A string is stretched between fixed supports separated by 73.0 cm. It is observed to have resonant frequencies of 380 and 475 Hz, and no other resonant frequencies between these two.
What is the lowest resonant frequency for this string?
To find the speed of the string, we can use the formula:
v = λf
where:
v is the speed of the wave on the string,
λ is the wavelength of the wave, and
f is the frequency of the wave.
When the string is stretched between fixed supports, only specific resonant frequencies are observed. These frequencies are given by:
f = (n/2L)√(T/μ)
where:
n is the harmonic number,
L is the length of the string (distance between the fixed supports),
T is the tension in the string, and
μ is the linear mass density of the string.
Given:
Distance between the fixed supports (L) = 73.0 cm = 0.73 m
Resonant frequency (f1) = 380 Hz
Resonant frequency (f2) = 475 Hz
Using the formula, we can calculate the speed of the wave on the string.
Step 1: Calculate the harmonic number for the first resonant frequency (f1).
Let's assume n1 = 1 for the first harmonic (fundamental frequency).
Step 2: Calculate the linear mass density (μ).
We are not given the density or mass of the string, so we cannot determine the linear mass density at this point.
Step 3: Calculate the wavelength (λ1) for the first resonant frequency.
Using the relationship between wavelength and frequency, we have:
λ1 = v/f1
Step 4: Calculate the speed of the wave (v).
Using the derived equation above, we have:
v = λ1*f1
Step 5: Calculate the wavelength (λ2) for the second resonant frequency.
Similarly, we can calculate the wavelength for the second resonant frequency using the formula:
λ2 = v/f2
Step 6: Calculate the speed of the wave (v) again using the derived equation:
v = λ2*f2
Now, we can answer specific questions or perform the calculations mentioned above.
To find the speed of the wave on the string, we can start by using the formula for the resonant frequencies of a string:
f = (n/2L) * v
Where:
f = frequency of the resonant mode
n = harmonic number (1, 2, 3, ...)
L = length of the string
v = speed of the wave on the string
We are given two resonant frequencies, 380 Hz and 475 Hz. Let's calculate the corresponding values of f/(n/2L) for each frequency:
For f1 = 380 Hz:
f1/(n/2L) = 380 Hz / (n/2L)
Similarly, for f2 = 475 Hz:
f2/(n/2L) = 475 Hz / (n/2L)
Since there are no other resonant frequencies between these two, we can assume that the ratio f1/(n/2L) is equal to the ratio f2/(n/2L). In other words:
f1/(n/2L) = f2/(n/2L)
Now, we can combine the two equations and solve for the speed of the wave (v):
380 Hz / (n/2L) = 475 Hz / (n/2L)
With the n/2L terms canceling out, we get:
380 Hz = 475 Hz
This means that the speed of the wave (v) is the same for both frequencies. Therefore, we can conclude that the speed of the wave on the string is constant and independent of the frequency. However, we cannot determine the exact value of the speed without additional information.