The length of a rod under longitudinal tensio | vedclass

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(D) Let the natural length of the rod be $L_0$.According to Hooke's law,the stress is proportional to strain within the elastic limit:$	ext{Stress} =

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$\frac{L_1 T_2-L_2 T_1}{T_2-T_1}$

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(D) Let the natural length of the rod be $L_0$.According to Hooke's law,the stress is proportional to strain within the elastic limit:$	ext{Stress} = Y 	imes 	ext{Strain}$$\frac{T}{A} = Y \frac{L - L_0}{L_0}$Rearranging for the change in length:$L - L_0 = \frac{T L_0}{A Y}$Let $k = \frac{L_0}{A Y}$,which is a constant for the rod.Then,$L = L_0 + k T$.For tension $T_1$,$L_1 = L_0 + k T_1$  --- $(i)$For tension $T_2$,$L_2 = L_0 + k T_2$  --- (ii)Subtracting $(i)$ from (ii):$L_2 - L_1 = k(T_2 - T_1) \Rightarrow k = \frac{L_2 - L_1}{T_2 - T_1}$Substitute $k$ into equation $(i)$:$L_0 = L_1 - k T_1 = L_1 - \left( \frac{L_2 - L_1}{T_2 - T_1} ight) T_1$$L_0 = \frac{L_1(T_2 - T_1) - T_1(L_2 - L_1)}{T_2 - T_1}$$L_0 = \frac{L_1 T_2 - L_1 T_1 - T_1 L_2 + L_1 T_1}{T_2 - T_1}$$L_0 = \frac{L_1 T_2 - L_2 T_1}{T_2 - T_1}$

The length of a rod under longitudinal tension $T_1$ is $L_1$ and that under longitudinal tension $T_2$ is $L_2$. What is the actual length of the rod,in the absence of tensions?

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