# 16.6 Standing waves and resonance  (Page 7/17)

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A 2.40-m wire has a mass of 7.50 g and is under a tension of 160 N. The wire is held rigidly at both ends and set into oscillation. (a) What is the speed of waves on the wire? The string is driven into resonance by a frequency that produces a standing wave with a wavelength equal to 1.20 m. (b) What is the frequency used to drive the string into resonance?

A string with a linear mass density of 0.0062 kg/m and a length of 3.00 m is set into the $n=100$ mode of resonance. The tension in the string is 20.00 N. What is the wavelength and frequency of the wave?

$\begin{array}{}\\ \\ {\lambda }_{100}=0.06\phantom{\rule{0.2em}{0ex}}\text{m}\hfill \\ \\ v=56.8\phantom{\rule{0.2em}{0ex}}\text{m/s,}\phantom{\rule{1em}{0ex}}{f}_{n}=n{f}_{1},\phantom{\rule{1em}{0ex}}n=1,2,3,4,5\text{...}\hfill \\ {f}_{100}=947\phantom{\rule{0.2em}{0ex}}\text{Hz}\hfill \end{array}$

A string with a linear mass density of 0.0075 kg/m and a length of 6.00 m is set into the $n=4$ mode of resonance by driving with a frequency of 100.00 Hz. What is the tension in the string?

Two sinusoidal waves with identical wavelengths and amplitudes travel in opposite directions along a string producing a standing wave. The linear mass density of the string is $\mu =0.075\phantom{\rule{0.2em}{0ex}}\text{kg/m}$ and the tension in the string is ${F}_{T}=5.00\phantom{\rule{0.2em}{0ex}}\text{N}.$ The time interval between instances of total destructive interference is $\text{Δ}t=0.13\phantom{\rule{0.2em}{0ex}}\text{s}.$ What is the wavelength of the waves?

$T=2\text{Δ}t,\phantom{\rule{1em}{0ex}}v=\frac{\lambda }{T},\phantom{\rule{1em}{0ex}}\lambda =2.12\phantom{\rule{0.2em}{0ex}}\text{m}$

A string, fixed on both ends, is 5.00 m long and has a mass of 0.15 kg. The tension if the string is 90 N. The string is vibrating to produce a standing wave at the fundamental frequency of the string. (a) What is the speed of the waves on the string? (b) What is the wavelength of the standing wave produced? (c) What is the period of the standing wave?

A string is fixed at both end. The mass of the string is 0.0090 kg and the length is 3.00 m. The string is under a tension of 200.00 N. The string is driven by a variable frequency source to produce standing waves on the string. Find the wavelengths and frequency of the first four modes of standing waves.

$\begin{array}{}\\ \\ {\lambda }_{1}=6.00\phantom{\rule{0.2em}{0ex}}\text{m},\phantom{\rule{1em}{0ex}}{\lambda }_{2}=3.00\phantom{\rule{0.2em}{0ex}}\text{m},\phantom{\rule{1em}{0ex}}{\lambda }_{3}=2.00\phantom{\rule{0.2em}{0ex}}\text{m},\phantom{\rule{1em}{0ex}}{\lambda }_{4}=1.50\phantom{\rule{0.2em}{0ex}}\text{m}\hfill \\ v=258.20\phantom{\rule{0.2em}{0ex}}\text{m/s}=\lambda f\hfill \\ {f}_{1}=43.03\phantom{\rule{0.2em}{0ex}}\text{Hz},\phantom{\rule{1em}{0ex}}{f}_{2}=86.07\phantom{\rule{0.2em}{0ex}}\text{Hz},\phantom{\rule{1em}{0ex}}{f}_{3}=129.10\phantom{\rule{0.2em}{0ex}}\text{Hz},\phantom{\rule{1em}{0ex}}{f}_{4}=172.13\phantom{\rule{0.2em}{0ex}}\text{Hz}\hfill \end{array}$

The frequencies of two successive modes of standing waves on a string are 258.36 Hz and 301.42 Hz. What is the next frequency above 100.00 Hz that would produce a standing wave?

A string is fixed at both ends to supports 3.50 m apart and has a linear mass density of $\mu =0.005\phantom{\rule{0.2em}{0ex}}\text{kg/m}.$ The string is under a tension of 90.00 N. A standing wave is produced on the string with six nodes and five antinodes. What are the wave speed, wavelength, frequency, and period of the standing wave?

$v=134.16\phantom{\rule{0.2em}{0ex}}\text{ms},\lambda =1.4\phantom{\rule{0.2em}{0ex}}\text{m},f=95.83\phantom{\rule{0.2em}{0ex}}\text{Hz},T=0.0104\phantom{\rule{0.2em}{0ex}}\text{s}$

Sine waves are sent down a 1.5-m-long string fixed at both ends. The waves reflect back in the opposite direction. The amplitude of the wave is 4.00 cm. The propagation velocity of the waves is 175 m/s. The $n=6$ resonance mode of the string is produced. Write an equation for the resulting standing wave.

Ultrasound equipment used in the medical profession uses sound waves of a frequency above the range of human hearing. If the frequency of the sound produced by the ultrasound machine is $f=30\phantom{\rule{0.2em}{0ex}}\text{kHz,}$ what is the wavelength of the ultrasound in bone, if the speed of sound in bone is $v=3000\phantom{\rule{0.2em}{0ex}}\text{m/s?}$

$\lambda =0.10\phantom{\rule{0.2em}{0ex}}\text{m}$

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