A rod of length 2 m is at a temperature of `20^@` C. find the free expansion of the rod, if the temperature is increased to `50^@`C, then find stress produced when the rod is (i) fully prevented to expand, (ii) permitted to expand by 0.4mm. `Y=2xx10^(11)N//m^(2), alpha=15xx10^(-6//^(@))C`.

Explanation

Free expansion of the rod `=alphaLDeltatheta`
`=15xx10^(-6@)Cxx2mxx(50-20)^@C`
`=9xx10^-4m=0.9mm`
i. If the expansion is fully prevented,
Then strain`=(9xx10^-4)/(2)implies4.5xx10^-4`
Temperature stress`=`strain`xxY`
`=4.5xx10^-4xx2xx10^(11)=9xx10^7 N//m^(2)`
ii. If 0.4 mm expansion is allowed, then length resticted to expand `=0.9-0.4=0.5mm`
Strain `=(5xx10^-4)/(2)=2.5xx10^-4`
Temperature stress=strain`xxY=2.5xx10^-4xx2xx10^(11)=5xx10^7 N//m^(2)`