Two identical half-open pipes each have a fundamental frequency of 450 Hz. What percentage change in the length of one of the pipes will cause a beat frequency of 13.0 Hz when they are sounded simultaneously? Assume the speed of sound in air is 342 m/s.
To solve this problem, we need to find the relationship between the beat frequency, the fundamental frequency, and the length of the pipe.
The beat frequency (f_b) is the difference between the frequencies of the two pipes. In this case, the beat frequency is given as 13.0 Hz.
Since the pipes are identical, their fundamental frequencies (f) are also the same. In this case, the fundamental frequency is given as 450 Hz.
The fundamental frequency of a pipe is related to its length (L) and the speed of sound (v) in the following way:
f = (v / 2L)
where v is the speed of sound and L is the length of the pipe.
To find the percentage change in the length of one of the pipes to produce the given beat frequency, we can rearrange this equation to solve for L:
L = v / (2f)
Now we can calculate the length of the pipe using the given values:
L = (342 m/s) / (2 * 450 Hz)
L ≈ 0.380 m
Next, we need to find the change in length of one of the pipes needed to produce the desired beat frequency.
Let's assume the change in length is ΔL, so the new length of the pipe would be (L + ΔL).
Using the same equation as before, the new fundamental frequency of the pipe would be:
f' = v / (2 * (L + ΔL))
The beat frequency is then given by the difference between the new fundamental frequency and the original fundamental frequency:
f_b = f' - f
13.0 Hz = (v / (2 * (L + ΔL))) - (v / (2L))
Now, let's solve this equation for the change in length (ΔL):
13.0 Hz = (v / (2 * (L + ΔL))) - (v / (2L))
Multiply through by 2L(L + ΔL) to eliminate the denominators:
26L(L + ΔL) Hz = v(L + ΔL) - v(L)
Expand and rearrange the equation:
26L^2 + 26LΔL = vΔL
Divide through by ΔL and rearrange:
26L^2 / ΔL + 26L = v
Now, we can substitute the values we have:
26 * (0.380 m) ^ 2 / ΔL + 26 * (0.380 m) = 342 m/s
Solving this equation will give us the change in length (ΔL) needed to produce the desired beat frequency. From there, we can calculate the percentage change in length:
Percentage Change = (ΔL / L) * 100
Solving the equation numerically will give us the required percentage change in the length of the pipe. Note that the solution will depend on the accuracy of the given values and the accuracy of the calculation.