Question

A steel wire 8m long and 4mm in diameter is fixed to two rigid supports.
Calculate the increase in tension when the temperature falls by 10°C.
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Answer 1

The increase in tension when the temperature falls by 10°C is 301.4 N.

Explanation:

The Young’s modulus for steel, γ = 2 x 10¹¹ Nm⁻²

Linear expansivity for steel, α  = 12 x 10⁻⁶ /℃

Temperature, T = 10 ℃

The length of the steel wire, L = 8 m

The diameter of the steel wire = 4 mm

So, the radius, r = d/2 = 4/2 = 2 mm = 2 * 10⁻³ m

∴ Area of cross-section,

A = πr² = 22/7 * (2* 10⁻³)² = 12.56 * 10⁻⁶ = 12.56 * 10⁻⁶ m²

We know,  

The increase in length of the wire, ∆L = L αT

So,  

Strain = ∆L/L = [L αT] / L = αT

And

Stress = γ * strain = γ αT

Thus,  

The increase in tension in the wire is given by,

= Stress * Area of cross-section

= [γ αT] * [πr²]

= [2 x 10¹¹ * 12 x 10⁻⁶ * 10] * [12.56 * 10⁻⁶]

= 301.4 N

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Answer 2

Answer:

The increase in tension when the temperature falls by 10°C is 301.4 N.

Explanation:

The Young’s modulus for steel, γ = 2 x 10¹¹ Nm⁻²

Linear expansivity for steel, α  = 12 x 10⁻⁶ /℃

Temperature, T = 10 ℃

The length of the steel wire, L = 8 m

The diameter of the steel wire = 4 mm

So, the radius, r = d/2 = 4/2 = 2 mm = 2 * 10⁻³ m

∴ Area of cross-section,

A = πr² = 22/7 * (2* 10⁻³)² = 12.56 * 10⁻⁶ = 12.56 * 10⁻⁶ m²

We know,  

The increase in length of the wire, ∆L = L αT

So,  

Strain = ∆L/L = [L αT] / L = αT

And

Stress = γ * strain = γ αT

Thus,  

The increase in tension in the wire is given by,

= Stress * Area of cross-section

= [γ αT] * [πr²]

= [2 x 10¹¹ * 12 x 10⁻⁶ * 10] * [12.56 * 10⁻⁶]

= 301.4 N

Hope it helps!