To determine the frequency of the fundamental note produced by an open pipe closed at one end, we need to understand the relationship between the length of the pipe, the speed of sound, and the frequency of the sound wave produced. ### Step-by-Step Explanation 1. **Understanding the Pipe Configuration**: - An open pipe closed at one end is a type of resonator. In this case, one end of the pipe is closed, and the other end is open to the air. - This configuration supports standing waves, which are formed by the interference of waves traveling in opposite directions. 2. **Fundamental Frequency**: - The fundamental frequency (or first harmonic) of a closed pipe occurs when there is one quarter of a wavelength (\(\lambda\)) fitting into the length of the pipe (\(L\)). - Mathematically, this can be expressed as: \[ L = \frac{\lambda}{4} \] - From this, we can derive the wavelength: \[ \lambda = 4L \] 3. **Relating Wavelength to Frequency**: - The frequency (\(f\)) of a wave is related to its speed (\(v\)) and wavelength (\(\lambda\)) by the formula: \[ f = \frac{v}{\lambda} \] - Substituting the expression for \(\lambda\) from above: \[ f = \frac{v}{4L} \] 4. **Final Expression for Frequency**: - Thus, the frequency of the fundamental note produced by the open pipe closed at one end is: \[ f = \frac{v}{4t} \] - Here, \(t\) is the length of the pipe, which we have denoted as \(L\). ### Correct Option - The correct answer is **C. \( \frac{v}{4t} \)**. ### Explanation of Other Options - **A. \( \frac{2v}{t} \)**: - This option suggests that the frequency is twice the speed of sound divided by the length of the pipe. This does not account for the fact that in a closed pipe, the fundamental frequency corresponds to a quarter wavelength fitting into the length of the pipe. - **B. \( \frac{v}{5t} \)**: - This option incorrectly suggests that the frequency is one-fifth of the speed of sound divided by the length of the pipe. This does not match the relationship established for a closed pipe. - **D. \( \frac{v}{2t} \)**: - This option implies that the frequency is half the speed of sound divided by the length of the pipe. This is incorrect for a closed pipe, as it does not consider the quarter wavelength condition. ### Common Pitfalls - **Misunderstanding Pipe Types**: Students often confuse open and closed pipes. Remember that closed pipes have a node at the closed end and an antinode at the open end. - **Wavelength Calculation**: Ensure you correctly derive the wavelength based on the pipe's length and the harmonic being considered. - **Formula Application**: Always double-check that you are using the correct formula for frequency based on the type of pipe. ### Revision Summary - The fundamental frequency of a closed pipe is given by \( f = \frac{v}{4L} \). - In this case, \(L\) is the length of the pipe, and \(v\) is the speed of sound in air. - The correct answer for the frequency of the note produced by the pipe is option C: \( \frac{v}{4t} \). - Remember the distinction between open and closed pipes when studying wave behavior in acoustics.