{\pi {{r}^{2}}\Upsilon }\] For the two wires F and \[\Upsilon \] are same, so \[\frac{{{(strain)}_{1}}}{{{(strain)}_{2}}}=\frac{r_{2}^{2}}{r_{1}^{2}}\] (iii) Extension,  \[\Delta l=\frac{F}{A}.\frac{l}{\Upsilon }=\frac{F}{\pi {{r}^{2}}}.\frac{l}{\Upsilon }\] For the two wires F, L and \[\Upsilon \] are same, so\[\frac{{{(\Delta l)}_{1}}}{{{(\Delta l)}_{2}}}=\frac{r_{2}^{2}}{r_{1}^{2}}\] Hence stress, strain and extension are all different for the two wires.