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.

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