A metal wire has a length of l₁ when subjected to a tension T₁ , and a length of l₂ when the tension is T₂ . The natural length of the wire is

Step-by-step solution

Let the natural length of the wire be L , cross-sectional area be A , and Young's modulus be Y . The extension in the wire under tension T₁ is l₁ = l₁ - L . From Hooke's law, Y = T₁ L A l₁ T₁ = YA L (l₁ - L) . Similarly, the extension under tension T₂ is l₂ = l₂ - L . T₂ = YA L (l₂ - L) . Dividing the two equations, we get: T₁ T₂ = l₁ - L l₂ - L T₁ l₂ - T₁ L = T₂ l₁ - T₂ L L(T₂ - T₁) = l₁ T₂ - l₂ T₁ L = l₁ T₂ - l₂ T₁ T₂ - T₁