Two wires of the same material (Young's modulus Y ) and same length L but radii R and 2 R respectively are joined end to end and a weight w is suspended from the combination as shown in the figure. The elastic potential energy in the system is

Step-by-step solution

k 1 = Yπ 2 R 2 L , k 2 = Yπ R 2 L Equivalent 1 k 1 + 1 k 2 = L 4 YπR 2 + L YπR 2 Since, k 1 x 1 = k 2 x 2 = w Elastic potential energy of the system U ⁡ = 1 2 k 1 x 1 2 + 1 2 k 2 x 2 2 U = 1 2 k 1 w k 1 2 + 1 2 k 2 w k 2 2 = 1 2 w 2 1 k 1 + 1 k 2 = 1 2 w 2 5 L 4 YπR 2 U ⁡ = 5 w 2 L 8 πYR 2