Each of the main cables supporting a suspension bridge can be roughly modeled as

Question

Each of the main cables supporting a suspension bridge can be roughly modeled as a one-dimensional string of length 300 m, with a mass of 7.5 times 10^6 kg, placed under tension of 10 MN (Mega-Newtons)= 1.0 times 10^7 N. Suddenly, an earthquake creates a transverse wave in the cable with an amplitude of 1.60 m. a. What is the wave speed? b. If the earthquake drives the wave with a frequency 1.3Hz, what is the wavelength? c. What is the wave number? d. What is the angular frequency? e. How much power is being delivered to the skyscraper by the earthquake?

Explanation / Answer

a)

wave speed v = sqrt(T/u)

T = tension force = 10^7 N

u = mass/length = 7.5*10^6/300 = 25000 kg /m

speed v = sqrt(10^7/25000) = 20 m/s

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(b)wavelength = speed/ frequency

wavelength = 20/(1.3) = 15.4 m

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(c)

wave number K = 2*pi/wavelength

K = 2*pi/15.4 = 0.41 m^-1

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(d)

angular frequency w = v*k = 2*pi*f = 8.2 rad/s

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(e)

power P = (1/2)*u*w^2*A^2*v

P = (1/2)*25000*8.2^2*1.6^2*20

P = 43033600 W

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