An aluminium rod and two steel rods, all having an identical length l₀ and equal cross-sectional areas, are rigidly joined at their ends as shown in the figure. At 0^ C , all the rods are in a state of zero tension. Determine the length of the system when its temperature is increased to . The coefficients of linear expansion for aluminium and steel are _a and _s , respectively, while their Young's moduli are Y_a and

Step-by-step solution

Let the final length of the composite system be l and the cross-sectional area of each rod be A . The natural length of the aluminium rod at temperature is l_ 0a = l₀(1 + _a ) . The natural length of each steel rod at temperature is l_ 0s = l₀(1 + _s ) . The mechanical strain in the aluminium rod is _a = l - l_ 0a l₀ . The mechanical strain in each steel rod is _s = l - l_ 0s l₀ . The tension in the aluminium rod is F_a = Y_a A _a = Y_a A ( l - l_ 0a l₀ ) . The tension in each steel rod is F_s = Y_s A _s = Y_s A (