The value of tension in a long thin metal wire has been changed from T₁ to T₂. The lengths of the metal wire at two different values of tension T₁ and T₂ are l₁ and l₂ respectively. The actual length of the metal wire is :

Solution

Physics sir · 4 steps

Extension grows in step with tension, so write both lengths as L plus an extension and eliminate.

1. Extension is linear. Within the elastic limit, Y = (T/A)/(Δ l/L) gives Δ l = L/YA T. Call k = L/YA. It is the same for both tensions, so l₁ = L + kT₁ and l₂ = L + kT₂, where L is the actual (unstretched) length.
2. Eliminate k. From each equation, k = (l₁ - L)/T₁ = (l₂ - L)/T₂. Cross-multiplying gives T₂ l₁ - T₂ L = T₁ l₂ - T₁ L, so L(T₁ - T₂) = T₁ l₂ - T₂ l₁.

   L = (T₁ l₂ - T₂ l₁)/(T₁ - T₂)

3. Quick check. Put T₁ = 0. Then the wire is unstretched, so L must equal l₁. The formula gives (-T₂ l₁)/-T₂ = l₁, as it should.
4. So. The actual length is (T₁ l₂ - T₂ l₁)/(T₁ - T₂), which is option (c).

The idea to keep. Length under tension is (natural length) + (constant × tension). Two readings give two equations in two unknowns. Test your answer by putting one tension to zero.

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