Question

steel wire 8m long and 4mm in diameter is fixed to two rigid b. [1] supports. Calculate the increase in tension when the temperature falls by 10°C. Given linear expansivity of steel, a 12x10 %/K, Young's modulus for steel, Y = 2x10¹¹ N/m².​

Answer 1

301.6 N

Step-by-step explanation:

The length of the wire L₀ = 8 m and its diameter, d = 4 mm = 4 × 10⁻³ m. Since its temperature drops by 10°C, it will have a change in length ΔL = L₀αΔθ where α = linear expansivity of steel, a 12 × 10⁻⁶ /K, and Δθ = temperature change = -10°C = -10 K(negative since it is a drop)

So, the strain, ε = ΔL/L₀ = αΔθ = 12 × 10⁻⁶ /K × 10 K = 12 × 10⁻⁵

Now the Young's modulus of steel, Y = σ/ε where σ = stress = T/A where T = increase in tension in steel wire and A = cross-sectional area of wire = πd²/4 where d = diameter of wire = 4 × 10⁻³ m and ε = strain = 12 × 10⁻⁵

So, σ = Yε

Since Y = 2 × 10¹¹ N/m².

Substituting the values of the variables into the equation, we have

σ = Yε

σ = 2 × 10¹¹ N/m² × 12 × 10⁻⁵

σ = 24 × 10⁶ N/m²

Since σ = T/A

T = σA

T = σπd²/4

Substituting the values of the variables into the equation, we have

T = σπd²/4

T = 24 × 10⁶ N/m² × π × (4 × 10⁻³ m)²/4

T = 24 × 10⁶ N/m² × π × 16 × 10⁻⁶ m²/4

T = 24 × 10⁶ N/m² × π × 4 × 10⁻⁶ m²

T = 96 N × π

T = 301.59 N

T ≅ 301.6 N

So, the increase in tension in the steel wire is 301.6 N