Question

The song Baby Shark has taken the world by storm, including Dr. Link's living room during his kids' daily dance parties. In the song, the melody of three notes, D (293.5 Hz), E (329.5 Hz), and G (392 Hz) are played repeatedly. Dr. Link decides to surprise his girls by learning to play it on his daughter' s Ukulele (a small guitar) but he needs your help. To play the note, Dr. Link needs to determine the tension in the G-string on the Ukulele. The nylon string on the Ukulele has total length 50 cm and total mass of 120 milligrams but only 34 cm of the string vibrates in its first harmonic.

a. Determine the tension in the G-string needed to play the G note.

b. Mrs Link wants to also learn how the play the song but on another identical Ukulele. When she plucks the G-string at the same time Dr. Link plucks his Ukulele string, it creates a beat frequency of 4 Hz. When she tightens the tension in the string, the beat frequency decreases. What is the original frequency of her Ukulele string?

c. Without tuning the Ukulele, Mrs. Link wants to know if it possible to play the D note. Please use the physics you know to explain if this is possible and if it is, tell her the length of the string that needs to vibrate. Recall that the longest the string can be that vibrates is the distance is 34 cm which is the unfingered length of the string.

d. If the G string somehow vibrated at its 5th harmonic, what would the wavelength of the standing wave?

Answer 1

Linear mass density of the string,

length of the string that can vibrate, l = 34 cm = 0.34 m

(a) Frequency of G note,

If T = tension in the string

(b) Beat frequency,  

Therefore frequency of Ukulele string,  

By tightening the string tension increases and beat frequency decreases, it means original frequency of Ukulele string is less than 392 Hz.

Therefore

(c) Frequency of D note,

If l is the length that needs to vibrate

then using equation (1)

So to play the D note, the length of the string that needs to vibrate is 45 cm, so it is not possible to play the D note as the maximum length of string that can vibrate is only 34 cm.

(d) The string in this case vibrated in its fifth harmonic, if is the wavelength, then

The wavelength of the wave in this case is 13.6 cm.