A steel wire of cross-sectional area 0.5 mm² is held between two fixed supports. If the wire is just taut at 20°C, determine the tension when the temperature falls to 0°C. Coefficient of linear expansion of steel is 1.2 × 10^–5 °C^–1 and its Young’s modulus is 2.0 × 10¹¹ N m^–2.

Question

Philoid · Accepted Answer

Given:

Cross-sectional area of the steel wire: A= 0.5 mm²=0.5×10^-6 m².
Temperature at which wire is just taut: T₁= 20 ° C.
Decrease in Temperature : T₂ = 0 °C
∴ Change in Temperature : Δ T = 20 ° C.
Coefficient of linear expansion of steel: α_s = 1.2 × 10^–5 °C^–1
Young’s modulus : Y= 2.0 × 10¹¹ N m^–2.
Formula used:

We know that, Young’s Modulus:

Here, F is the force or tension between the wire and the fixed supports and A is the cross -section area.
Also, Formula for Linear Expansion is ( in this case it’s compression as temperature is reduced):

Where Δ L is the Change in Length due to decrease in temperature and L is the original length at T₁.
Substituting the value of ΔL in the formula for Y, we get:

Hence , the tension in the wire when the temperature falls to 0°C is 24 N.

A steel wire of cross-sectional area 0.5 mm2 is held between two fixed supports. If the wire is just taut at 20°C, determine the tension when the temperature falls to 0°C. Coefficient of linear expansion of steel is 1.2 × 10–5 °C–1 and its Young’s modulus is 2.0 × 1011 N m–2.

A steel wire of cross-sectional area 0.5 mm² is held between two fixed supports. If the wire is just taut at 20°C, determine the tension when the temperature falls to 0°C. Coefficient of linear expansion of steel is 1.2 × 10^–5 °C^–1 and its Young’s modulus is 2.0 × 10¹¹ N m^–2.