' \ufeffA steel wire 8 \ufeffmetres long and 4 mm in diameter…

Question

' \ufeffA steel wire  8  \ufeffmetres long and  4 mm in diameter…

Anu Bhardwaj · Accepted Answer

Given values: Length of the steel wire, 𝐿 =8 m Diameter of the steel wire, 𝑑 =4 mm Radius of the wire, 𝑟 = 𝑑/ 2=4/2×10^−3 m=2×10^−3 m Temperature change, Δ 𝑇 =10 ∘ C Coefficient of linear expansion for steel, α = 12×10^{-6}  Young's modulus for steel, 𝛾 =210×10^9 N/m^2 Calculate the area of cross-section: 𝐴 = 𝜋𝑟 2= 𝜋 ×(2×10^−3)^2 A = π × ( 4 × 1 0 − 6 ) 𝐴 ≈ 3.14159 × 4 × 1 0 − 6 A ≈ 3.14159 × 4 × 1 0 − 6 𝐴 ≈ 12.56636 × 1 0 − 6 A ≈ 12.56636 × 1 0 − 6 𝐴 ≈ 12.56636 × 1 0 − 6 m 2 A ≈ 12.56636 × 1 0 − 6 m 2 =12.56×10^−6 m^2 Calculate the increase in length of the wire: Δ 𝐿 = 𝐿 × 𝛼 ×Δ 𝑇 =8×12×10^−6×10=0.00096 m Calculate strain: Strain=Δ 𝐿/𝐿 =0.00096/8=12×10^−5 Calculate stress: Stress= 𝛾 ×strain=210×10^9×12×10^−5=252×10^5 N/m^2 Calculate the increase in tension: Increase in tension=Stress×Area of cross-section Increase in tension=Stress×Area of cross-section =252×10^5×12.56×10^−6=316.512N So, the increase in tension in the wire when the temperature falls by 10 ∘ C is 316.512N. Final Answer:- the increase in tension in the wire when the temperature falls by 10 ∘ C is 316.512N.

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Mechanical engineering

Anu Bhardwaj

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