Question

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.)

Answer 1

Answer:

361.46 N

Explanation:

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

Y = Young's modulus for steel = $20\times 10^{10}\ Pa$

A = Area = $3.1\times 10^{-6}\ m^2$

$L_0$ = Original length = 1.28 km

$\Delta T$ = Change in temperature = $45-(-10)$

Length contraction is given by

$\Delta L=\alpha L_0\Delta T$

Also,

$\Delta L=\dfrac{L_0T}{YA}$

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

The tension in the wire is 361.46 N