Question

In: Physics

Parasaurolophus was a dinosaur whose distinguishing feature was a hollow crest on the head. The 1.4...

Parasaurolophus was a dinosaur whose distinguishing feature was a hollow crest on the head. The 1.4 m long hollow tube in the crest had connections to the nose and throat, leading some investigators to hypothesize that the tube was a resonant chamber for vocalization

If you model the tube as an open-closed system, what are the first three resonant frequencies? (Use 350 m/s for the speed of sound.)

Enter your answers using two significant figures in ascending order separated by commas.
f = _______Hz, ________Hz, ________Hz


Solutions

Expert Solution

Concepts and reason

The concepts required to solve the given problem are frequency, wavelength, and velocity.

Initially, derive the expressions for the first three resonant frequencies by using the relation between velocity, frequency, and wavelength. Finally, calculate the first three resonant frequencies by using the expressions of the frequencies of open closed pipe.

Fundamentals

The lowest frequency which is produced by the oscillation of an object is called as fundamental frequency.

The fundamental frequency of open closed pipe is,

f=v4Lf = \frac{v}{{4L}}

Here, f is the frequency of sound, v is the velocity of sound, and L is the length of the tube.

The velocity of the sound in air is,

v=fλv = f\lambda

Here, λ\lambda is the wavelength.

For the first harmonic of the closed pipe, the length of the tube is same as the length of ¼ wavelength.

L=λ4λ=4L\begin{array}{c}\\L = \frac{\lambda }{4}\\\\\lambda = 4L\\\end{array}

The velocity of the sound in air is,

v=f1λv = {f_1}\lambda

Here, f1{f_1} is the first resonant frequency.

Substitute 4L for λ\lambda .

v=f1(4L)f1=v4L\begin{array}{c}\\v = {f_1}\left( {4L} \right)\\\\{f_1} = \frac{v}{{4L}}\\\end{array}

For the third harmonic of the closed end pipe, the length of the tube is,

L=34λλ=4L3\begin{array}{c}\\L = \frac{3}{4}\lambda \\\\\lambda = \frac{{4L}}{3}\\\end{array}

Since 3/4 of a wavelength fit into the tube.

The velocity of the sound in air is,

v=f2λv = {f_2}\lambda

Here, f2{f_2} is the second resonant frequency.

Substitute 4L3\frac{{4L}}{3} for λ\lambda .

v=f2(4L3)f2=3v4L\begin{array}{c}\\v = {f_2}\left( {\frac{{4L}}{3}} \right)\\\\{f_2} = \frac{{3v}}{{4L}}\\\end{array}

For the Fifth harmonic of the closed end pipe, the length of the tube is,

L=54λλ=4L5\begin{array}{c}\\L = \frac{5}{4}\lambda \\\\\lambda = \frac{{4L}}{5}\\\end{array}

Since 5/4 of a wavelength fit into the tube.

The velocity of the sound in air is,

v=f3λv = {f_3}\lambda

Here, f3{f_3} is the third resonant frequency.

Substitute 4L5\frac{{4L}}{5} for λ\lambda .

v=f3(4L5)f3=5v4L\begin{array}{c}\\v = {f_3}\left( {\frac{{4L}}{5}} \right)\\\\{f_3} = \frac{{5v}}{{4L}}\\\end{array}

The first resonant frequency is,

f1=v4L{f_1} = \frac{v}{{4L}}

Substitute 350 m/s for v and 1.4 m for L.

f1=350m/s4(1.4m)=62.5Hz\begin{array}{c}\\{f_1} = \frac{{350\;{\rm{m/s}}}}{{4\left( {1.4\;{\rm{m}}} \right)}}\\\\ = 62.5\;{\rm{Hz}}\\\end{array}

The second resonant frequency is,

f2=3v4L{f_2} = \frac{{3v}}{{4L}}

Substitute 350 m/s for v and 1.4 m for L.

f2=3(350m/s)4(1.4m)=187.5Hz\begin{array}{c}\\{f_2} = \frac{{3\left( {350\;{\rm{m/s}}} \right)}}{{4\left( {1.4\;{\rm{m}}} \right)}}\\\\ = 187.5\;{\rm{Hz}}\\\end{array}

The third resonant frequency is,

f3=5v4L{f_3} = \frac{{5v}}{{4L}}

Substitute 350 m/s for v and 1.4 m for L.

f3=5(350m/s)4(1.4m)=312.5Hz\begin{array}{c}\\{f_3} = \frac{{5\left( {350\;{\rm{m/s}}} \right)}}{{4\left( {1.4\;{\rm{m}}} \right)}}\\\\ = 312.5\;{\rm{Hz}}\\\end{array}

Ans:

The first three resonant frequencies are 62 Hz, 190 Hz, and 310 Hz.


Related Solutions

(1) Parasaurolophus was a dinosaur whose distinguishing feature was a hollow crest on the head. The...
(1) Parasaurolophus was a dinosaur whose distinguishing feature was a hollow crest on the head. The 1.2m long hollow tube in the crest had connections to the nose and throat, leading some investigators to hypothesize that the tube was a resonant chamber for vocalization. (Figure 1) If you model the tube as an open-closed system, what are the first three resonant frequencies? (Use 350 m/s for the speed of sound.) Enter your answers using two significant figures in ascending order...
6: The key distinguishing feature of the ________________ is that securities of up to a year...
6: The key distinguishing feature of the ________________ is that securities of up to a year in maturity are traded there.      a. money market      b. primary market      c. capital market      d. secondary market 7: Which of the following would not be an indirect transfer?      a. M&I Bank takes in deposits from Bill and Teresa. M&I uses most of the money to make a loan to Josephine.      b. Northwestern Mutual Life Insurance sells a life insurance policy to Sharon. The premium...
explain some distinguishing feature of non parametric tests?
explain some distinguishing feature of non parametric tests?
The distinguishing feature of a corporation is that A. Provides limited liability only to small shareholders...
The distinguishing feature of a corporation is that A. Provides limited liability only to small shareholders B. There is no legal difference between the corporation and its owners C. It spreads liability for its corporate obligations to all shareholders D. It is a legally defined, artificial being, separate from its owners
The distinguishing feature of international business is that international firms operate in environments that are highly...
The distinguishing feature of international business is that international firms operate in environments that are highly uncertain and where the rules of the game are often ambiguous, contradictory, and subject to rapid change, as compared to the domestic environment. It is crucial for businesses to understand these differences and find ways to hurdle the barriers these differences present in a socially responsible manner. TASK: Select any organization of your choice. Carry out a research on the following aspects and present...
Rhodesian Ridgebacks are known for the distinguishing feature of the ridge of hair running along their...
Rhodesian Ridgebacks are known for the distinguishing feature of the ridge of hair running along their back in the opposite direction from the rest of the coat. Recent studies have shown this to be due to an autosomal dominant mutation in the "Ridge" gene (i.e., those homozygous recessive for the mutation are ridgeless). You have two ridged dogs (Dog 1 and Dog 2) that you have decided to breed, but you are uncertain of their exact genotypes. 1) After breeding...
One distinguishing feature of organizations in the professional service industry is the difficulty in measuring the quantity and quality of their output
  11- One distinguishing feature of organizations in the professional service industry is the difficulty in measuring the quantity and quality of their output True False 12- Effective cash management is not an objective of the financing process True False 13- The human resources management process is the same thing as the payroll process True False
4. List 5 examples of tropical fruits, including their family, Latin binomial, and a distinguishing feature.
4. List 5 examples of tropical fruits, including their family, Latin binomial, and a distinguishing feature.
A hollow ball of diameter D is suspended from a string of negligible mass whose length...
A hollow ball of diameter D is suspended from a string of negligible mass whose length is equal to the ball’s diameter. The string is attached to the surface of the ball. Find an expression for the period of this physical pendulum in the small-amplitude approximation.
A hollow ball of diameter D is suspended from a string of negligible mass whose length...
A hollow ball of diameter D is suspended from a string of negligible mass whose length is equal to the ball’s diameter. The string is attached to the surface of the ball. Find an expression for the period of this physical pendulum in the small-amplitude approximation.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT