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

361.46 N

Step-by-step explanation:

`\alpha` = Coefficient of thermal expansion =
`11* 10^(-6)\ /^(\circ)C`

Y = Young's modulus for steel =
`20* 10^(10)\ Pa`

A = Area =
`3.1* 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=(L_0T)/(YA)`

`\alpha L_0\Delta T=(L_0T)/(YA)\\\Rightarrow T=\alpha \Delta TYA\\\Rightarrow T=11* 10^(-6)* (43-(-10))* 20* 10^(10) * 3.1* 10^(-6)\\\Rightarrow T=361.46\ N`

The tension in the wire is 361.46 N