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?

Key Concepts

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. 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 periodic variation in amplitude. 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 3 Hz different from the original frequency of 200 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 the tension lowers it. 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 relationship helps determine the new frequency after the tension adjustment.