A violin string has a length of 0.350 m and is tuned to concert G, with f = 392 Hz.
Where must the violinist place her finger to play concert A, with f = 430 Hz?
If this position is to remain correct to half the width of a finger (that is, to within 0.600 cm), what is the maximum allowable percentage change in the string tension?
I have no idea how to solve this problem or even the equations involved with it. It is due in a few hours. Any help would be much appreciated! Thank you!
To find the position where the violinist must place her finger to play concert A, we can use the equation that relates the frequency of a vibrating string to its length. This equation is known as the wave equation:
f = v / λ
where:
- f is the frequency of the wave
- v is the velocity of the wave, which is equal to the speed of sound (approximately 343 m/s for air)
- λ is the wavelength of the wave, which is twice the length of the vibrating segment of the string (since each side of the segment vibrates in opposite phase)
We can rearrange this equation to solve for the wavelength:
λ = v / f
Now, let's solve for the wavelength of concert G first:
λ_G = 343 m/s / 392 Hz = 0.875 m
Since concert A has a frequency of 430 Hz, we can find the corresponding wavelength:
λ_A = 343 m/s / 430 Hz = 0.797 m
The length of the vibrating segment of the string for concert G is 0.350 m. To find the position where the violinist should place her finger to play concert A, we need to determine the difference in string length between concerts G and A. We can do this by subtracting the new wavelength from the original length:
ΔL = λ_G / 2 - λ_A / 2
ΔL = (0.875 m - 0.797 m) / 2 = 0.039 m
Therefore, the violinist should place her finger approximately 0.039 m from the bridge to play concert A.
Now let's move on to finding the maximum allowable percentage change in string tension. To do this, we need to use the equation for the tension in a string:
T = μ * f^2 * L
where:
- T is the tension in the string
- μ is the linear mass density of the string (mass per unit length)
- f is the frequency of the string
- L is the length of the vibrating segment
Since we are only concerned with the change in tension, we can ignore μ and L, which remain constant. Thus, the equation becomes:
T ∝ f^2
where ∝ denotes proportionality.
We can now find the ratio of the tensions between concerts A and G:
T_A / T_G = (f_A / f_G)^2
T_A / T_G = (430 Hz / 392 Hz)^2 = 1.098
In order to find the maximum allowable percentage change in string tension, we need to determine the difference from unity and convert it to a percentage:
ΔT = |T_A / T_G - 1| = |1.098 - 1| = 0.098
Max allowable percentage change = ΔT * 100% = 0.098 * 100% = 9.8%
Therefore, the maximum allowable percentage change in string tension is 9.8%.