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Chapter 14: Problem 36

A tuning rork \(A\) of frequency 512 Hz produces 4 beats per second when sounded with a tuning fork \(B\). Due to filing of the prongs of the tuning fork \(B\), the number of the beats per second becomes \(6 .\) The actual frequency of \(B\) is (a) \(516 \mathrm{~Hz}\) (b) \(508 \mathrm{~Hz}\) (c) \(512 \mathrm{~Hz}\) (d) none of these

Short Answer

Expert verified
The actual frequency of tuning fork B is 508 Hz.

Step by step solution

01

Understanding Beats

When two tuning forks produce beats, this indicates they have frequencies that are close but not identical. The number of beats per second is equal to the absolute difference in frequencies of the two waves involved: \(|f_A - f_B| = \text{beats per second}\). Here, \(f_A = 512\, \text{Hz}\) and the initial beat frequency is 4 per second, so \(|512 - f_B| = 4\).
02

Establish Possible Frequencies

Given \(|512 - f_B| = 4\), we have two possible frequencies for \(f_B\). The possible frequencies are: \(f_B = 512 + 4 = 516\, \text{Hz}\) or \(f_B = 512 - 4 = 508\, \text{Hz}\). This means initially, \(f_B\) could be either 516 Hz or 508 Hz.
03

Effect of Filing on Beat Frequency

Filing the prongs of a tuning fork generally results in a higher frequency because the effective length of the prong decreases (less mass). Thus, due to filing, the frequency of tuning fork \(B\) has increased, which results in the increase of beat frequency from 4 to 6 beats per second.
04

Determine the Actual Frequency

Since beating frequency became 6, we use \(|512 - f_B| = 6\). If filing increased \(f_B\), the original \(f_B\) must have been 508 Hz to increase to something closer or past 512 Hz causing more beat frequency. Mathematically filing increases frequency because \(f_B\) of 508 results in a frequency slightly higher than 512 Hz.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tuning Fork Concept
Tuning forks are fascinating tools used to produce specific pitches accurate enough for tuning musical instruments. These consist of a metal fork with two flat prongs that vibrate at a specific pitch when struck.
The primary role of a tuning fork in physics is to serve as a source of sound waves of a precise frequency. This frequency is determined by the length, mass, and shape of its prongs. When struck, the tuning fork's prongs vibrate back and forth, displacing air molecules and creating sound waves at the fork's natural frequency.
In the context of sound and physics, when two tuning forks with slightly different frequencies are struck together, they produce a phenomenon known as "beats." These are fluctuations in sound intensity that result from the interference of two sound waves close in frequency. This is why a tuning fork can be very useful in exercises and experiments related to sound waves and frequencies.
Frequency Calculation
Frequency measures how often a wave's crests pass a point in a given time frame, usually a second. The unit of frequency is Hertz (Hz).
To compute the frequency of tuning forks producing beats, we observe the number of beats, which are fluctuations heard per second due to interference. The formula for beats is given by:
  • Beats per second = |Frequency of Fork A - Frequency of Fork B|
These beats occur because the frequencies do not exactly match, leading to alternating constructive and destructive interference.
In the earlier exercise, you noticed that the frequency of fork A was 512 Hz. Initially, it produced 4 beats per second with fork B, indicating:
  • |512 - f_B| = 4
This gives us two potential frequencies for fork B: 508 Hz or 516 Hz. Upon filing, the frequency changes, affecting the number of beats, as we explore further.
Effect of Filing on Frequency
Filing a tuning fork changes its physical structure, particularly the mass and shape of its prongs. When you file the tuning fork, you remove some material, leading to a change in its natural frequency.
In general, because the mass of the prongs is reduced, the frequency increases. Imagine a guitar string: a shorter string or one with less mass will vibrate faster, producing a higher pitch. This same principle applies when filing a tuning fork.
In the problem, filing the prongs of tuning fork B increased its frequency, changing the beats per second from 4 to 6 when played alongside fork A (512 Hz). This fact confirms that the original lower frequency of fork B was 508 Hz. Filing increased its frequency to a value higher than or close to 512 Hz, causing a higher beat frequency.
Knowing this, one can predict how changes in material or structure will impact sound properties, offering an essential tool for understanding acoustics and sound manipulation.

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Most popular questions from this chapter

An organ pipe closed at one end resonates with a tuning fork of frequencies \(180 \mathrm{~Hz}\) and \(300 \mathrm{~Hz}\). It will also resonate with tuning fork of frequencies: (a) \(360 \mathrm{~Hz}\) (b) \(420 \mathrm{~Hz}\) (c) \(480 \mathrm{~Hz}\) (d) \(540 \mathrm{~Hz}\)

If copper has modulus of rigidity \(4 \times 10^{10} \mathrm{~N} / \mathrm{m}^{2}\) and Bulk modulus \(1.2 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}\) and density \(9 \mathrm{~g} / \mathrm{cm}^{3}\), then the velocity of longitudinal wave, when set-up in solid copper, is : (a) \(4389 \mathrm{~m} / \mathrm{s}\) (b) \(5000 \mathrm{~m} / \mathrm{s}\) (c) \(4000 \mathrm{~m} / \mathrm{s}\) (d) \(4300 \mathrm{~m} / \mathrm{s}\)

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