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?
To find the answers to these questions, we need to use the relevant formulas related to waves and strings.
(a) Speed of the wave:
The speed (v) of a wave on a string is given by the formula:
v = √(T/μ)
where T is the tension and μ is the mass per unit length.
Given:
Tension (T) = 46.000 N
Mass per unit length (μ) = 0.11000 g/cm = 1.100 g/m = 0.0011 kg/m
Substituting these values into the formula:
v = √(46.000 N) / (0.0011 kg/m)
We can calculate the value of v by performing the calculation inside the square root and then take the square root of the result.
(b) Fundamental wavelength:
The fundamental wavelength (λ) for a standing wave on a string is given by the formula:
λ = 2L/n
where L is the length of the string and n is the harmonic number.
Given:
Length of the string (L) = 60.00 cm = 0.60 m
We need to find the value of n for the fundamental mode, which is the first harmonic mode. For the first harmonic, n = 1.
Substituting these values into the formula:
λ = 2(0.60 m) / 1
(c) Fundamental frequency:
The fundamental frequency (f) for a string is given by the formula:
f = v / λ
where v is the speed of the wave and λ is the wavelength.
We have already calculated the value of v in part (a) and the value of λ in part (b). We can substitute these values into the formula:
f = v / λ
(d) Highest resonant frequency:
The highest resonant frequency is the frequency of the highest harmonic that can be heard by a person capable of hearing frequencies up to 20,000 Hz. The fundamental frequency corresponds to the first harmonic. Each subsequent harmonic is an integer multiple of the fundamental frequency. We need to find the highest harmonic frequency that is still within the human hearing range.
Given:
Maximum frequency audible (f_max) = 20,000 Hz
To find the highest resonant frequency, we need to find the highest harmonic number (n_max) that gives a frequency lower than or equal to f_max. The fundamental frequency is given by f = n_max * f.
To find n_max, we rearrange the formula:
n_max = f_max / f
Substituting the known values into the formula, we can find the highest resonant frequency.
Please note that for this particular problem, we have found the fundamental frequency (part c) and the highest resonant frequency (part d) assuming the relationship between frequency and harmonic number.