A brass rod of length 50 cm and diameter 3.0 mm is joined to a steel rod of the same length and diameter. What is the change in length of the combined rod at 250 °C, if the original lengths are at 40.0 °C? Is there a ‘thermal stress’ developed at the junction? The ends of the rod are free to expand (Co-efficient of linear expansion of brass = 2.0 × 10–5 K–1, steel = 1.2 × 10–5 K–1 ).
Initial temperature, T1=40oC Final temperature, T2=250oC Change in temperature, △T=T2−T1=210oC Length of the brass rod at T1,l1=50cm Diameter of the brass rod at T1,d1=3.0cm Length of the steel rod at T2,l2=50cm Diameter of the steel rod at T2,d2=3.0mm Coefficient of linear expansion of brass, α1=2.0×10−5K−1 Coefficient of linear expansion of steel, α2=1.2×10−5K−1 For the expansion in the brass rod, we have: Change in length (△l1) / Original length (l1)=α1△T ∴△l1=50×(2.1×10−5)×210 =0.2205cm For the expansion in the steel rod, we have: Change in length (△l2)/ Original length (l2)=α2△T ∴△l1=50×(1.2×10−5)×210 =0.126cm Total change in the lengths of brass and steel, △l=△l1+△l2 =0.2205+0.126 =0.346cm Total change in the length of the combined rod =0.346cm Since the rod expands freely from both ends, no thermal stress is developed at the junction.
