The largest tension that can be sustained by a stretched string of linear mass density μ, even in principle, is given by τ = μc2, where c is the speed of light in vacuum. (This is an enormous value. The breaking tensions of all ordinary materials are about 12 orders of magnitude less than this.)
a) What is the speed of a traveling wave on a string under such tension?
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b) If a 1.190-m-long guitar string, stretched between anchored ends, were made of this hypothetical material, what frequency would its first harmonic have?
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c) If that guitar string were plucked at its midpoint and given a displacement of 2.17 mm there to produce the fundamental frequency, what would be the maximum speed attained by the midpoint of the string?
a) The speed of a traveling wave on a string under tension is given by the formula:
v = sqrt(τ / μ)
where v is the speed of the wave, τ is the tension in the string, and μ is the linear mass density of the string.
In this case, the tension is given by τ = μc^2, so we can substitute this into the formula:
v = sqrt((μc^2) / μ)
Simplifying, we get:
v = sqrt(c^2)
Since c is the speed of light in vacuum, which is approximately 3 x 10^8 m/s, the speed of the traveling wave on the string under this tension is also approximately 3 x 10^8 m/s.
b) To find the frequency of the first harmonic of a guitar string made of this hypothetical material, we can use the formula:
f = v / λ
where f is the frequency, v is the speed of the wave on the string, and λ is the wavelength of the harmonic.
In this case, the wavelength of the first harmonic is twice the length of the string, so:
λ = 2 * 1.190 m = 2.380 m
Substituting the values into the formula, we get:
f = (3 x 10^8 m/s) / (2.380 m)
Calculating, the frequency of the first harmonic of the guitar string made of this hypothetical material is approximately 1.26 x 10^8 Hz.
c) To find the maximum speed attained by the midpoint of the string when it is plucked to produce the fundamental frequency, we can use the formula:
v = A ω
where v is the maximum speed, A is the amplitude of the displacement, and ω is the angular frequency.
In this case, the displacement of the midpoint is given as 2.17 mm, which is equivalent to 0.00217 m.
The angular frequency is related to the fundamental frequency by the formula:
ω = 2πf
Substituting the value of the fundamental frequency calculated earlier, we get:
ω = 2π * (1.26 x 10^8 Hz)
Calculating, the angular frequency is approximately 7.92 x 10^8 rad/s.
Finally, we can calculate the maximum speed:
v = (0.00217 m) * (7.92 x 10^8 rad/s)
Calculating, the maximum speed attained by the midpoint of the string is approximately 1,717.44 m/s.
a) To find the speed of a traveling wave on a string under such tension, we can use the equation for wave speed on a string, which is given by the formula:
v = √(F/μ)
Where v is the wave speed, F is the tension in the string, and μ is the linear mass density of the string.
In this case, the tension is given by τ = μc^2, so we can substitute this into the equation:
v = √(τ/μ) = √(μc^2/μ) = c
Therefore, the speed of a traveling wave on a string under this tension would be equal to the speed of light in a vacuum, c.
b) To find the frequency of the first harmonic of a guitar string made of this hypothetical material, we can use the formula for the frequency of a standing wave on a stretched string:
f = v / λ
Where f is the frequency, v is the wave speed, and λ is the wavelength.
In this case, we are given the length of the string, which is 1.190 m. The first harmonic of a string has a wavelength equal to twice the length of the string, so λ = 2 * 1.190 m = 2.380 m.
We already determined in part a) that the wave speed is equal to the speed of light in a vacuum, c. Therefore, we can substitute these values into the formula:
f = c / λ = c / 2.380 m
You can solve this equation by substituting the numeric value for the speed of light in a vacuum to find the frequency of the first harmonic.
c) To find the maximum speed attained by the midpoint of the guitar string when it is plucked at its midpoint and given a displacement of 2.17 mm to produce the fundamental frequency, we can use the formula for the speed of a particle in simple harmonic motion:
v = Aω
Where v is the speed, A is the amplitude (given as 2.17 mm or 0.00217 m), and ω is the angular frequency.
The angular frequency can be calculated using the formula:
ω = 2πf
Where f is the fundamental frequency.
Once you have the angular frequency, you can substitute the values into the equation to find the maximum speed attained by the midpoint of the string.