Question

The steel railway track is to be stopped from expanding at the joints due to temperature increases. If the area of cross section of the track is $100c{{m}^{2}}$ the elastic temperature increases ${{20}^{{}^\circ }}C$ to ${{40}^{{}^\circ }}C$, then the work done per unit length will be (if $\alpha =1.2\times {{10}^{-5}}{{{}^\circ }}C$ and $Y={{10}^{11}}N/{{m}^{2}}$ )
(A) 57.6 J
(B) 14.4 J
(C) 28.8 J
(D) 7.2 J

Answer 1

We know that the Young's modulus of the material of a wire decreases with increase in temperature. When the temperature increases, the atomic thermal vibrations increase, and this will cause the changes of lattice potential energy and curvature of the potential energy curve, so the Young's modulus will also change. And with the increase of temperature, the material will have a volume expansion. High temperature reduces material stiffness and strength, while low temperature increases material stiffness and strength. Almost all materials creep over time if exposed to elevated temperatures under applied load. Based on this concept we have to solve this question.

Complete step by step answer:
We know that the work done is denoted by W which is given as:
$W=F(x)dx$
So, now we can write the force as:
$F=\dfrac{TA\Delta l}{L}$
Now we can write that:
$W=\dfrac{YA\Delta l}{L}\times \Delta l=YA{{\left( \dfrac{\Delta l}{L} \right)}^{2}}\times L$
So, we can write the work done per unit length as:
$\dfrac{W}{L}=YA{{(\dfrac{\Delta l}{L})}^{2}}=YA{{(\alpha \Delta T)}^{2}}$
Now we have to put the values to get the answer as:
$\dfrac{W}{L}={{10}^{11}}\times 100\times {{(1.2\times {{10}^{5}}\times 20)}^{2}}=57.6J$
Hence, we can say that work done per unit length will be 57.6 J.

So, the correct answer is option A.

Note: We should know that the Young's Modulus or Elastic Modulus is in essence the stiffness of a material. In other words, it is how easily it is bended or stretched. To be more exact, the physics and numerical values are worked out like this: Young's Modulus = Stress / Strain. Young’s modulus is not constant because it's a material property. It is not always the same in all orientation of a material. For the metals and ceramics that are isotopic, in such cases Young's modulus will have constant value since their mechanical properties are same in all orientations.
It should also be known as Poisson’s ratio is the ratio of transverse contraction strain to longitudinal extension strain in the direction of stretching force. Strain e is defined in elementary form as the change in length divided by the original length.