Problem 67 - Pearson Physics

Question

Tuning a Piano To tune middle C on a piano, a tuner hits the key and at the same time sounds a \(261-\mathrm{Hz}\) tuning fork. If the tuner hears three beats per second, what are the possible frequencies of the piano key?

Step-by-Step Solution

Answer

The possible frequencies of the piano key are 258 Hz or 264 Hz.

Step 1: Understanding Beats

When two frequencies are close but not identical, they produce a phenomenon called 'beats,' which is perceived as a fluctuation in volume of the sound. The beat frequency is the difference between the two frequencies.

Step 2: Given Information

We know that the frequency of the tuning fork is \(261\, \text{Hz}\) and the beats frequency is \(3\, \text{Hz}\). The equation to find the unknown frequency \(f\) is \(|f - 261| = 3\).

Step 3: Setting Up the Equation

Using the beat frequency equation, create two scenarios. Either the piano frequency is higher or lower than the tuning fork frequency. Hence, \(f - 261 = 3\) or \(261 - f = 3\).

Step 4: Solving the Equations

Solve the two equations:1. \(f - 261 = 3\) \(f = 261 + 3 = 264\, \text{Hz}\)2. \(261 - f = 3\) \(f = 261 - 3 = 258\, \text{Hz}\)

Step 5: Conclude the Possible Frequencies

Based on the solutions, the possible frequencies of the piano key are either \(258\, \text{Hz}\) or \(264\, \text{Hz}\).

Key Concepts

Tuning a Piano

Tuning a piano is both an art and a science. When you tune a piano, you're adjusting the tension of its strings to achieve the correct pitch. The tuner typically uses a tuning fork, which produces a specific frequency, like 261 Hz for middle C, as a reference point. This practice ensures that the piano produces harmonious sounds as intended by the music's composer. To start, the tuner strikes the piano key and the tuning fork simultaneously. If they don't match perfectly, you'll hear a pulsating sound called beats. This indicates that the piano string's frequency differs from the desired frequency and needs adjustment. The objective is to minimize or eliminate these beats by fine-tuning the string until its frequency aligns with the tuning fork. This process requires a keen ear and precise adjustments, as even slight differences in frequency can greatly affect the sound quality.

Frequency Difference

When two sound waves of slightly different frequencies interfere, they create a beat frequency. This beat frequency is the absolute difference between the two frequencies. In our case, the tuning fork's frequency is 261 Hz, and the tuner hears three beats per second.To find the piano's possible frequencies, we can use the beat frequency formula:- \( |f - 261| = 3 \)This formula arises because the beats we hear result from the superposition of the two slightly different sound waves. Therefore, the possible frequencies for the piano key would be either:- 258 Hz (if the piano frequency is lower)- 264 Hz (if the piano frequency is higher)These frequencies of 258 Hz or 264 Hz create the beat frequency with the tuning fork's 261 Hz, resulting in the three beats per second observed by the tuner.

Acoustics

Acoustics, the science of sound, plays a vital role in understanding phenomena like beats and frequency differences. It examines how sound waves behave in different environments and how they're perceived by our ears. When tuning a piano, acoustics helps explain why we hear certain sounds or beats and how they translate into musical harmony. The interaction of sound waves — whether they're combining, canceling each other out, or enhancing — dictates how we perceive the resulting sound. This is particularly critical in musical settings, where clarity and precision dictate the overall auditory experience. In essence, acoustics allows us to manipulate sound to achieve desired auditory outputs. It ensures that the concepts of frequency and beat frequency are put to practical use, facilitating tasks like tuning musical instruments, designing concert halls, and developing audio equipment.