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An electric transmission line can become coated with ice and the increased diameter may cause vortex formation in prevailing winds. The electric line will oscillate due to the pressure variations, particularly if the frequency of the oscillations is close to one of the resonant frequencies of the wire. The oscillations of long lines may be enough to snap wires or even pull down support poles. If a wire is 347 m long (from pole to pole) and has a tension of 65.2 x 106 N and has a linear density of 3.35 kg/m, (a) find the fundamental mode, and (b) find the frequency difference between successive modes.

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Answer:

a.   f₀ = 6.355 Hz ; b. Δf = 6.35 Hz

Explanation:

Given Data:

Length of wire = l =347 m             ;

Tension in wire = T = 65.2 * 10⁶N ;

Linear density = μ = 3.35 kg/m     ;

Solution:

a)

Fundamental mode = f₀ = (1/2l)*(sqr.root(T/μ))

By putting the values, we get

                                    f₀ = (1/2(347))*[tex]\sqrt[]{\frac{65.2 * 10^6}{3.35} }[/tex]

                                    f₀ = 6.355 Hz

b)

To find the frequency difference between successive modes we need to find frequency of second harmonic first

                                    f₁ = (2/2l)*(sqr.root(T/μ))

                                    f₁ = (2/2(347))*[tex]\sqrt[]{\frac{65.2 * 10^6}{3.35} }[/tex]

                                    f₁ = 12.71 Hz

Difference is:

                                 Δf = f₁ - f₀ = 12.71 - 6.355

                                      = 6.35 Hz

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