The 2nd harmonic of a string of length 60cm and linear mass density 1.7g/m has the same frequency as the 5th possible harmonic of a closed pipe of length 1.2m.
Find the tension in the string.
To find the tension in the string, we can use the equation for the frequency of a vibrating string:
fn = (n/2L) * sqrt(T/μ)
where fn is the frequency of the nth harmonic, L is the length of the string, T is the tension in the string, and μ is the linear mass density of the string.
We are given the length of the string (L = 60 cm = 0.6 m), the linear mass density of the string (μ = 1.7 g/m = 0.0017 kg/m), and we need to find the tension in the string (T).
Let's start by finding the frequency of the 2nd harmonic of the string using the given parameters:
f2 = (2/2L) * sqrt(T/μ)
Next, let's find the frequency of the 5th harmonic of the closed pipe. For a closed pipe, the frequency of the nth harmonic is given by:
f = (2n-1) * f1
where f1 is the fundamental frequency of the closed pipe.
Given that the length of the closed pipe is 1.2 m, the fundamental frequency (f1) can be calculated using the formula:
f1 = V / (2L)
where V is the speed of sound in air.
Now, we need to find the speed of sound in air. The speed of sound in air can be approximated as:
V ≈ 331 + 0.6T
where T is the temperature in degrees Celsius.
Since the temperature is not given, we cannot calculate the exact speed of sound in air. However, we can make an estimate using a reasonable temperature value.
Assuming a temperature of 20 degrees Celsius, we can find the speed of sound in air:
V ≈ 331 + 0.6 * 20 = 331 + 12 = 343 m/s
Now, we can calculate the frequency of the 5th harmonic of the closed pipe:
f5 = (2*5-1) * f1 = 9 * f1
Finally, we set the frequencies of the 2nd harmonic of the string and the 5th harmonic of the closed pipe equal to each other:
f2 = f5
(2/2L) * sqrt(T/μ) = 9 * f1
Simplifying the equation, we have:
sqrt(T/μ) = (9 * f1) / (2/2L)
T/μ = [(9 * f1) / (2/2L)]^2
T = μ * [(9 * f1) / (2/2L)]^2
Now, we can substitute the known values and calculate the tension in the string.