A string has a mass per unit length of 7.00x10-3 kg/m and a length of 0.230 m. What must be the tension in the string if its second harmonic has the same frequency as the second resonance mode of a 2.05 m long pipe open at one end?
To find the tension in the string, we need to use the formula for the frequency of a string's harmonic modes. The formula is given by:
f = (nh/2L) * √(T/μ)
where:
f is the frequency of the harmonic mode
n is the harmonic number (in this case, n=2 for the second harmonic)
h is the speed of sound in air (which we need to find)
L is the length of the pipe
T is the tension in the string (what we want to find)
μ is the mass per unit length of the string
We are given:
L = 2.05 m (length of the pipe)
n = 2 (harmonic number)
μ = 7.00x10^-3 kg/m (mass per unit length of the string)
We need to find h and T.
First, let's solve for h:
Since the second harmonic of the string has the same frequency as the second resonance mode of the pipe, we know that the frequencies of both are equal.
Thus, we have:
f_string = f_pipe
Using the formula for the frequency of the pipe's resonance mode:
f_pipe = (nv)/(4L)
where v is the speed of sound in air
Substituting the given values and solving for v:
f_pipe = (2v)/(4L)
v = (2f_pipe)(L)
Now, let's substitute the known values into the formula for the string's frequency:
f_string = (n*(((2*(2f_pipe)*L)/(2L))*√(T/μ))
Since f_string = f_pipe, we can equate the two frequencies:
(n*(((2*(2f_pipe)*L)/(2L))*√(T/μ)) = (2v)/(4L)
Now, we can solve for T:
(n*(((2*(2f_pipe)*L)/(2L))*√(T/μ) = (2v)/(4L)
(n*(((2*(2f_pipe)*L)/(2L))*√(T/μ) = (2(2f_pipe)*L)
Now, we can isolate T by dividing both sides by (√(T/μ)):
n*((2*(2f_pipe)*L)/(2L)) = 2(2f_pipe)*L/(√(T/μ))
n*((2*(2f_pipe)*L)/(2L)) = 2(2f_pipe)*√(μ/T)
Finally, we can solve for T by isolating it:
T = (μ*(2*f_pipe)^2) / (n^2)
Substituting the known values:
T = (7.00x10^-3 kg/m * (2*f_pipe)^2) / (2^2)
Now, we have everything we need to find the tension in the string. To get the value of f_pipe, we need the frequency value for the second resonance mode of a pipe open at one end. However, that value is not provided in the question.