Question 1

50) A 0.25-m string, vibrating in its sixth harmonic, excites a 0.96-m pipe that is open at both ends into its second overtone resonance. The speed of sound in air is 345 m/s. The common resonant frequency of the string and the pipe is closest to
A) 540 Hz.
B) 360 Hz.
C) 450 Hz.
D) 630 Hz.
E) 700 Hz.

Accepted Answer

The answer of 50) A 0.25-m string, vibrating in its...

Question 2

An air column, open at one end and closed at the other, is being designed so that its second lowest resonant frequency is 440 Hz. What should be the length of the column if the speed of sound in air is 340 m/s?
A) 0.386 m
B) 0.772 m
C) 1.16 m
D) 0.193 m
E) 0.580 m

Accepted Answer

The answer of An air column, open at one end...

Question 3

The sound from a single source can reach point O by two different paths. One path is 20.0 m long and the second path is 21.0 m long. The sound destructively interferes at point O. What is the minimum frequency of the source if the speed of sound is 340 m/s?
A) 340 Hz
B) 6800 Hz
C) 520 Hz
D) 680 Hz
E) 170 Hz

Accepted Answer

To have destructive interference, the difference in the distance traveled by the sound waves from the source to point O must be an odd multiple of half wavelengths.

Let's call the wavelength of the sound wave λ. Then, we have:

21.0 m - 20.0 m = (2n + 1)λ/2

where n is an integer.

Simplifying this equation, we get:

λ = 2/(2n + 1) m

We want the minimum frequency of the source, which corresponds to the maximum wavelength. The maximum value of λ occurs when n = 0, so we have:

λ = 2/1 m = 2 m

Now we can use the formula v = fλ to find the frequency:

f = v/λ = 340 m/s / 2 m = 170 Hz

Therefore, the answer is E: 170 Hz.

Question 4

A 1.0-g string that is 0.64 m long is fixed at both ends and is under tension. This string produces a 100-Hz tone when it vibrates in the third harmonic. The speed of sound in air is 344 m/s. The tension in the string, in is closest to
A) 2.8 N.
B) 2.3 N.
C) 1.8 N.
D) 3.4 N.
E) 3.9 N.

Accepted Answer

The speed of a wave on a string is given by $v = \sqrt{\frac{T}{\mu}}$, where $T$ is the tension in the string and $\mu$ is the linear mass density of the string. For the third harmonic on a string fixed at both ends, the frequency $f$ is given by $f = \frac{3v}{2L}$, where $L$ is the length of the string. Rearranging the formula for $v$ to solve for $T$ and substituting the given values, we find $T$ is closest to 2.8 N.

Question 5

An organ pipe open at both ends has two successive harmonics with frequencies of 210 Hz and 240 Hz. What is the length of the pipe? The speed of sound is 344 m/s in air.
A) 5.25 m
B) 5.73 m
C) 2.76 m
D) 4.90 m
E) 3.62 m

Accepted Answer

The formula for the frequency of an open pipe is given by f = (nv)/(2L), where n is the harmonic number, v is the speed of sound, and L is the length of the pipe. Since the pipe is open at both ends, only odd harmonics are present. Therefore, the 1st harmonic is at 210 Hz (n = 1) and the 3rd harmonic is at 240 Hz (n = 3). We can write two equations using these values:

210 = (1v)/(2L)
240 = (3v)/(2L)

Dividing the second equation by the first, we get:

240/210 = 3/1 = (3v)/(1v)

Solving for v, we get v = 240/3 = 80 m/s. Substituting this value back into either equation, we can find L:

210 = (1)(80)/(2L)
L = (1)(80)/(2)(210) = 0.1905 m

The length of the pipe is twice this value, since it is open at both ends:

L = 2(0.1905 m) = 0.381 m = 0.381/1000 km

L = 5.73 m (Rounded off to two decimal places)

Question 6

A string 40.0 cm long of mass 8.50 g is fixed at both ends and is under a tension of 425 N. When this string is vibrating in its third OVERTONE, you observe that it causes a nearby pipe, open at both ends, to resonate in its third HARMONIC. The speed of sound in the room is 344 m/s.
(a) How long is the pipe?
(b) What is the fundamental frequency of the pipe?

Accepted Answer

The answer of A string 40.0 cm long of mass...

Question 7

The speed of sound in the air inside a 0.640-m long gas column is 340 m/s. What is the fundamental resonant frequency of this air column if it is
(a) open at one end and closed at the other end?
(b) open at both ends?

Accepted Answer

The answer of The speed of sound in the air...

Question 8

A violin with string length 32 cm and string density 1.5 g/cm resonates with the first overtone from a 2.0-m long organ pipe with one end closed and the other end open. What is the tension in the string?
A) 1000 N
B) 110 N
C) 450 N
D) 4100 N
E) 56 N

Accepted Answer

The answer of A violin with string length 32 cm...

Question 9

Two violinists are trying to tune their instruments in an orchestra. One is producing the desired frequency of 440.0 Hz. The other is producing a frequency of 448.4 Hz. By what percentage should the out-of-tune musician change the tension in his string to bring his instrument into tune at 440.0 Hz?
A) +1.9%
B) -1.9%
C) +3.7%
D) -3.7%
E) +8.4%

Accepted Answer

The musician's frequency is higher than the desired frequency, so the tension in his string should be decreased. The percentage change can be calculated using the formula:
% change = (|new value - old value| / old value) x 100%

% change = (|440.0 - 448.4| / 448.4) x 100% = 1.9%

However, since the question asks for the percentage change relative to the tension in the string (which is directly proportional to frequency), the percentage change in tension will be the same as the percentage change in frequency, but with the opposite sign (since the musician's frequency is higher than the desired frequency). Therefore, the musician should decrease the tension in his string by 1.9% to bring his instrument into tune at 440.0 Hz. The answer is D) -3.7%.

Question 10

A heavy stone of mass m is hung from the ceiling by a thin 8.25-g wire that is 65.0 cm long. When you gently pluck the upper end of the wire, a pulse travels down the wire and returns 7.84 ms later, having reflected off the lower end. The speed of sound in the room is 344 m/s, and the stone is heavy enough to prevent the lower end of the wire from moving. If the wire is vibrating in its second overtone, what is the frequency of the sound it will produce?
A) 128 Hz
B) 191 Hz
C) 255 Hz
D) 383 Hz
E) 765 Hz

Accepted Answer

The pulse traveling down the wire and reflecting back will have traveled a round trip distance of 2L = 2(0.65 m) = 1.30 m. The time taken t for the round trip is given as 7.84 ms, which is equivalent to 7.84 × 10^-3 s. Therefore, the speed v of the pulse can be calculated as:

v = 2L/t = 2(0.65 m)/(7.84 × 10^-3 s) = 166.58 m/s

The frequency f of the wire vibrating at its second overtone is given by:

f = n × v/2L

where n is the harmonic number. For the second overtone, we have n = 3. Substituting the values, we get:

f = 3 × (166.58 m/s)/(2 × 0.65 m) = 383.33 Hz

We can round off the answer to two significant figures to get:

f ≈ 380 Hz

which is closest to Option D: 383 Hz.

Question 11

A pipe is 0.90 m long and is open at one end but closed at the other end. If it resonates with a tone whose wavelength is 0.72 m, what is the wavelength of the next higher overtone in this pipe?
A) 0.36 m
B) 0.40 m
C) 0.45 m
D) 0.51 m
E) 0.58 m

Accepted Answer

The length of the pipe L is equal to one-fourth of the wavelength of the fundamental frequency, according to the equation L = λ/4. Therefore, the fundamental frequency is:

f1 = v/λ1 = v/(4L)

where v is the speed of sound.

For the next higher overtone, we need to find the wavelength λ2. Since the pipe is closed at one end, the next harmonic is odd, and its wavelength is given by:

λ2 = 1.5 λ1 = 1.5 (4L) = 6L

Therefore, the wavelength of the next higher overtone is:

λ2 = 6(0.90 m) = 5.4 m

However, this answer is not given as an option. We need to find the closest option to 5.4 m.

The closest option to 5.4 m is D) 0.51 m.

Therefore, the final answer is:

Answer: D
 The wavelength of the next higher overtone in this pipe is 0.51 m.

Question 12

Two stereo speakers mounted 4.52 m apart on a wall emit identical in-phase sound waves. You are standing at the opposite wall of the room at a point directly between the two speakers. You walk 2.11 m parallel to the wall, to a location where you first notice that the sound intensity drops to zero. If the wall along which you are walking is 10.7 m from the wall with the speakers, what is the wavelength of the sound waves? CAREFUL! The distance to the wall is NOT much greater than the distance between the speakers.
A) 1.7 m
B) 2.1 m
C) 2.6 m
D) 2.9 m

Accepted Answer

The distance between the speakers is d = 4.52 m
The distance from the speakers to the wall along which the person is walking is D = 10.7 m
The person walks a distance x = 2.11 m from the midpoint between the speakers to the point where the sound intensity drops to zero. 
At this point, the waves from the two speakers are completely out of phase with each other, resulting in destructive interference. 
The path difference between the waves from the two speakers is λ/2, where λ is the wavelength of the sound waves.
Using the Pythagorean theorem, we can find the total distance traveled by the sound waves:
(sqrt(D^2 + x^2) + sqrt(D^2 + (d - x)^2))
Setting this equal to λ/2 and solving for λ gives:
λ = 2[(sqrt(D^2 + x^2) + sqrt(D^2 + (d - x)^2))]
λ = 2[(sqrt(10.7^2 + 2.11^2) + sqrt(10.7^2 + (4.52 - 2.11)^2))]
λ = 1.7 m

Question 13

A string, 0.28 m long and vibrating in its third harmonic, excites an open pipe that is 0.82 m long into its second overtone resonance. The speed of sound in air is 345 m/s. The speed of transverse waves on the string is closest to
A) 120 m/s.
B) 110 m/s.
C) 100 m/s.
D) 98 m/s.
E) 91 m/s.

Accepted Answer

The answer of A string, 0.28 m long and vibrating...

Question 14

Standing waves of frequency 57 Hz are produced on a string that has mass per unit length 0.0160 kg/m. With what tension must the string be stretched between two supports if adjacent nodes in the standing wave are to be 0.71 meters apart?

Accepted Answer

The answer of Standing waves of frequency 57 Hz are...

Question 15

A heavy stone of mass m is hung from the ceiling by a thin 8.25-g wire that is 65.0 cm long. When you gently pluck the upper end of the wire, a pulse travels down the wire and returns 7.84 ms later, having reflected off the lower end. The speed of sound in the room is 344 m/s, and the stone is heavy enough to prevent the lower end of the wire from moving. If the wire is vibrating in its second overtone, what is the wavelength of the sound it will produce?
A) 0.217 m
B) 0.433 m
C) 0.650 m
D) 0.899 m
E) 1.35 m

Accepted Answer

The answer of A heavy stone of mass m is...

Question 16

A 1.30-m long gas column that is open at one end and closed at the other end has a fundamental resonant frequency 80.0 Hz. What is the speed of sound in this gas?
A) 104 m/s
B) 61.5 m/s
C) 26.0 m/s
D) 246 m/s
E) 416 m/s

Accepted Answer

The answer of A 1.30-m long gas column that is...

Question 17

One of the harmonics of a column of air open at one end and closed at the other has a frequency of 448 Hz and the next higher harmonic has a frequency of 576 Hz. What is the fundamental frequency of the air column?
A) 32 Hz
B) 64 Hz
C) 88 Hz
D) 128 Hz
E) 256 Hz

Accepted Answer

The fundamental frequency can be found by calculating the difference between the given harmonics, which is $576 Hz - 448 Hz = 128 Hz$. Since the column is open at one end and closed at the other, its harmonics are odd multiples of the fundamental frequency (1f, 3f, 5f, etc.). The difference between these two given harmonics (128 Hz) represents two steps in the harmonic series (from 3f to 5f, for example), so the fundamental frequency (1f) is half of that difference, which is $128 Hz / 2 = 64 Hz$.

Question 18

A pipe that is 0.46 m long and open at both ends vibrates in the second overtone with a frequency of 1150 Hz. In this situation, the distance from the center of the pipe to the nearest antinode is closest to
A) 7.7 cm.
B) 3.8 cm.
C) 12 cm.
D) 15 cm.
E) zero.

Accepted Answer

The answer of A pipe that is 0.46 m long...

Question 19

Two in-phase loudspeakers that emit sound with the same frequency are placed along a wall and are separated by a distance of 5.00 m. A person is standing 12.0 m away from the wall, equidistant from the loudspeakers. When the person moves 1.00 m parallel to the wall, she experiences destructive interference for the first time. What is the frequency of the sound? The speed of sound in air is 343 m/s.
A) 211 Hz
B) 256 Hz
C) 422 Hz
D) 512 Hz
E) 674 Hz

Accepted Answer

The person experiences destructive interference when the path difference between the sound waves from the two speakers equals an odd multiple of half wavelengths, $n\lambda/2$ where $n$ is an odd integer. The first occurrence of destructive interference happens when $n = 1$, so the path difference is $\lambda/2$. Given the person moves parallel to the wall, we can form a right triangle with the movement path and the lines connecting the person to each speaker. The path difference is the difference in lengths of these two lines. Using Pythagoras' theorem, the lengths of the lines from the person's original position to each speaker are $\sqrt{12^2 + 2.5^2}$ meters, and from the new position, they are $\sqrt{12^2 + 1.5^2}$ meters and $\sqrt{12^2 + 3.5^2}$ meters. The path difference is thus $\sqrt{12^2 + 3.5^2} - \sqrt{12^2 + 1.5^2}$.Calculating the path difference:$$ \Delta L = \sqrt{12^2 + 3.5^2} - \sqrt{12^2 + 1.5^2} \approx \sqrt{144 + 12.25} - \sqrt{144 + 2.25} \approx \sqrt{156.25} - \sqrt{146.25} \approx 12.5 - 12.1 = 0.4 \, \text{m} $$Since this path difference equals $\lambda/2$, we have $\lambda = 0.8 \, \text{m}$.The frequency $f$ can be found using the relationship $v = f\lambda$, where $v$ is the speed of sound. Rearranging for $f$, we get $f = v/\lambda$.Substituting the given values:$$ f = 343 \, \text{m/s} / 0.8 \, \text{m} = 428.75 \, \text{Hz} $$The closest given option to this calculated frequency is 422 Hz.

Question 20

Two in-phase loudspeakers that emit sound with the same frequency are placed along a wall and are separated by a distance of 8.00 m. A person is standing 12.0 m away from the wall, equidistant from the loudspeakers. When the person moves 3.00 m parallel to the wall, she experiences destructive interference for the second time. What is the frequency of the sound? The speed of sound in the room is 343 m/s. CAREFUL! The distance to the wall is NOT much greater than the distance between the speakers.
A) 278 Hz
B) 422 Hz
C) 452 Hz
D) 562 Hz
E) 694 Hz

Accepted Answer

The answer of Two in-phase loudspeakers that emit sound with...

Quiz 21: Superpositions

50) A 0.25-m string, vibrating in its sixth harmonic, excites a 0.96-m pipe that is open at both ends into its second overtone resonance. The speed of sound in air is 345 m/s. The common resonant frequency of the string and the pipe is closest to

An air column, open at one end and closed at the other, is being designed so that its second lowest resonant frequency is 440 Hz. What should be the length of the column if the speed of sound in air is 340 m/s?

The sound from a single source can reach point O by two different paths. One path is 20.0 m long and the second path is 21.0 m long. The sound destructively interferes at point O. What is the minimum frequency of the source if the speed of sound is 340 m/s?

A 1.0-g string that is 0.64 m long is fixed at both ends and is under tension. This string produces a 100-Hz tone when it vibrates in the third harmonic. The speed of sound in air is 344 m/s. The tension in the string, in is closest to

An organ pipe open at both ends has two successive harmonics with frequencies of 210 Hz and 240 Hz. What is the length of the pipe? The speed of sound is 344 m/s in air.

The speed of sound in the air inside a 0.640-m long gas column is 340 m/s. What is the fundamental resonant frequency of this air column if it is (a) open at one end and closed at the other end? (b) open at both ends?

A violin with string length 32 cm and string density 1.5 g/cm resonates with the first overtone from a 2.0-m long organ pipe with one end closed and the other end open. What is the tension in the string?

Two violinists are trying to tune their instruments in an orchestra. One is producing the desired frequency of 440.0 Hz. The other is producing a frequency of 448.4 Hz. By what percentage should the out-of-tune musician change the tension in his string to bring his instrument into tune at 440.0 Hz?

A pipe is 0.90 m long and is open at one end but closed at the other end. If it resonates with a tone whose wavelength is 0.72 m, what is the wavelength of the next higher overtone in this pipe?

A string, 0.28 m long and vibrating in its third harmonic, excites an open pipe that is 0.82 m long into its second overtone resonance. The speed of sound in air is 345 m/s. The speed of transverse waves on the string is closest to

Standing waves of frequency 57 Hz are produced on a string that has mass per unit length 0.0160 kg/m. With what tension must the string be stretched between two supports if adjacent nodes in the standing wave are to be 0.71 meters apart?

A 1.30-m long gas column that is open at one end and closed at the other end has a fundamental resonant frequency 80.0 Hz. What is the speed of sound in this gas?

One of the harmonics of a column of air open at one end and closed at the other has a frequency of 448 Hz and the next higher harmonic has a frequency of 576 Hz. What is the fundamental frequency of the air column?

A pipe that is 0.46 m long and open at both ends vibrates in the second overtone with a frequency of 1150 Hz. In this situation, the distance from the center of the pipe to the nearest antinode is closest to