question_answer 47)

                  A rail track made of steel having length 10 m is clamped on a railway line at its two ends. On a summer day due to rise in temperature by \[{{20}^{o}}C\], it is deformed as shown in figure. Find \[x\] (displacement of the centre) if \[{{\alpha }_{steel}}\,=1.2\,\times \,{{10}^{-5}}{{/}^{o}}C.\]                

Answer:

                  From given figure                 \[x=\,\sqrt{{{\left( \frac{L+\Delta L}{2} \right)}^{2}}-{{\left( \frac{L}{2} \right)}^{2}}}\]                 \[=\,\frac{1}{2}\,\sqrt{(L+\Delta L)-{{L}^{2}}}\,\,=\frac{1}{2}\,[\Delta {{L}^{2}}+2L\,\Delta L]\]                 Since \[\Delta L\] is very small, so neglect \[\Delta {{L}^{2}}\]                 \[\therefore \]\[x=\frac{1}{2}\,\sqrt{2L\,\Delta L}\]                  ?.. (i)                 Now \[\Delta L=\,L\alpha \,\Delta T=\,10\times \,1.2\,\times \,{{10}^{-5}}\,\times \,20\,\]                 \[\therefore \]\[x=\,\frac{1}{2}\,\sqrt{2\times \,10\times \,24\times \,{{10}^{-4}}}\]                 \[=\frac{1}{2}\,\times \,21.9\,\times \,{{10}^{-2}}\]                 \[=0.1095m\simeq 0.11\,m\]