How Does Water Leakage Affect the Frequency of a Vibrating Guitar String?

• chrisakatibs
In summary, the conversation discusses a bucket hanging from a steel guitar string and the changes that occur when the bucket springs a leak and water begins to drain out. The first part of the conversation discusses the resonant frequency of the string when the wind blows. The second part involves using various equations to determine the changes in wave speed, wavelength, and frequency as the water drains from the bucket. The final question asks about the time it takes for the resonant frequency to no longer be audible.
chrisakatibs

Homework Statement

A 210 g bucket containing 12.5 kg of water is hanging from a steel guitar string with mass m = 8.58 g, density p = 7750 kg/m3, and diameter d = 1.00 mm. A) When the wind blows, it causes the cord to vibrate at what resonant frequency? B) Suddenly the bucket springs a leak in the bottom such that water drops out at a steady rate of 2.00 g/s. At what rate are the (i) wave speed, (ii) wavelength of the fundamental mode of vibration, and (iii) frequency changing? C) How long will it take until the vibration is no longer audible (between 20 Hz and 20,000 Hz)?

Homework Equations

Many, wave equaitons, string tension etc.

The Attempt at a Solution

First parts are here to confirm, need help with last part, I believe it should be just plugging 20 Hz into an equation.

V=sqrt(T*L/mass)
T=mg=(12.5 0.210)*9.81=124.6851 N
L=volume /area =(0.00858/7750)*10^(6)/(3.14*0.5^2)=1.41031436m
a)V=sqrt(124.6851*1.41031436/0.00858)=143.15997 m/s
frequency=velocity /2L=143.15997/(2*1.41031)=50.754 Hz
B)i) dv/dt= -0.0196*sqrt(L/(Tm))=0.01968*sqrt(1.41031436/(124.6851*0.00858))=0.0225960m/s^2
wave speed change rate = 0.01129 m /s^2
ii)wavelength of the fundamental mode of vibration change=1/(2*50.754*124.6851)=0.00007901056 m/s
iii)frequency changing=0.01129 /(2*1.4103143)=0.00400265387

It helps if you explain your reasoning instead of just listing a bunch of equations.
The last question concerns the rate that the resonant frequency excited by the wind will change as water leaves the bucket.

in B - you appear to have the wavelength increasing with time
I think you should check your reasoning there - what sets the fundamental wavelength on a string fixed at both ends?

You left off the units in B(iii) too - the fundamental frequency appears to be increasing in time, as does the wave speed.
What is it about the string that changes as the bucket drains?

1. What is string tension and how does it affect waves?

String tension is the amount of force applied to a string, usually by tightening or loosening it. In terms of waves, string tension affects the frequency and wavelength of a wave. Higher string tension creates waves with a higher frequency and shorter wavelength, while lower string tension creates waves with a lower frequency and longer wavelength.

2. How is string tension measured?

String tension is commonly measured in units of force, such as pounds or newtons. It can also be measured indirectly by measuring the frequency of a wave produced by the string and using mathematical equations to calculate the tension.

3. What is the relationship between string tension and pitch?

The pitch of a sound produced by a string is directly related to its tension. As string tension increases, the pitch of the sound also increases. This is because higher tension results in higher frequencies, which our ears perceive as a higher pitch.

4. Can string tension affect the tone or quality of a sound?

Yes, string tension can play a significant role in the tone and quality of a sound produced by a string. Higher string tension can produce a brighter and more crisp sound, while lower string tension can create a softer and more mellow sound.

5. How does changing string tension affect the playing experience?

Changing string tension can greatly impact the playing experience for a musician. Higher string tension can make it more difficult to press down on the strings and may require more force to produce a sound. Lower string tension can make it easier to play, but may also result in a less responsive instrument.

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