Problem 63 - College Physics

Question

If the frequency of the fifth harmonic of a vibrating string is \(425 \mathrm{~Hz},\) what is the frequency of the second harmonic?

Step-by-Step Solution

Answer

The frequency of the second harmonic is 170 Hz.

Step 1: Understand the Relationship between Harmonics

The frequency of harmonics in a vibrating string is determined by the fundamental frequency. The nth harmonic frequency is given by \( f_n = n \times f_1 \), where \( f_1 \) is the fundamental frequency and \( n \) is the harmonic number.

Step 2: Identify the Known Harmonic

We are given the frequency of the fifth harmonic, \( f_5 = 425 \text{ Hz} \). From the relationship above, \( f_5 = 5 \times f_1 \).

Key Concepts

Vibrating String

When we talk about vibrating strings, we're diving into a foundational concept in physics and music. Vibrating strings, like those of a guitar or a piano, produce sound by moving back and forth at different speeds. This concept also applies to strings in scientific experiments or engineering applications. When a string vibrates, it does not just move randomly. Instead, the movement creates a pattern called a waveform. These waveforms are formed due to the tension and fixed endpoints of the string.

• The string vibrates in distinct modes, each corresponding to a specific pattern.
• These modes are called harmonics and are related to the natural frequencies of the string.

Understanding the behavior of a vibrating string is crucial. It forms the basis of many musical instruments. Yet, it's also vital in physics for understanding waves and oscillations.

Harmonic Frequencies

Harmonic frequencies arise when a vibrating string vibrates at frequencies that are whole-number multiples of a fundamental frequency. If a string is fixed at both ends, it supports a series of harmonics based on its length, tension, and mass density.

• The harmonic series begins with the fundamental frequency, followed by the second harmonic, third harmonic, and so on.
• Each harmonic produces a unique mode of vibration.
• The frequency of the nth harmonic is described by the equation: \( f_n = n \times f_1 \).

Harmonics are important because they contribute to the overall sound produced by the string. They provide richness and complexity to the tone, with each harmonic adding its own unique pitch and intensity. Recognizing harmonic frequencies is key in fields like acoustics and musical theory.

Fundamental Frequency

At the heart of understanding vibrating strings and harmonics is the concept of the fundamental frequency. The fundamental frequency, often referred to as the first harmonic or the natural frequency, is the lowest frequency at which a string can vibrate.

• The fundamental frequency determines the pitch of the sound produced by the string.
• All other harmonic frequencies are integer multiples of this fundamental frequency.
• In the equation \( f_n = n \times f_1 \), \( f_1 \) represents the fundamental frequency.

So, if we know the fundamental frequency, we can find other harmonic frequencies simply by multiplying. For example, if the fundamental frequency is 100 Hz, the second harmonic would be \( 2 \times 100 = 200 \) Hz. This simple relationship makes it much easier to work with harmonic frequencies, especially in musical tuning and sound engineering.