The portion of string between the bridge and upper end of the fingerboard (the part of the string that is free to vibrate) of a certain musical instrument is 60.0 cm long [.6m] and has a mass of 2.23 g [.00223kg]. The string sounds an A4 note (440 Hz) when played.
Where must the player put a finger (at what distance from the bridge) to play a D5 note (587 Hz)? For both notes, the string vibrates in its fundamental mode.
x= ?? cm
Without retuning, is it possible to play a G4 note (392 Hz) on this string?
To find the distance from the bridge where the player must put a finger to play a D5 note, we can use the formula for the speed of a wave on a string:
v = f * λ
where:
- v is the speed of the wave
- f is the frequency of the wave
- λ is the wavelength of the wave
In the fundamental mode of vibration, the wavelength can be calculated as twice the length of the string:
λ = 2 * L
where:
- L is the length of the string between the bridge and the upper end of the fingerboard
Let's calculate the wavelength for the A4 note (440 Hz):
λ_A4 = 2 * 0.6 m = 1.2 m
Now, we can calculate the speed of the wave for the A4 note:
v_A4 = f_A4 * λ_A4
= 440 Hz * 1.2 m
= 528 m/s
Next, we can calculate the wavelength for the D5 note (587 Hz) using the speed of the wave for the A4 note:
λ_D5 = v_A4 / f_D5
= 528 m/s / 587 Hz
≈ 0.899 m
Finally, to find the distance from the bridge where the player must put a finger to play a D5 note, we can divide the wavelength by 2:
x = λ_D5 / 2
≈ 0.899 m / 2
≈ 0.45 m
Therefore, the player needs to put a finger approximately 0.45 meters from the bridge to play a D5 note.
Now, to determine if it is possible to play a G4 note (392 Hz) on this string without retuning, we can follow a similar process.
Using the same equation for wavelength:
λ_G4 = v_A4 / f_G4
= 528 m/s / 392 Hz
≈ 1.346 m
Dividing the wavelength by 2:
x_G4 = λ_G4 / 2
≈ 1.346 m / 2
≈ 0.673 m
Therefore, the player needs to put a finger approximately 0.673 meters from the bridge to play a G4 note. Since this distance is greater than the length of the string (0.6 meters), it is not possible to play a G4 note on this string without retuning.