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Chapter 9: Problem 80

The end correction of a resonance column is \(1.0 \mathrm{~cm}\). If the shortest length resonating with a tuning fork is \(14.0 \mathrm{~cm}\), the next resonating length is (A) \(44 \mathrm{~cm}\) (B) \(45 \mathrm{~cm}\) (C) \(46 \mathrm{~cm}\) (D) \(47 \mathrm{~cm}\)

Short Answer

Expert verified
(A) \(44 \mathrm{~cm}\)

Step by step solution

01

Understanding resonance in columns

Resonance in columns happens when a sound wave reflects off the closed end and interferes constructively with the incoming sound wave. When this occurs, standing waves are formed, and these standing waves have specific lengths called resonating lengths. It's essential to know the end correction factor, which accounts for the effective length being slightly larger than the actual length of the column. The formula to consider the end correction is: Effective length = Actual length + End correction
02

Apply the given data to find the first resonating length

The end_correction is given as 1.0 cm, and the shortest resonating length is 14.0 cm. We can calculate the effective length for the first resonating length: Effective length = Actual length + End correction Effective length = 14.0 cm + 1.0 cm Effective length = 15.0 cm
03

Calculate the next resonating length

In resonance columns, the difference between consecutive resonating lengths is half the wavelength (位/2) of the sound wave. So, we can find the next resonating length using this knowledge: Next effective length = First effective length + 位/2 Since we don't know the wavelength, we can instead use the difference in consecutive lengths, which corresponds to 位/2 of the sound wave: Difference in consecutive lengths = Next resonating length - First resonating length 位/2 = Next resonating length - 14.0 cm Next resonating length = 14.0 cm + 位/2 Now, we subtract the end correction from the next effective length to find the next resonating length: Next resonating length = Next effective length - End correction Next resonating length = (14.0 cm + 位/2) - 1.0 cm
04

Analyze the choices and find the correct answer

By analyzing the choices provided, we can see that the differences between them are approximately: (A) 44 cm - 14 cm = 30 cm (B) 45 cm - 14 cm = 31 cm (C) 46 cm - 14 cm = 32 cm (D) 47 cm - 14 cm = 33 cm We can notice that only choice (A) has a difference of 30 cm, which is an even value, and thus is more likely to be correct as the difference corresponds to half a wavelength (位/2). Therefore, the correct answer is: (A) \(44 \mathrm{~cm}\)

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