11-U6L8-Air Columns - Homework 10. A 37.0 cm long closed air column resonates at a fundamental frequency of 230 Hz. If the length was then increased to 45.0 cm then what is the new fundamental frequency?

Here's how to solve this problem:<br /><br />**Understanding the Concept**<br /><br />A closed air column resonates at its fundamental frequency when the length of the column is one-quarter of the wavelength of the sound wave. This can be expressed as:<br /><br />L = λ/4<br /><br />Where:<br /><br />* L is the length of the air column<br />* λ is the wavelength<br /><br />We also know the relationship between frequency (f), wavelength (λ), and the speed of sound (v):<br /><br />v = fλ<br /><br />**Solving the Problem**<br /><br />1. **Find the speed of sound:**<br /><br />First, we need to find the wavelength associated with the initial length and frequency. Rearranging the first equation:<br /><br />λ = 4L = 4 * 0.37 m = 1.48 m<br /><br />Now, we can find the speed of sound using the second equation:<br /><br />v = fλ = 230 Hz * 1.48 m = 340.4 m/s<br /><br />We assume the speed of sound remains constant when the length of the air column changes.<br /><br />2. **Find the new fundamental frequency:**<br /><br />With the new length (L = 0.45 m), we can find the new wavelength:<br /><br />λ = 4L = 4 * 0.45 m = 1.8 m<br /><br />Now, we can find the new fundamental frequency (f'):<br /><br />f' = v/λ = 340.4 m/s / 1.8 m ≈ 189.1 Hz<br /><br />**Answer:**<br /><br />The new fundamental frequency is approximately 189.1 Hz.<br />