A copper rod $15$ mm diameter, $0\mathrm{.}8$ m long is heated through $50$C$.$ What is its expansion when free to expand? Suppose the expansion is prevented by gripping it at both ends, find the stress, its nature and the force applied by the grips, when:

$($i$)$ The grips do not yield.

$($ii$)$ One grip yields back by $0\mathrm{.}5$ mm$.$

Take $\_$c$=18\mathrm{.}5$ x $10^-6/$C and E$\_\{$C$\}=1\mathrm{.}25*10^5$ N $/$ mm $^2$