The ratio of the lengths of two wires A and B of the same material is 1 : 2 and the ratio of their diameter is 2 : 1. They are stretched by the same force, then the ratio of the increase in length will be

Explanation

To determine the ratio of increase in length for wires A and B, we use the formula for elongation: ΔL = (F*L)/(A*Y), where F is the force, L is the initial length, A is the cross-sectional area, and Y is the Young's modulus. Given that both wires are made of the same material and are subjected to the same force, their Young's modulus is constant and can be ignored when comparing elongations.For wire A, let its length be L, and its diameter be d. Thus, its cross-sectional area is A = π(d/2)2.For wire B, the length is 2L, and the diameter is d/2. Its cross-sectional area is A = π(d/4)2 = πd2/16.The ratio of increase in length for wires A and B is given by the ratio of their elongations: ΔLA/ΔLB = (F*L/AA)/(F*2L/AB) = (L/AA)/(2L/AB).Substituting for the areas, we get (1/π(d/2)2)/(2/π(d/4)2), which simplifies to (1/(d2/4))/(2/(d2/16)) = 16/8 = 1/8. Hence, the correct answer is 1 : 8.