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The Golden Gate Bridge in San Francisco has a main span of length 1.28 km, one of the longest in the world. Imagine that a steel wire with this length and a cross-sectional area of 3.10 ✕ 10^−6 m^2 is laid on the bridge deck with its ends attached to the towers of the bridge, on a summer day when the temperature of the wire is 43.0°C. When winter arrives, the towers stay the same distance apart and the bridge deck keeps the same shape as its expansion joints open. When the temperature drops to −10.0°C, what is the tension in the wire? Take Young's modulus for steel to be 20.0 ✕ 10^10 N/m^2. (Assume the coefficient of thermal expansion of steel is 11 ✕ 10−6 (°C)−1.)

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boffeemadrid

Answer:

361.46 N

Explanation:

[tex]\alpha[/tex] = Coefficient of thermal expansion = [tex]11\times 10^{-6}\ /^{\circ}C[/tex]

Y = Young's modulus for steel = [tex]20\times 10^{10}\ Pa[/tex]

A = Area = [tex]3.1\times 10^{-6}\ m^2[/tex]

[tex]L_0[/tex] = Original length = 1.28 km

[tex]\Delta T[/tex] = Change in temperature = [tex]45-(-10)[/tex]

Length contraction is given by

[tex]\Delta L=\alpha L_0\Delta T[/tex]

Also,

[tex]\Delta L=\dfrac{L_0T}{YA}[/tex]

[tex]\alpha L_0\Delta T=\dfrac{L_0T}{YA}\\\Rightarrow T=\alpha \Delta TYA\\\Rightarrow T=11\times 10^{-6}\times (43-(-10))\times 20\times 10^{10} \times 3.1\times 10^{-6}\\\Rightarrow T=361.46\ N[/tex]

The tension in the wire is 361.46 N

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