Question 1

What is the frequency of the fundamental mode of vibration of a steel piano wire stretched to a tension of  440 N? The wire is  0.600 m long and has a mass of  5.60 g.

Accepted Answer

The answer of What is the frequency of the fundamental...

Question 2

A  25-g string is stretched with a tension of  43 N between two fixed points  12 m apart. What is the frequency of the second harmonic?

Accepted Answer

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Question 3

A string that is  2.0  meters long is fixed at both ends and tightened until the wave speed is  18 m/s. What is the frequency of the standing wave shown in the figure?

Accepted Answer

The answer of A string that is  2.0 ...

Question 4

Standing waves of frequency  17 Hz are produced on a string that has a mass per unit length of  0.0160 kg/m. To what tension must the string be stretched between two supports if adjacent nodes in the standing wave are to be  0.87 m apart?

Accepted Answer

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Question 5

One of the harmonics of a string fixed at both ends has a frequency of  52.2 Hz and the next higher harmonic has a frequency of  60.9 Hz. What is the fundamental frequency of the string?

Accepted Answer

The difference between any two consecutive harmonics of a string fixed at both ends is equal to the fundamental frequency. The difference between 60.9 Hz and 52.2 Hz is 8.7 Hz, which is the fundamental frequency of the string.

Question 6

Find the first three harmonics of a string of linear mass density  2.00 g/m and length  0.600 m when the tension in it is  50.0 N.

Accepted Answer

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Question 7

A string of linear density  1.5 g/m is under a tension of  20 N. What should be its length if its fundamental resonance frequency is  220 Hz?

Accepted Answer

The fundamental frequency of a string is given by $ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} $. Solving for L, $ L = \frac{1}{2f} \sqrt{\frac{T}{\mu}} $. Substituting $ f = 220 \, \text{Hz} $, $ T = 20 \, \text{N} $, and $ \mu = 1.5 \, \text{g/m} = 0.0015 \, \text{kg/m} $, we find $ L \approx 0.26 \, \text{m} $.

Question 8

If the frequency of a violin string is to be increased by  20% , what change in tension must be applied?

Accepted Answer

The frequency of a string is given by $f = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$, where $f$ is the frequency, $L$ is the length of the string, $T$ is the tension, and $\mu$ is the linear mass density of the string. Since the frequency is to be increased by 20%, and the frequency is proportional to the square root of the tension ($f \propto \sqrt{T}$), to achieve a 20% increase in frequency, the tension must be increased by a factor that, when square rooted, gives 1.20. Solving for the new tension factor: $ \sqrt{T_{new}} = 1.20 \sqrt{T_{old}} $, squaring both sides gives $ T_{new} = 1.44 T_{old} $, which is a 44% increase in tension.

Question 9

A string of length  2.5 m is fixed at both ends. When the string vibrates at a frequency of  85 Hz, a standing wave with five loops is formed.
(a) Determine the distance between two adjacent nodes.
(b) Determine the wavelength of the waves that travel on the string.
(c) Determine the speed of traveling waves on this string.
(d) Determine the fundamental frequency of this string.

Accepted Answer

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Question 10

A  4.0-g  string is  0.39 m long and is under tension. The string vibrates at  600 Hz in its third harmonic. What is the tension in this string?

Accepted Answer

The tension in the string can be calculated using the formula for the frequency of a vibrating string in its nth harmonic, which is $f_n = \frac{n}{2L} \sqrt{\frac{T}{\mu}}$, where $f_n$ is the frequency of the nth harmonic, $n$ is the harmonic number, $L$ is the length of the string, $T$ is the tension in the string, and $\mu$ is the linear mass density of the string. Given that the string vibrates at 600 Hz in its third harmonic (n=3), is 0.39 m long, and has a mass of 4.0 g (which is 0.004 kg), we can solve for $T$.First, calculate the linear mass density, $\mu = \frac{m}{L} = \frac{0.004 \, \text{kg}}{0.39 \, \text{m}}$.Then, rearrange the formula to solve for $T$: $T = \mu \left(\frac{2Lf_n}{n}\right)^2$.Substituting the given values and solving for $T$ yields a tension of approximately 250 N.

Question 11

A standing wave is oscillating at  950 Hz  on a string, as shown in the figure. What is the wave speed?

Accepted Answer

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Question 12

A  4.0-g string is  0.36 m long and is under tension. The string vibrates at  500 Hz in its third harmonic. What is the wavelength of the standing wave in the string?

Accepted Answer

The answer of A  4.0-g string is  0.36...

Question 13

The speed of propagation of a transverse wave on a  2.0-m long string fixed at both ends is  200 m/s Which one of the following is not a resonant frequency of this string?

Accepted Answer

The resonant frequencies of a string fixed at both ends are given by $f_n = n\frac{v}{2L}$, where $v$ is the speed of the wave, $L$ is the length of the string, and $n$ is a positive integer (1, 2, 3, ...). For a string of length 2.0 m and speed 200 m/s, the fundamental frequency ($n=1$) is $f_1 = \frac{200}{2*2} = 50$ Hz. The second harmonic ($n=2$) is $100$ Hz, the third harmonic ($n=3$) is $150$ Hz, and so on. Thus, 25 Hz is not a resonant frequency because it does not fit the formula for any value of $n$.

Question 14

A guitar-like stringed instrument has a string that is  16 cm long. It sounds the musical note  A(440 Hz) when played without fingering. By what distance should you shorten it to play the note C (523 Hz)?

Accepted Answer

The frequency of a string is inversely proportional to its length. To find the new length for 523 Hz, use the ratio 440/523 = L/16. Solving for L gives approximately 13.5 cm. The string should be shortened by 16 - 13.5 = 2.5 cm.

Quiz 11: Oscillations and Waves

What is the frequency of the fundamental mode of vibration of a steel piano wire stretched to a tension of 440 N? The wire is 0.600 m long and has a mass of 5.60 g.

A 25-g string is stretched with a tension of 43 N between two fixed points 12 m apart. What is the frequency of the second harmonic?

A string that is 2.0 meters long is fixed at both ends and tightened until the wave speed is 18 m/s. What is the frequency of the standing wave shown in the figure?

Standing waves of frequency 17 Hz are produced on a string that has a mass per unit length of 0.0160 kg/m. To what tension must the string be stretched between two supports if adjacent nodes in the standing wave are to be 0.87 m apart?

One of the harmonics of a string fixed at both ends has a frequency of 52.2 Hz and the next higher harmonic has a frequency of 60.9 Hz. What is the fundamental frequency of the string?

Find the first three harmonics of a string of linear mass density 2.00 g/m and length 0.600 m when the tension in it is 50.0 N.

A string of linear density 1.5 g/m is under a tension of 20 N. What should be its length if its fundamental resonance frequency is 220 Hz?

If the frequency of a violin string is to be increased by 20% , what change in tension must be applied?

A 4.0-g string is 0.39 m long and is under tension. The string vibrates at 600 Hz in its third harmonic. What is the tension in this string?

A standing wave is oscillating at 950 Hz on a string, as shown in the figure. What is the wave speed?

A 4.0-g string is 0.36 m long and is under tension. The string vibrates at 500 Hz in its third harmonic. What is the wavelength of the standing wave in the string?

The speed of propagation of a transverse wave on a 2.0-m long string fixed at both ends is 200 m/s Which one of the following is not a resonant frequency of this string?

A guitar-like stringed instrument has a string that is 16 cm long. It sounds the musical note A(440 Hz) when played without fingering. By what distance should you shorten it to play the note C (523 Hz) ?