A 60.00 cm guitar string under a tension of 46.000 N has a mass per unit length of 0.11000 g/ cm.
(a) What is the speed of the wave? (b) What is the fundamental wavelength for this string? (c) What is the fundamental frequency for this string? (d)What is the highest resonant frequency that can be heard by a person capable of hearing frequencies up to 20,000 Hz?
Mass per unit length m(o) =0.011 kg/m T=46 N
Speed in the stretched string is v = sqrt(T/m(o)) =sqrt(46/0.011)= 64.7 m/s
λ=2L=1.2 m
f=v/ λ=64.7/1.2=53.9 Hz
Resonant frequency F(n)=n•v/2L=n(64.7/1.2)=53.9•n=20000
n=20000/53.9 =370 => f(n)= n•v/2L=370•64.7/1.2=19949 HZ
Its help
To find the answers to these questions, we can use the following formulas:
(a) The speed of a wave on a string is given by the equation:
v = √(T/μ)
where v is the speed of the wave, T is the tension in the string, and μ is the mass per unit length of the string.
(b) The fundamental wavelength for a vibrating string fixed at both ends is given by the formula:
λ = 2L
where λ is the wavelength and L is the length of the string.
(c) The fundamental frequency of a vibrating string fixed at both ends is given by:
f = v/λ
where f is the fundamental frequency.
(d) The highest resonant frequency that can be heard by a person capable of hearing frequencies up to 20,000 Hz is equal to the fundamental frequency of the string.
Let's plug in the given values and solve for each part:
(a) Speed of the wave (v):
v = √(T/μ)
v = √(46.000 N / (0.11000 g/cm * 100 cm/m))
v = √(46.000 N / (0.011000 g/m))
v = √(46.000 N / 0.011000 kg/m)
v = √(4181.81818182 m^2/s^2)
v ≈ 64.64 m/s
(b) Fundamental wavelength (λ):
λ = 2L
λ = 2 * 60.00 cm
λ = 120.00 cm
(c) Fundamental frequency (f):
f = v/λ
f = 64.64 m/s / (120.00 cm * 0.01 m/cm)
f = 64.64 m/s / 1.20 m
f ≈ 53.87 Hz
(d) Highest resonant frequency:
The highest resonant frequency is equal to the fundamental frequency, which is approximately 53.87 Hz.
To find the answers to these questions, we need to use the formulas related to wave characteristics and string properties. Let's break down the steps to solve each part:
(a) Speed of the wave:
The speed of a wave on a string is given by the equation:
v = sqrt(T/μ)
Where:
v = speed of the wave
T = tension in the string
μ = mass per unit length
Substituting the given values:
T = 46.000 N
μ = 0.11000 g/cm
To convert grams per centimeter (g/cm) to kilograms per meter (kg/m), we divide by 100:
μ = 0.0011 kg/m
Now, substituting the values into the equation:
v = sqrt(46.000 N / 0.0011 kg/m)
Calculating the square root of T/μ, we find the speed of the wave.
(b) Fundamental wavelength:
The fundamental wavelength for a string is given by the equation:
λ = 2L
Where:
λ = wavelength
L = length of the string
Substituting the given value:
L = 60.00 cm
Convert centimeters (cm) to meters (m) by dividing by 100:
L = 0.60 m
Now, substituting the value into the equation:
λ = 2 * 0.60 m
Calculating 2L will give us the fundamental wavelength.
(c) Fundamental frequency:
The fundamental frequency for a string is given by the equation:
f = v/λ
Where:
f = frequency
v = speed of the wave
λ = wavelength
Using the values we have already calculated for speed and wavelength, substitute them into the equation to find the fundamental frequency.
(d) Highest resonant frequency:
The highest resonant frequency for a string is determined by the human hearing range, which is up to 20,000 Hz. This means that the highest frequency the string can produce and be heard is 20,000 Hz.
Once we have the fundamental frequency, we can calculate the highest resonant frequency by determining the frequencies of the higher harmonics and finding the highest one that is still within the human hearing range.
Remember that higher harmonics are achieved by multiplying the fundamental frequency by whole numbers (2, 3, 4, etc.).
By calculating the fundamental frequency and finding the highest resonant frequency within the human hearing range, we can determine the answer to part (d).
It's important to follow these steps and ensure the correct conversion of units to obtain accurate answers.