question_answer 40)

                  A wire of length L and radius r is clamped rigidly at one end. When the other end of the wire is pulled by a force \[f\] its length increases by \[l\]. Another wire of the same material of length 2L and radius 2r, is pulled by a force \[2f\]. Find the increase in length of this wire.                

Answer:

                  \[Y=\frac{f/\pi {{r}^{2}}}{l/L}=\frac{fL}{\pi {{r}^{2}}L}\]                 In second case,                 \[Y=\frac{2f/4\pi {{r}^{2}}}{x/2L}=\,\frac{fL}{\pi {{r}^{2}}x},\] where \[x\] is the increase in length of second wire.                 Since both the wire are made of same material, so Y is same in both the cases.                 \[\therefore \] \[\frac{fL}{\pi {{r}^{2}}l}\,=\frac{fL}{\pi {{r}^{2}}x}\] or \[x=\,l\]