14. Which of the following wavelengths will produce standing waves on a string that is 3.5 m long?
a. 1.75 m
b. 3.5 m
c. 5.0 m
d. 7.0 m
To determine which wavelength will produce standing waves on a string that is 3.5 m long, you need to find the possible wavelengths that fit into the length of the string.
The general formula for the possible wavelengths on a string is:
λ = 2L / n
Where:
λ is the wavelength
L is the length of the string
n is an integer representing the number of nodes or half-wavelengths
For standing waves, the first harmonic (also called the fundamental frequency) has only one-half of a wavelength, giving n = 1. Therefore, the formula simplifies to:
λ = 2L
Now, let's calculate the wavelength for each option:
a. 1.75 m:
λ = 2(3.5 m) = 7 m
This wavelength does not match the length of the string, so it does not produce standing waves.
b. 3.5 m:
λ = 2(3.5 m) = 7 m
This wavelength matches the length of the string, so it produces standing waves.
c. 5.0 m:
λ = 2(3.5 m) = 7 m
This wavelength is longer than the length of the string, so it does not produce standing waves.
d. 7.0 m:
λ = 2(3.5 m) = 7 m
This wavelength matches the length of the string, so it produces standing waves.
In conclusion, options b (3.5 m) and d (7.0 m) will produce standing waves on a string that is 3.5 m long.
To determine which of the given wavelengths will produce standing waves on a string that is 3.5 m long, we need to consider the conditions for standing waves on a string.
The conditions for standing waves on a string are that the length of the string must be an integer multiple of half the wavelength. Therefore, we need to find a wavelength that satisfies this condition for a string length of 3.5 m.
Let's go through each of the given wavelengths:
a. 1.75 m: Half of this wavelength is 0.875 m, and it is not an integer multiple of the string length of 3.5 m.
b. 3.5 m: Half of this wavelength is 1.75 m, and it is an integer multiple of the string length of 3.5 m.
c. 5.0 m: Half of this wavelength is 2.5 m, and it is not an integer multiple of the string length of 3.5 m.
d. 7.0 m: Half of this wavelength is 3.5 m, and it is an integer multiple of the string length of 3.5 m.
From the given options, both options b (3.5 m) and d (7.0 m) will produce standing waves on a string that is 3.5 m long.