The temperature of a metal strip having coeff | vedclass

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(D) Let the initial length at temperature $T_1$ be $L_0$.For the first temperature change from $T_1$ to $T_2$,the increase in length is $\Delta L_1 =

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$\Delta L_1[1 + \alpha \Delta T]$

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(D) Let the initial length at temperature $T_1$ be $L_0$.For the first temperature change from $T_1$ to $T_2$,the increase in length is $\Delta L_1 = L_0 \alpha (T_2 - T_1) = L_0 \alpha \Delta T$.The length of the strip at temperature $T_2$ is $L_2 = L_0 + \Delta L_1 = L_0(1 + \alpha \Delta T)$.For the second temperature change from $T_2$ to $T_3$,the increase in length is $\Delta L_2 = L_2 \alpha (T_3 - T_2)$.Given $T_3 + T_1 = 2T_2$,we can write $T_3 - T_2 = T_2 - T_1 = \Delta T$.Substituting $L_2$ and $(T_3 - T_2)$ into the expression for $\Delta L_2$:$\Delta L_2 = [L_0(1 + \alpha \Delta T)] \alpha \Delta T = (L_0 \alpha \Delta T)(1 + \alpha \Delta T)$.Since $\Delta L_1 = L_0 \alpha \Delta T$,we get $\Delta L_2 = \Delta L_1(1 + \alpha \Delta T)$.

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