The 2nd harmonic of a string of length 60 cm and linear mass density 1.1 g/m has the same frequency as the 5th resonance mode of a closed pipe of length 0.8 m. Find the tension in the string.
Why did the string go to therapy? Because it had tension issues!
To find the tension in the string, we can use the relationship between the fundamental frequency of a string and the linear mass density and tension.
The fundamental frequency of a string can be calculated using the formula:
f = (1/2L) * sqrt(T/μ)
where:
f is the fundamental frequency,
L is the length of the string,
T is the tension in the string, and
μ is the linear mass density of the string.
First, let's find the fundamental frequency of the string. Given that the length of the string is 60 cm (0.6 m) and the linear mass density is 1.1 g/m (0.0011 kg/m), we have:
f_string = (1/2 * 0.6) * sqrt(T/0.0011)
Next, let's find the frequency of the 5th resonance mode of the closed pipe. The resonance frequencies of a closed pipe can be calculated using the formula:
f_closed_pipe = n * (v/2L)
where:
f_closed_pipe is the resonance frequency of the closed pipe,
n is the harmonic number (in this case, 5th resonance mode corresponds to n = 5),
v is the speed of sound in air (approximately 343 m/s at room temperature), and
L is the length of the closed pipe.
Given that the length of the closed pipe is 0.8 m and the harmonic number is 5, we have:
f_closed_pipe = 5 * (343/2*0.8)
Now, since the 2nd harmonic of the string has the same frequency as the 5th resonance mode of the closed pipe, we equate the two frequencies:
(1/2 * 0.6) * sqrt(T/0.0011) = 5 * (343/2*0.8)
Now, we can solve this equation for T, the tension in the string.