   Chapter 10, Problem 18P

Chapter
Section
Textbook Problem

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 4.00 × 10−6 m2 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 35.0°C. (a) 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 × 1010 N/m2. (b) Permanent deformation occurs if the stress in the steel exceeds its elastic limit of 3.00 × 108 N/m2. At what temperature would the wire reach its elastic limit? (c) Explain how your answers to (a) and (b) would change if the Golden Gate Bridge were twice as long.

(a)

To determine
The tension in the wire.

Explanation

Given Info: The length of the bridge is 1.29 km. Cross sectional area (A) of the steel wire is 4.00×106m2 . The temperature drops from T1=35.0οC to T2=10.0οC . The Young’s modulus (Y) for steel is 20.0×1010Nm2 . The co-efficient of linear expansion ( α ) of steel is 11×106/οC .

Formula to calculate the tension is,

F=AY(ΔLL0)       (I)

Formula to calculate the change in length is,

ΔLL0=α(T2T1)       (II)

From Equations (I) and (II),

F=AYα|(T2T1)|

Substitute 4.00×106m2 for A, 20

(b)

To determine
The temperature of the wire when it reaches the elastic limit.

(c)

To determine
The tension in the wire and temperature of the wire when it reaches the elastic limit.

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