The A string of a violin is 31 cm long between fixed points with a fundamental frequency of 440 Hz and a mass per unit length of 5.8×10^−4 kg/m.
What are the wave speed in the string?
What are the tension in the string?
What is the length of the tube of a simple wind instrument (say, an organ pipe) closed at one end whose fundamental is also 440 Hz if the speed of sound is 343 m/s in air?
What is the frequency of the first overtone of each instrument?
The wavelength is twice the string length, or 0.62 m.
The frequency is 440 Hz.
The wave speed must be the product of the3 two, or 268.40 m/s.
Get the tension T from the formula
Wave speed = sqrt (T/5.84*10^-4 kg/m)
The tension will be in Newtons
Thank you, do you know how to get the other 2 answers?
Step 1: To find the wave speed in the string, use the formula:
Wave speed = square root of (tension / mass per unit length)
Given:
Length of the A string (L) = 31 cm = 0.31 m
Fundamental frequency (f) = 440 Hz
Mass per unit length (μ) = 5.8×10^−4 kg/m
Wave speed = square root of (tension / mass per unit length)
Wave speed = square root of (f * λ / μ)
Step 2: To find the tension in the string, use the formula:
Tension = (wave speed)^2 * mass per unit length
Given:
Wave speed = square root of (tension / mass per unit length)
Mass per unit length (μ) = 5.8×10^−4 kg/m
Tension = (wave speed)^2 * mass per unit length
Step 3: To find the length of the tube of a simple wind instrument, use the formula:
Length of tube = (wave speed) / (4 * fundamental frequency)
Given:
Speed of sound in air (v) = 343 m/s
Fundamental frequency (f) = 440 Hz
Length of tube = (wave speed) / (4 * fundamental frequency)
Step 4: To find the frequency of the first overtone of each instrument, use the formula:
Frequency of first overtone = 2 * fundamental frequency
Now, let's calculate each step one by one.
Step 1:
Wave speed = square root of (tension / mass per unit length)
Wave speed = square root of (440 Hz * λ / 5.8×10^−4 kg/m)
Step 2:
Tension = (wave speed)^2 * mass per unit length
Tension = (wave speed)^2 * 5.8×10^−4 kg/m
Step 3:
Length of tube = (wave speed) / (4 * fundamental frequency)
Length of tube = (343 m/s) / (4 * 440 Hz)
Step 4:
Frequency of first overtone = 2 * fundamental frequency
Frequency of first overtone = 2 * 440 Hz
To find the wave speed in the string, we can use the formula:
Wave speed (v) = √(T/μ)
Where T is the tension in the string and μ is the mass per unit length.
Given:
Length of A string = 31 cm = 0.31 m
Fundamental frequency (ν) = 440 Hz
Mass per unit length (μ) = 5.8 × 10^−4 kg/m
To find the tension in the string, we need to use the formula:
Fundamental frequency (ν) = (1/2L) √(T/μ), where L is the length of the string.
Given:
Length of A string = 31 cm = 0.31 m
Fundamental frequency (ν) = 440 Hz
Mass per unit length (μ) = 5.8 × 10^−4 kg/m
Rearranging the formula, we get:
T = (4L^2μν^2)/π^2
To find the length of the tube of a simple wind instrument, we can use the formula:
Length of tube = (v/fundamental frequency)/4
Given:
Speed of sound (v) = 343 m/s
Fundamental frequency (ν) = 440 Hz
To find the frequency of the first overtone of each instrument, we can use the formula:
Frequency of first overtone = 2 * fundamental frequency
Now let's calculate these values:
Wave speed (v) = √(T/μ) = √(T/(5.8 × 10^−4 kg/m))
T = (4 * (0.31 m)^2 * (5.8 × 10^−4 kg/m) * (440 Hz)^2) / π^2
Length of tube = (v/fundamental frequency)/4 = (343 m/s) / (440 Hz) / 4
Frequency of first overtone = 2 * fundamental frequency = 2 * 440 Hz