A classical mechanics problem by Jaber Al-arbash

At a temperature of $60 ^\circ F$ , a 0.04 in. gap exists between the ends of the two bars shown. Bar (1) is an aluminum alloy [ $\alpha = 12.5 \times 10^{-6} / ^\circ F$ ] bar with a width of 3 in and a thickness of 0.75 in. Bar (2) [ $\alpha = 9.6 \times 10^{-6} / ^\circ F$ ] bar with a width of 2 in and a thickness of 0.75 in.

Assume the supports at $A$ and $C$ are rigid. What is the lowest temperature at which the two bars contact each other?

Equations to be used:

a) Elongation or lengthening $\delta = \alpha \Delta TL$ .

b) $\delta_1 + \delta_2 =0.04$ .

Note : Since there is a gap of 0.04 in between the two bars, the sum of the elongations of bar (1) and (2) is given as shown in equation (b).

Symbols:

$\alpha$ : Thermal expansion of a bar

$L$ : Length

$T$ : Temperature

$\delta$ : Elongation.