Question

If a transmission line in a cold climate collects ice, the increased diameter tends to cause vortex formation in a passing wind. The air pressure variations in the vortexes tend to cause the line to oscillate (gallop), especially if the frequency of the variations matches a resonant frequency of the line. In long lines, the resonant frequencies are so close that almost any wind speed can set up a resonant mode vigorous enough to pull down support towers or cause the line to short out with an adjacent line. If a transmission line has a length of $347 \mathrm{m},$ a linear density of $3.35 \mathrm{kg} / \mathrm{m},$ and a tension of 65.2 $\mathrm{MN}$ , what are (a) the frequency of the fundamental mode and ( b ) the frequency difference between successive modes?

Name: Problem 48, Chapter 16: Fundamentals of Physics
Uploaded: 2019-08-07T16:07:57-08:00
Duration: 4 min 12 s
Channel: Ben Nicholson

   If a transmission line in a cold climate collects ice, the increased diameter tends to cause vortex formation in a passing
wind. The air pressure variations in the vortexes tend to cause the
line to oscillate (gallop), especially if the frequency of the variations matches a resonant frequency of the line. In long lines, the
resonant frequencies are so close that almost any wind speed can
set up a resonant mode vigorous enough to pull down support towers or cause the line to short out with an adjacent line. If a transmission line has a length of $347 \mathrm{m},$ a linear density of $3.35 \mathrm{kg} / \mathrm{m},$ and a
tension of 65.2 $\mathrm{MN}$ , what are (a) the frequency of the fundamental
mode and ( b ) the frequency difference between successive modes?

Fundamentals of Physics

David Halliday,… 10th Edition

Chapter 16, Problem 48

Instant Answer

Solved by Expert Ben Nicholson

08/07/2019

Step 1

The wave speed is given by the formula $v=\sqrt{\frac{T}{\mu}}$, where $T$ is the tension and $\mu$ is the linear density. Substituting the given values, we get $v=\sqrt{\frac{65.2 \times 10^6 \mathrm{N}}{3.35 \mathrm{kg/m}}}=4412 \mathrm{m/s}$. Show more…

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If a transmission line in a cold climate collects ice, the increased diameter tends to cause vortex formation in a passing wind. The air pressure variations in the vortexes tend to cause the line to oscillate (gallop), especially if the frequency of the variations matches a resonant frequency of the line. In long lines, the resonant frequencies are so close that almost any wind speed can set up a resonant mode vigorous enough to pull down support towers or cause the line to short out with an adjacent line. If a transmission line has a length of $347 \mathrm{m},$ a linear density of $3.35 \mathrm{kg} / \mathrm{m},$ and a tension of 65.2 $\mathrm{MN}$ , what are (a) the frequency of the fundamental mode and ( b ) the frequency difference between successive modes?

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Key Concepts

Vibrations of a Stretched String

This concept involves understanding how a string under tension supports standing waves or vibrational modes. The natural or resonant frequencies are determined by the string’s length (L), tension (T), and linear density (?), and are quantified by the relation f? = n/(2L) × ?(T/?), where n is the mode number. This explains why a structure, such as a transmission line, oscillates at specific frequencies when disturbed.

Resonance

Resonance occurs when an external disturbance, such as wind-induced vibrations, has a frequency that matches one of the natural frequencies of a system. This matching can lead to large amplitude oscillations because energy is efficiently transferred into the vibrational mode. In systems with closely spaced natural frequencies, even a wide range of external frequencies can trigger resonant behavior, highlighting the importance of considering resonance in structural design and safety.

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