Two wires are made of the same material and have the same volume. The first wire has cross-sectional area A and the second wire has cross-sectional area 3 A . If the length of the first wire is increased by l on applying a force F , how much force is needed to stretch the second wire by the same amount?

Step-by-step solution

Let the length and area of the first wire be L₁ and A₁ = A . Let the length and area of the second wire be L₂ and A₂ = 3A . Since the volumes are equal, V₁ = V₂ , we have A₁ L₁ = A₂ L₂ . Substituting the given values, A L₁ = 3A L₂ , which implies L₁ = 3 L₂ . The Young modulus Y is the same for both wires. The force F required to produce an extension l is given by F = Y A l L . For the first wire, F = Y A l L₁ . For the second wire, let the force be F' . Then F' = Y A₂ l L₂ = Y (3A) l L₂ . Substituting L₂ = L₁ 3 int