A brass wire of length 2 m and radius 1 mm at 27°C is held taut between two rigid supports. Initially it was cooled to a temperature of -43°C creating a tension T in the wire. The temperature to which the wire has to be cooled in order to increase the tension in it to 1.4T, is ____ °C.

Solution

Physics sir · 4 steps

Tension comes from the temperature drop, not the temperature itself, so scale the drop by 1.4.

1. Thermal tension. The supports stop the wire from contracting, so a stretching strain α Δθ is locked in. The tension is stress times area: T = YAα Δθ.
2. Tension tracks the drop. Y, A and α stay the same, so T ∝ Δθ. That is why the length 2 m and radius 1 mm are never needed.
3. First cooling. Cooling from 27°C to -43°C is a drop of 70°C, and it gives tension T.

   Δθ₂/Δθ₁ = 1.4T/T ⇒ Δθ₂ = 1.4 × 70 = 98°C

4. So. The drop is measured from 27°C, the temperature at which the wire had no tension. The final temperature is 27 - 98 = -71°C, which is option (a).

The idea to keep. Thermal stress depends on the temperature change from the stress-free state: T = YAα Δθ. Always measure the drop from the temperature at which the wire was clamped with no tension.

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