Determine Increase in Length of Metal Strip under Temperature Rise

To find the increase in length $\Delta L_2$ during the second temperature rise, we analyze the thermal expansion in two stages:

Stage 1: Temperature increases from $T_1$ to $T_2$ Let $L_1$ be the original length of the metal strip at temperature $T_1$. The temperature difference for the first stage is given as: $\Delta T = T_2 - T_1$

The increase in length $\Delta L_1$ during this stage is given by the linear expansion formula: $\Delta L_1 = L_1 \alpha \Delta T$

The new length of the strip at temperature $T_2$, denoted as $L_2$, becomes: $L_2 = L_1 + \Delta L_1 = L_1 (1 + \alpha \Delta T)$

Stage 2: Temperature increases from $T_2$ to $T_3$ We are given that $T_3 + T_1 = 2T_2$, which simplifies to: $T_3 - T_2 = T_2 - T_1 = \Delta T$

The temperature difference for this second stage is also $\Delta T$. The increase in length $\Delta L_2$ during this stage depends on the length $L_2$ at temperature $T_2$: $\Delta L_2 = L_2 \alpha (T_3 - T_2) = L_2 \alpha \Delta T$

Expressing $\Delta L_2$ in terms of $\Delta L_1$: Substitute the expression for $L_2$ into the formula for $\Delta L_2$: $\Delta L_2 = L_1 (1 + \alpha \Delta T) \alpha \Delta T$

Rearranging the terms: $\Delta L_2 = (L_1 \alpha \Delta T) (1 + \alpha \Delta T)$

Since $\Delta L_1 = L_1 \alpha \Delta T$, we substitute $\Delta L_1$ into the equation: $\Delta L_2 = \Delta L_1 [1 + \alpha \Delta T]$

Correct Answer: Option D ($\Delta L_1 [1+ \alpha \Delta T]$)