At a temperature of $20C,$ a $2mm.$ gap exists between the ends of the two bars shown in Fig. Bar $(1)$ is an aluminum alloy $\{$:GPa,$v=0\mathrm{.}32,=12\mathrm{.}510-\frac{6}{}C]$ bar with a width of $40mm.$ and a thickness of $10mm.$ Bar $(2)$ is a stainless steel $0\mathrm{.}12,=9\mathrm{.}610-\frac{6}{}C]$ bar with a width of $30mm$ and a thickness of $10mm.$ The supports at A and $C$ are rigid. Determine:

$($a$)$ the lowest temperature at which the two bars contact each other.

$($b$)$ the normal stress in the two bars at a temperature of $150C.$

$($c$)$ the normal strain in the two bars at $150C.$

$($d$)$ the change in width of the aluminum bar at a temperature of $150C.$