A stretched string fixed at both ends is 2.0 m long. What are the three wavelengths that will produce standing waves on this string? Name at least one wavelength that would not produce a standing wave pattern, and explain your answer.
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To find the wavelengths that will produce standing waves on a stretched string, we need to understand the concept of standing waves first. A standing wave is formed when two waves of the same frequency traveling in opposite directions superpose (or combine) and create a pattern where certain points appear to be standing still.
In the case of a stretched string fixed at both ends, the standing waves are formed by the superposition of the incident and reflected waves. For a standing wave to occur, the length of the string needs to be an integer multiple of the half-wavelength.
To find the possible wavelengths, we can use the formula:
λ = 2L/n
Where:
λ = wavelength
L = length of the string (in this case, 2 meters)
n = integer (1, 2, 3, ...)
Let's calculate the three possible wavelengths:
For n = 1: λ = 2(2)/1 = 4 meters
For n = 2: λ = 2(2)/2 = 2 meters
For n = 3: λ = 2(2)/3 ≈ 1.33 meters
Therefore, the three possible wavelengths that will produce standing waves on this string are 4 meters, 2 meters, and approximately 1.33 meters.
Now, to answer the second part of your question, let's consider a wavelength that would not produce a standing wave pattern. If the wavelength is such that it doesn't satisfy the condition of being an integer multiple of the half-wavelength, it won't produce a standing wave.
For example, let's consider a wavelength of 3 meters. Since 3 meters is not an integer multiple of the half-wavelength (1 meter in this case), it will not produce a standing wave pattern.
I hope this explanation clarifies your understanding! Let me know if you have any further questions.
To find the three wavelengths that will produce standing waves on the string, we can use the formula:
λ = 2L/n
where λ is the wavelength, L is the length of the string, and n is the harmonic number.
Given that the string is 2.0 m long, we have:
λ = 2(2.0m)/n
= 4.0m/n
Now, let's calculate the wavelengths for the first three harmonics (n = 1, 2, 3):
For n = 1:
λ1 = 4.0m/1
= 4.0m
For n = 2:
λ2 = 4.0m/2
= 2.0m
For n = 3:
λ3 = 4.0m/3
= 1.333m
So, the three wavelengths that will produce standing waves on this string are 4.0 m, 2.0 m, and 1.333 m.
As for a wavelength that would not produce a standing wave pattern, any wavelength that does not satisfy the condition of λ = 2L/n will not produce a standing wave. For example, if we consider a wavelength of 3.0 m, it will not produce a standing wave because it does not satisfy the equation.
Therefore, 3.0 m is an example of a wavelength that would not produce a standing wave pattern on this string.
There are more than three wavelengths that will produce a standing wave.
Any wavelength that will fit an integral number of half-waves between the endpoints can produce a standing wave pattern.
Is this supposed to be a multiple choice question?