Two rods A and B of equal length L and identical cross-sectional area are joined in series. The thermal conductivities of rods A and B are 3K and 2K respectively, and their coefficients of linear expansion are 2 and 3 respectively. Initially, the entire composite rod is at a uniform temperature of 0^ C . The free end of rod A is then maintained at 100^ C and the free end of rod B is maintained at 0^ C . Assuming no h

Step-by-step solution

First, find the junction temperature T_J in the steady state by equating the rate of heat flow. 3K A (100 - T_J) L = 2K A (T_J - 0) L 300 - 3T_J = 2T_J 5T_J = 300 T_J = 60^ C In steady state, the temperature gradient is linear. The average temperature of each rod determines its thermal expansion. Average temperature of rod A, T_A = 100 + 60 2 = 80^ C Average temperature of rod B, T_B = 60 + 0 2 = 30^ C Change in length of rod A: L_A = L _A T_A = L (2 ) (80 - 0) = 160 L Change in length of rod B: L_B = L _B T_B = L