The length of metallic wire is l₁ when tension in it is T₁. It is l₂ when the tension is T₂. The original length of the wire will be :

Solution

Physics sir · 4 steps

Extension is proportional to tension, so write it for both tensions and let the constant cancel.

1. Hooke's law. Let the original length be L. The extension is Δ l = TL/AY, and L, A and Y are the same in both cases. So the extension is proportional to the tension: l₁ - L = kT₁ and l₂ - L = kT₂.

   (l₁ - L)/(l₂ - L) = T₁/T₂

2. Solve for L. Cross-multiply: T₂ l₁ - T₂ L = T₁ l₂ - T₁ L. Collect the L terms: L(T₂ - T₁) = T₂ l₁ - T₁ l₂.
3. Quick check. Take L = 1, k = 1, T₁ = 1, T₂ = 2, so l₁ = 2 and l₂ = 3. The formula gives (2 × 2 - 1 × 3)/(2 - 1) = 1, which is correct.
4. So. L = (T₂ l₁ - T₁ l₂)/(T₂ - T₁), which is option (d).

The idea to keep. When you know the length at two tensions, write stretched length minus original length equal to k times tension for each case, then divide. The unknown constant cancels and only L is left.

Answer from the NTA final answer key. Solution by PrepWiser.