Closed Pipe Questions - Use $343 \mathrm{~m} / \mathrm{s}$ as the speed of sound.
Example: A closed pipe is made that is $40 \mathrm{~cm}$ long. The closed pipe needs to play a $C$ note of frequency $261.64 \mathrm{~Hz}$. Calculate how much water needs to be added to the bottom of the pipe in order to make this note?
Answer
Final Answer: \(\boxed{0}\). No water needs to be added. Instead, the pipe needs to be shortened.
Step 1 ：Given that the speed of sound, \(v = 343 \mathrm{~m/s}\), the frequency of the note, \(f = 261.64 \mathrm{~Hz}\), and the original length of the pipe, \(L_{\mathrm{original}} = 0.4 \mathrm{~m}\).
Step 2 ：We can calculate the length of the pipe that will produce this frequency using the formula \(f = \frac{v}{4L}\). Rearranging for \(L\), we get \(L = \frac{v}{4f}\).
Step 3 ：Substituting the given values, we find that \(L_{\mathrm{new}} = \frac{343}{4 \times 261.64} = 0.32774040666564747 \mathrm{~m}\).
Step 4 ：To find out how much water needs to be added, we subtract the original length of the pipe from the new length: \(water_{\mathrm{needed}} = L_{\mathrm{new}} - L_{\mathrm{original}} = -0.07225959333435256 \mathrm{~m}\).
Step 5 ：The result is negative, which means that the original pipe is too long to produce the desired frequency. Therefore, we need to shorten the pipe, not add water to it.
Step 6 ：This is contrary to the original question, which asked how much water needs to be added. Therefore, the answer is that no water needs to be added; instead, the pipe needs to be shortened.
Step 7 ：Final Answer: \(\boxed{0}\). No water needs to be added. Instead, the pipe needs to be shortened.
