You have five tuning forks that vibrate at close but different frequencies. What are the (a) maximum and (b) minimum number of different beat frequencies you can produce by sounding the forks two at a time, depending on how the frequencies differ?
You have five tuning forks that vibrate at close but different frequencies. What are the (a) maximum and (b) minimum number of different beat frequencies you can produce by sounding the forks two at a time, depending on how the frequencies differ?
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![**Problem Statement**
You have five tuning forks that vibrate at close but different frequencies. What are the (a) maximum and (b) minimum number of different beat frequencies you can produce by sounding the forks two at a time, depending on how the frequencies differ?
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**Solution Explanation:**
To determine the maximum and minimum number of different beat frequencies you can produce by sounding the forks two at a time, we need to explore the nature of beat frequencies. When two tuning forks with slightly different frequencies are sounded together, a beat frequency is heard, which is the absolute difference between the two frequencies.
**(a) Maximum Number of Different Beat Frequencies:**
The maximum number of different beat frequencies is obtained when all the frequency differences are unique.
Given five tuning forks, the number of unique pairs you can select is given by the combination formula \( C(n, 2) \), where \( n \) is the number of tuning forks:
\[ C(5, 2) = \frac{5!}{2!(5-2)!} = 10 \]
Thus, you can produce a maximum of 10 different beat frequencies by considering each possible pair of tuning forks.
**(b) Minimum Number of Different Beat Frequencies:**
The minimum number of different beat frequencies occurs when the differences in the frequencies are not unique. If the tuning forks are tuned such that some pairs produce the same beat frequency, the number of unique beat frequencies will be reduced.
In the extreme case, if the frequencies of the tuning forks are such that every pair produces the same beat frequency, the minimum number of different beat frequencies will be 1.
In practice, you can achieve this by having the forks grouped in a way that the frequency difference between each adjacent fork is identical.
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**Conclusion:**
- **Maximum Number of Different Beat Frequencies**: 10
- **Minimum Number of Different Beat Frequencies**: 1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F827c1aef-2518-4e31-b017-9f3e8476ed5b%2Fb2436955-ad89-4a54-8675-d133266f1f22%2F0e370o8_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
You have five tuning forks that vibrate at close but different frequencies. What are the (a) maximum and (b) minimum number of different beat frequencies you can produce by sounding the forks two at a time, depending on how the frequencies differ?
---
**Solution Explanation:**
To determine the maximum and minimum number of different beat frequencies you can produce by sounding the forks two at a time, we need to explore the nature of beat frequencies. When two tuning forks with slightly different frequencies are sounded together, a beat frequency is heard, which is the absolute difference between the two frequencies.
**(a) Maximum Number of Different Beat Frequencies:**
The maximum number of different beat frequencies is obtained when all the frequency differences are unique.
Given five tuning forks, the number of unique pairs you can select is given by the combination formula \( C(n, 2) \), where \( n \) is the number of tuning forks:
\[ C(5, 2) = \frac{5!}{2!(5-2)!} = 10 \]
Thus, you can produce a maximum of 10 different beat frequencies by considering each possible pair of tuning forks.
**(b) Minimum Number of Different Beat Frequencies:**
The minimum number of different beat frequencies occurs when the differences in the frequencies are not unique. If the tuning forks are tuned such that some pairs produce the same beat frequency, the number of unique beat frequencies will be reduced.
In the extreme case, if the frequencies of the tuning forks are such that every pair produces the same beat frequency, the minimum number of different beat frequencies will be 1.
In practice, you can achieve this by having the forks grouped in a way that the frequency difference between each adjacent fork is identical.
---
**Conclusion:**
- **Maximum Number of Different Beat Frequencies**: 10
- **Minimum Number of Different Beat Frequencies**: 1
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