Problem 64 - Pearson Physics

Question

(a) Is the beat frequency produced when a \(245-\mathrm{Hz}\) tone and a \(240-\mathrm{Hz}\) tone are played together greater than, less than, or equal to the beat frequency produced when a \(140-\mathrm{Hz}\) tone and a \(145-\mathrm{Hz}\) tone are played together? (b) Choose the best explanation from among the following: A. The beat frequency is determined by the difference between the frequencies and is independent of their actual values. B. The higher frequencies will produce a greater beat frequency. C. The percent change in frequency for \(240 \mathrm{~Hz}\) and \(245 \mathrm{~Hz}\) is less than for \(140 \mathrm{~Hz}\) and \(145 \mathrm{~Hz}\), resulting in a smaller beat frequency.

Step-by-Step Solution

Answer

(a) Equal, (b) A. Beat frequency is determined by the difference between frequencies.

Step 1: Understand the Concept of Beat Frequency

Beat frequency is the difference between two frequencies when they are played together. Mathematically, this is expressed as \( f_{beat} = |f_1 - f_2| \), where \( f_1 \) and \( f_2 \) are the individual frequencies.

Step 2: Calculate the Beat Frequency for 245 Hz and 240 Hz

Calculate the difference between the two frequencies: \[ f_{beat1} = |245 \, \text{Hz} - 240 \, \text{Hz}| = 5 \, \text{Hz} \].

Step 3: Calculate the Beat Frequency for 140 Hz and 145 Hz

Calculate the difference between the frequencies: \[ f_{beat2} = |145 \, \text{Hz} - 140 \, \text{Hz}| = 5 \, \text{Hz} \].

Step 4: Compare the Two Beat Frequencies

The beat frequencies calculated in Step 2 and Step 3 are both 5 Hz. Thus, they are equal.

Step 5: Evaluate the Explanation Options

Evaluate each explanation option: - Option A: Correct, as the beat frequency depends on the difference between the frequencies and not their actual values. - Option B: Incorrect, as the frequencies themselves do not determine the beat frequency. - Option C: Incorrect, as beat frequency is not about percent change. Thus, Option A is the best explanation.

Key Concepts

Difference in Frequencies

Frequency is the number of times a wave repeats in a second. It is measured in Hertz (Hz). When we talk about the difference in frequencies, we are looking at the space or gap between two different frequencies. Imagine two musical notes being played together, each having its own frequency. The difference in frequencies is simply a subtraction of one frequency from the other. This is an important concept, particularly when dealing with sound waves, as it leads to interesting phenomena, such as beats. You don't need to worry about whether the result of your subtraction is positive or negative; it's always about the absolute value of the difference.

• For example, with frequencies of 245 Hz and 240 Hz, the difference is 5 Hz.
• Similarly, with frequencies of 145 Hz and 140 Hz, the difference remains 5 Hz.

Understanding the difference in frequencies helps us to compute the beat frequency, which we will discuss next.

Concept of Beat Frequency

When two sound waves of slightly different frequencies are played together, they interfere with each other. This interference creates a pulsating effect or a "beat" sound, which is simply a fluctuation of loudness. The concept of beat frequency is built around this effect. The beats occur because when two frequencies are close to each other, the waves periodically come in and out of phase. During certain moments, they add up and make the sound louder (constructive interference). At other moments, they cancel each other out and make the sound softer (destructive interference). This produces a periodic variation in volume, known as beating, which is easy to hear. It's calculated as the absolute difference between the two frequencies causing the sound waves:\[f_{\text{beat}} = |f_1 - f_2|\] The key takeaway is that the beat frequency is independent of the actual values of the frequencies; it only depends on how far apart they are.

Calculation of Beat Frequency

Calculating the beat frequency is straightforward. You only need to find the absolute difference between the two frequencies involved. This calculation is vital for understanding how often the beats occur, which can be essential for tuning musical instruments or diagnosing issues with mechanical systems. To find the beat frequency:- Subtract the lower frequency from the higher one.- Take the absolute value of this result.Let's apply this:

• For frequencies 245 Hz and 240 Hz, calculate: \[ |245 \text{ Hz} - 240 \text{ Hz}| = 5 \text{ Hz} \]
• For frequencies 145 Hz and 140 Hz, calculate: \[ |145 \text{ Hz} - 140 \text{ Hz}| = 5 \text{ Hz} \]

As shown, both sets of frequencies produce a beat frequency of 5 Hz. This demonstrates how understanding and calculating beat frequency helps in comparing different pairs of sound waves to see if they produce the same beat effect.