Ch 17: Superposition

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 17: Superposition

Problem 33

Chapter 17, Problem 33

Two strings are adjusted to vibrate at exactly 200 Hz. Then the tension in one string is increased slightly. Afterward, three beats per second are heard when the strings vibrate at the same time. What is the new frequency of the string that was tightened?

Verified step by step guidance

Step 1: Understand the concept of beats. Beats occur when two sound waves of slightly different frequencies interfere with each other. The beat frequency is equal to the absolute difference between the two frequencies: \( f_{beat} = |f_1 - f_2| \).

Step 2: Identify the given values. The original frequency of both strings is \( f_1 = 200 \, \text{Hz} \), and the beat frequency after tightening one string is \( f_{beat} = 3 \, \text{Hz} \).

Step 3: Use the beat frequency formula to determine the new frequency of the tightened string. Since \( f_{beat} = |f_1 - f_2| \), the new frequency \( f_2 \) could either be \( f_1 + f_{beat} \) or \( f_1 - f_{beat} \).

Step 4: Calculate the possible values for \( f_2 \). Substitute \( f_1 = 200 \, \text{Hz} \) and \( f_{beat} = 3 \, \text{Hz} \) into the formula: \( f_2 = 200 + 3 \) or \( f_2 = 200 - 3 \).

Step 5: Determine the correct value for \( f_2 \). Since the tension in the string was increased, the frequency of the tightened string must increase. Therefore, the new frequency of the tightened string is \( f_2 = 200 + 3 \, \text{Hz} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Frequency

Frequency is the number of cycles of a periodic wave that occur in one second, measured in hertz (Hz). In this context, both strings initially vibrate at a frequency of 200 Hz, meaning they complete 200 cycles per second. When the tension in one string is increased, its frequency will change, which is crucial for understanding the resulting beat frequency.

Beats

Beats occur when two sound waves of slightly different frequencies interfere with each other, resulting in a fluctuating sound intensity. The beat frequency is equal to the absolute difference between the two frequencies. In this scenario, hearing three beats per second indicates that the frequency of the tightened string is either 203 Hz or 197 Hz, as the difference from the original frequency of 200 Hz must equal 3 Hz.

Tension and Frequency Relationship

The frequency of a vibrating string is directly related to the tension in the string. Increasing the tension raises the frequency, while decreasing it lowers the frequency. This relationship is described by the formula f = (1/2L)√(T/μ), where f is frequency, L is the length of the string, T is tension, and μ is the linear mass density. Understanding this concept is essential to determine the new frequency after the tension adjustment.

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