When a metal pipe (open at both ends) is cut into two pieces, the lowest resonance frequency in one piece is 273 Hz and that for the other is 401 Hz. What resonant frequency would have been produced by the original length of pipe?
To find the resonant frequency of the original length of the metal pipe, we need to understand the relationship between the length of a pipe and its resonant frequency.
In an open-pipe (such as the metal pipe mentioned), the resonant frequencies are determined by the length of the pipe. The fundamental frequency (lowest resonant frequency) of an open-pipe can be found using the formula:
f = v / (2L)
Where:
f is the resonant frequency (Hz)
v is the speed of sound in air (approximately 343 m/s at room temperature)
L is the length of the pipe (in meters)
Let's solve the problem step by step:
Step 1: Calculate the resonant frequency for each of the two pieces.
- Resonant frequency for the first piece (f1) = 273 Hz
- Resonant frequency for the second piece (f2) = 401 Hz
Step 2: Use the formula mentioned above to calculate the length of each piece.
- For the first piece:
f1 = v / (2L1)
Rearranging the formula:
L1 = v / (2f1)
- For the second piece:
f2 = v / (2L2)
Rearranging the formula:
L2 = v / (2f2)
Step 3: Substitute the values of frequency and speed of sound into the formulas to calculate the lengths of each piece.
- For the first piece:
L1 = 343 m/s / (2 * 273 Hz)
L1 ≈ 0.629 m
- For the second piece:
L2 = 343 m/s / (2 * 401 Hz)
L2 ≈ 0.427 m
Step 4: Add the lengths of both pieces to find the original length of the pipe.
- Original length (L) = L1 + L2
- L ≈ 0.629 m + 0.427 m
- L ≈ 1.056 m
So, the resonant frequency produced by the original length of the pipe would be determined by its length, which is approximately 1.056 m. Using the speed of sound in air (343 m/s), you can calculate the corresponding resonant frequency using the formula mentioned earlier:
f = v / (2L)
f = 343 m/s / (2 * 1.056 m)
f ≈ 162.42 Hz
Therefore, the resonant frequency produced by the original length of the metal pipe would be approximately 162.42 Hz.