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198.


Problem 22.49 (RHK)

The distance between the towers of the main span of Golden Gate Bridge near San Francisco is 4200 ft. The sag of the cable halfway between the towers at is 470 ft. Assuming that the coefficient of linear expansion for the cable is , we have to compute the change in length of the cable for a temperature change from 10 to . We can assume that there is no bending or separation of the towers and that the shape of the cable is parabolic.


Solution:           Click For PDF Version

 Let the equation of the parabola be

.

We will calculate the length of the parabolic curve from the coordinate (-x,y) to (0,0) to (x,y). We call this length . It is given by the integral


We will fix the function describing the parabola by calculating the constant a using the data that the point and is on the parabolic curve.

Therefore,


We can now calculate the length of the cable joining the towers at temperature . Substituting and

in the expression for l, we find



Using the value of the coefficient of linear expansion for the cable, , we estimate the length of the cable at . We find


Therefore, the change in length of the cable when the temperature changes from to will be