Question 1

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 answer of One of the harmonics of a string...

Question 2

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 $n$ is a positive integer (the harmonic number), $v$ is the speed of the wave on the string, and $L$ is the length of the string. For a 2.0-m long string with a wave speed of 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. Since $25$ Hz does not fit into this series (it would correspond to $n=0.5$, which is not a positive integer), it is not a resonant frequency of this string.

Question 3

A 0.588-m string is tightly clamped at both ends.If the lowest standing wave frequency of the string is $$326 \mathrm {~Hz}$$ how fast do waves travel on this string?

Accepted Answer

The lowest standing wave frequency is given by:$$f_1 = \frac{v}{2L}$$where:* $f_1$ is the lowest standing wave frequency* $v$ is the wave speed* $L$ is the length of the stringSolving for $v$, we get:$$v = 2f_1L$$Substituting the given values, we get:$$v = 2(326 \mathrm{~Hz})(0.588 \mathrm{~m}) = 383 \mathrm{~m/s}$$Therefore, the correct answer is A.

Question 4

A guitar string 0.65 m long has a tension of 61 N and a mass per unit length of 3.0 g/m.
(a)What is the speed of waves on the string when it is plucked?
(b)What is the string's fundamental frequency of vibration when plucked?
(c)At what other frequencies will this string vibrate?

Accepted Answer

The answer of A guitar string 0.65 m long has...

Question 5

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 frequency of the fundamental mode of vibration of a stretched string is given by:

$f_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$

where $L$ is the length of the wire, $T$ is the tension in the wire, and $\mu$ is the linear mass density of the wire (mass per unit length).

Here, $L = 0.600$ m, $T = 440$ N, and $\mu = m/L$, where $m$ is the mass of the wire. Therefore:

$\mu = \frac{5.60	imes10^{-3}\, 	ext{kg}}{0.600\, 	ext{m}} = 9.33	imes10^{-3}\, 	ext{kg/m}$

Substituting these values into the formula gives:

$f_1 = \frac{1}{2(0.600\, 	ext{m})} \sqrt{\frac{440\, 	ext{N}}{9.33	imes10^{-3}\, 	ext{kg/m}}} \approx 181\, 	ext{Hz}$

Therefore, the best choice is C, 181 Hz.

Question 6

The vertical displacement y(x,t)of a horizontal string aligned along the x-axis is given by the equation y(x,t)= (6.00 mm)cos[(3.25 m^-1)x - (7.22 s^-1)t].What are the (a)speed and (b)amplitude of this wave?

Accepted Answer

The answer of The vertical displacement y(x,t)of a horizontal string...

Question 7

Find the speed of an ocean wave whose displacement is given by the equation $y = 3.7 \cos ( 2.2 x - 5.6 t )$ where x and y are in meters and t is in seconds.

Accepted Answer

The speed of the wave can be determined by looking at the coefficient of the time variable $t$ in the wave equation. The general form of a wave equation is $y = A \cos(kx - \omega t)$, where $\omega$ is the angular frequency and relates to the wave speed $v$ through the equation $v = \omega / k$. In the given equation, $\omega = 5.6$ and $k = 2.2$, so the wave speed $v = 5.6 / 2.2 = 2.545$, which rounds to $2.5$ m/s.

Question 8

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 ($\mu = \frac{m}{L}$, with $m$ being the mass of the string).Given:- $f_3 = 600 \, \text{Hz}$ (since it's the third harmonic, $n = 3$)- $L = 0.39 \, \text{m}$- $m = 4.0 \, \text{g} = 0.004 \, \text{kg}$ (conversion from grams to kilograms)First, calculate the linear mass density ($\mu$):$\mu = \frac{m}{L} = \frac{0.004 \, \text{kg}}{0.39 \, \text{m}} \approx 0.010256 \, \text{kg/m}$Then, rearrange the formula to solve for $T$:$T = \mu \left(\frac{2Lf_n}{n}\right)^2$Plug in the values:$T = 0.010256 \left(\frac{2 \times 0.39 \times 600}{3}\right)^2 \approx 250 \, \text{N}$Therefore, the tension in the string is approximately 250 N.

Question 9

Standing waves of frequency 17 Hz are produced on a string that has a mass per unit length of $$0.0160 \mathrm {~kg} / \mathrm { m } .$$ 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

The answer of Standing waves of frequency 17 Hz are...

Question 10

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

Accepted Answer

The answer of A string that is 2.0 meters long...

Question 11

The vertical displacement y(x,t)of a horizontal string aligned along the x-axis is given by the equation y(x,t)= (6.00 mm)cos[(3.25 m^-1)x - (7.22 s^-1)t].What are the (a)wavelength and (b)period of this wave?

Accepted Answer

The answer of The vertical displacement y(x,t)of a horizontal string...

Question 12

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

Accepted Answer

The answer of A standing wave is oscillating at 950...

Question 13

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

The frequency of the second harmonic (n=2) in a stretched string can be calculated using the formula $f_n = \frac{n}{2L} \sqrt{\frac{T}{\mu}}$, where $f_n$ is the frequency of the nth harmonic, $L$ is the length of the string, $T$ is the tension in the string, and $\mu$ is the mass per unit length of the string. Given $L = 12$ m, $T = 43$ N, and the mass of the string is $25$ g ($0.025$ kg), the mass per unit length $\mu = \frac{0.025}{12}$ kg/m. Plugging these values into the formula for the second harmonic ($n=2$) gives a frequency of approximately $12$ Hz.

Question 14

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

The answer of A string of length 2.5 m is...

Question 15

What is the wave speed in a brass wire with a radius of 0.500 mm stretched with a tension of 125 N? The density of brass is 8.60 × 10³ kg/m³.

Accepted Answer

The wave speed in a stretched wire can be calculated using the equation:
v = sqrt(Tension/linear density)
where,
Tension = 125 N 
linear density = (density of brass) * (cross-sectional area)
cross-sectional area = pi * r^2 = pi * (0.5 x 10^-3)^2 = 7.85 x 10^-7 m^2
linear density = (8.60 x 10^3 kg/m^3) * (7.85 x 10^-7 m^2) = 0.00675 kg/m
v = sqrt(125/0.00675) = 136 m/s
Therefore, the answer is C.

Question 16

What are the wavelength (in meters)and frequency (in hertz)of a wave whose displacement is given by the equation y = 0.5 sin(0.20x + 120t),where x and y are in meters and t is in seconds?

Accepted Answer

The answer of What are the wavelength (in meters)and frequency...

Question 17

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

The answer of Find the first three harmonics of a...

Question 18

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 calculate the new length of the string, we can use the formula: 
new length = original length * original frequency / new frequency 
Plugging in the values, we get: 
new length = 16 cm * 440 Hz / 523 Hz 
new length = 13.5 cm 
So, we need to shorten the string by: 
16 cm - 13.5 cm = 2.5 cm 
Therefore, the answer is A) 2.5 cm.

Question 19

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 wavelength of the standing wave in the string can be determined by understanding that in the third harmonic, the string supports three half wavelengths. Therefore, the length of the string is equal to 3/2 of the wavelength. Given the length of the string is 0.36 m, we can set up the equation: 0.36 m = 3/2 * wavelength. Solving for the wavelength gives us 0.24 m.

Question 20

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 answer of A string of linear density 1.5 g/m...

Quiz 21: Mechanical Waves

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?

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 0.588-m string is tightly clamped at both ends.If the lowest standing wave frequency of the string is $326 \mathrm {~Hz}$ how fast do waves travel on this string?

A guitar string 0.65 m long has a tension of 61 N and a mass per unit length of 3.0 g/m. (a)What is the speed of waves on the string when it is plucked? (b)What is the string's fundamental frequency of vibration when plucked? (c)At what other frequencies will this string vibrate?

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.

The vertical displacement y(x,t)of a horizontal string aligned along the x-axis is given by the equation y(x,t)= (6.00 mm)cos[(3.25 m^-1)x - (7.22 s^-1)t].What are the (a)speed and (b)amplitude of this wave?

Find the speed of an ocean wave whose displacement is given by the equation $y = 3.7 \cos ( 2.2 x - 5.6 t )$ where x and y are in meters and t is in seconds.

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?

Standing waves of frequency 17 Hz are produced on a string that has a mass per unit length of $0.0160 \mathrm {~kg} / \mathrm { m } .$ 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?

The vertical displacement y(x,t)of a horizontal string aligned along the x-axis is given by the equation y(x,t)= (6.00 mm)cos[(3.25 m^-1)x - (7.22 s^-1)t].What are the (a)wavelength and (b)period of this wave?

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

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?

What is the wave speed in a brass wire with a radius of 0.500 mm stretched with a tension of 125 N? The density of brass is 8.60 × 10³ kg/m³.

What are the wavelength (in meters) and frequency (in hertz) of a wave whose displacement is given by the equation y = 0.5 sin(0.20x + 120t) ,where x and y are in meters and t is in seconds?

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 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) ?

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?

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?

Quiz 21: Mechanical Waves

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?

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 0.588-m string is tightly clamped at both ends.If the lowest standing wave frequency of the string is 326 Hz326 \mathrm {~Hz}326 Hz how fast do waves travel on this string?

A guitar string 0.65 m long has a tension of 61 N and a mass per unit length of 3.0 g/m. (a)What is the speed of waves on the string when it is plucked? (b)What is the string's fundamental frequency of vibration when plucked? (c)At what other frequencies will this string vibrate?

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.

The vertical displacement y(x,t)of a horizontal string aligned along the x-axis is given by the equation y(x,t)= (6.00 mm)cos[(3.25 m-1)x - (7.22 s-1)t].What are the (a)speed and (b)amplitude of this wave?

Find the speed of an ocean wave whose displacement is given by the equation y=3.7cos⁡(2.2x−5.6t) y = 3.7 \cos ( 2.2 x - 5.6 t ) y=3.7cos(2.2x−5.6t) where x and y are in meters and t is in seconds.

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?

Standing waves of frequency 17 Hz are produced on a string that has a mass per unit length of 0.0160 kg/m.0.0160 \mathrm {~kg} / \mathrm { m } .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?

A string that is 2.0 meters long is fixed at both ends and tightened until the wave speed is 18 m/s18 \mathrm {~m} / \mathrm { s }18 m/s What is the frequency of the standing wave shown in the figure?

The vertical displacement y(x,t)of a horizontal string aligned along the x-axis is given by the equation y(x,t)= (6.00 mm)cos[(3.25 m-1)x - (7.22 s-1)t].What are the (a)wavelength and (b)period of this wave?

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

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?

What is the wave speed in a brass wire with a radius of 0.500 mm stretched with a tension of 125 N? The density of brass is 8.60 × 103 kg/m3.

What are the wavelength (in meters) and frequency (in hertz) of a wave whose displacement is given by the equation y = 0.5 sin(0.20x + 120t) ,where x and y are in meters and t is in seconds?

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 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) ?

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?

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?

A 0.588-m string is tightly clamped at both ends.If the lowest standing wave frequency of the string is $326 \mathrm {~Hz}$ how fast do waves travel on this string?

The vertical displacement y(x,t)of a horizontal string aligned along the x-axis is given by the equation y(x,t)= (6.00 mm)cos[(3.25 m^-1)x - (7.22 s^-1)t].What are the (a)speed and (b)amplitude of this wave?

Find the speed of an ocean wave whose displacement is given by the equation $y = 3.7 \cos ( 2.2 x - 5.6 t )$ where x and y are in meters and t is in seconds.

Standing waves of frequency 17 Hz are produced on a string that has a mass per unit length of $0.0160 \mathrm {~kg} / \mathrm { m } .$ 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?

The vertical displacement y(x,t)of a horizontal string aligned along the x-axis is given by the equation y(x,t)= (6.00 mm)cos[(3.25 m^-1)x - (7.22 s^-1)t].What are the (a)wavelength and (b)period of this wave?

What is the wave speed in a brass wire with a radius of 0.500 mm stretched with a tension of 125 N? The density of brass is 8.60 × 10³ kg/m³.