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Answer :
Wavelength of standing wave: 0.8 m
Tension in the string: 480 N
Finger placement for 650 Hz: approximately 30.8 cm from the bridge.
To solve the student's question regarding a violin string, let's break it down into manageable parts:
1. Finding the Wavelength of the Standing Wave
When a string vibrates in its fundamental mode, the wavelength ( [tex]\lambda[/tex]) can be determined using the formula:
[tex]\lambda = 2L[/tex]
where [tex]L[/tex] is the length of the string. Given that the length of the string is 40 cm (or 0.4 m), we can calculate:
[tex]\lambda = 2 \times 0.4 \, \text{m} = 0.8 \, \text{m}[/tex]
2. Finding the Tension in the String
Next, we need to find the tension in the string ( [tex]T[/tex]). We use the formula that connects frequency ( [tex]f[/tex]), wavelength ( [tex]\lambda[/tex]), mass ( [tex]m[/tex]), and tension:
[tex]v = f \times \lambda[/tex]
where [tex]v[/tex] is the speed of the wave on the string and is also given by:
[tex]v = \sqrt{\frac{T}{\mu}}[/tex]
and [tex]\mu[/tex] (linear mass density) is calculated as follows:
[tex]\mu = \frac{m}{L} = \frac{0.0012 \, \text{kg}}{0.4 \, \text{m}} = 0.003 \, \text{kg/m}[/tex]
Now, we can find the wave speed using the frequency (500 Hz) and the wavelength (0.8 m):
[tex]v = 500 \, \text{Hz} \times 0.8 \, \text{m} = 400 \, \text{m/s}[/tex]
Next, we can use the wave speed to find the tension:
[tex]v = \sqrt{\frac{T}{\mu}}[/tex]
[tex]400 = \sqrt{\frac{T}{0.003}}[/tex]
Squaring both sides gives us:
[tex]160000 = \frac{T}{0.003}[/tex]
Thus, multiplying by 0.003 gives:
[tex]T = 160000 \, \text{N/m} \times 0.003 = 480 \text{N}[/tex]
3. Where to Place Your Finger to Increase Frequency to 650 Hz
To increase the frequency from 500 Hz to 650 Hz, we can find the ratio of the new frequency to the old frequency:
[tex]\frac{f_{new}}{f_{old}} = \frac{650 \, \text{Hz}}{500 \, \text{Hz}} = 1.3[/tex]
This indicates that the frequency is increased by a factor of 1.3. The frequency of a string can be altered by shortening its effective vibrating length. The new length required can be found from:
[tex]\text{New Length} = \frac{L_{old}}{1.3} \approx \frac{0.4 \, \text{m}}{1.3} \approx 0.308 \, \text{m} \approx 30.8 \, \text{cm}[/tex]
So, you should place your finger about 30.8 cm from the bridge of the violin to decrease its effective length and increase the frequency to 650 Hz.
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