Question
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?
Solution
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Orion
Master · Tutor for 5 years
Answer
Here's how to solve this problem:**Understanding the Concept**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:L = λ/4Where:* L is the length of the air column* λ is the wavelengthWe also know the relationship between frequency (f), wavelength (λ), and the speed of sound (v):v = fλ**Solving the Problem**1. **Find the speed of sound:**First, we need to find the wavelength associated with the initial length and frequency. Rearranging the first equation:λ = 4L = 4 * 0.37 m = 1.48 mNow, we can find the speed of sound using the second equation:v = fλ = 230 Hz * 1.48 m = 340.4 m/sWe assume the speed of sound remains constant when the length of the air column changes.2. **Find the new fundamental frequency:**With the new length (L = 0.45 m), we can find the new wavelength:λ = 4L = 4 * 0.45 m = 1.8 mNow, we can find the new fundamental frequency (f'):f' = v/λ = 340.4 m/s / 1.8 m ≈ 189.1 Hz**Answer:**The new fundamental frequency is approximately 189.1 Hz.