Beats: Two strings of identical material and radius are stretched with the same tension with their ends fixed, but one string is 8.0 mm longer than the other. Waves on these strings propagate at 420 m/s. The fundamental frequency of the longer string is 528 Hz. What is the beat frequency when each string is vibrating at its fundamental frequency?
A) 22 Hz
B) 11 Hz
C) 16 Hz
D) 5.5 Hz
E) 27 Hz

Question

Quizplus · Accepted Answer

The beat frequency is the absolute difference between the frequencies of two sounds. Given the fundamental frequency of the longer string is 528 Hz, we can find the fundamental frequency of the shorter string using the relationship between frequency, velocity, and length. For strings under the same tension, the frequency is inversely proportional to the length ($f \propto \frac{1}{L}$). If the longer string has a length $L$ and frequency $f$, the shorter string, being 8.0 mm shorter, will have a slightly higher frequency due to its shorter length. However, without knowing the exact lengths, we can't directly calculate the frequencies using the lengths. Instead, we use the given information to infer the relationship.Given that the velocity of waves on the strings is constant and the same for both strings, and knowing the fundamental frequency of one string, we can deduce that the difference in frequencies (the beat frequency) arises due to the difference in lengths. The exact calculation of the second frequency requires knowing the lengths, which isn't provided directly. However, since the question asks for the beat frequency and provides options, we can deduce the correct answer based on the information given.The correct approach involves understanding that the frequency of the shorter string would be higher than 528 Hz because the length of the string inversely affects the frequency for a given tension and mass per unit length. The beat frequency is the difference between these two frequencies. Without the exact lengths or the ability to calculate the second frequency directly from the given information, we rely on understanding the physical principles involved rather than detailed calculation.Given the options and the nature of the question, which seems to require an understanding of the concept rather than detailed calculations, we infer the correct answer based on the provided information and the fundamental principles of waves on strings. The key is recognizing that the beat frequency results from the difference in fundamental frequencies of the two strings, which is directly influenced by their length difference under the same tension and material properties.

Beats: Two Strings of Identical Material and Radius Are Stretched