In a manufacturing operation, a 1-mm-thick tubular steel collar is to be shrink-fitted onto a 3-mm-diameter pin of circular cross section. Prior to initiating the shrink-fit operation, the collar is heated to \(200^{\circ} \mathrm{C}\) and the aluminum alloy pin is cooled to \(-100^{\circ} \mathrm{C}\). After these temperatures are achieved, the hot collar is quickly slid over the cold pin. The collar subsequently cools and shrinks as the pin heats and expands, ultimately forming a tight fit between the two. If the tight fit is achieved when the temperature difference between the collar and pin is reduced to \(50^{\circ} \mathrm{C}\), how much time elapses before formation of the tight fit? Prior to the tight fit, a thermal contact resistance of \(5 \times\) \(10^{-4} \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\) exists at the rod-collar interface. Plot the steel and aluminum temperatures over the time interval \(0 \leq t \leq 2 \mathrm{~s}\). The aluminum alloy density and specific heat are \(2800 \mathrm{~kg} / \mathrm{m}^{3}\) and \(1000 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), respectively, while the corresponding steel properties are \(7800 \mathrm{~kg} / \mathrm{m}^{3}\) and \(510 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\).