A string of mass 35 g and length 3.0 m is clamped at the ends. It is found that standing waves can be set up on this string when it is driven by an oscillator at a frequency of 28 cycles/second. When the oscillator frequency is slowly increased, another standing wave pattern appears at 35 cycles/second. a) What is the fundamental frequency of the string? b) What is the tension in the string?
A string of mass 35 g and length 3.0 m is clamped at the ends. It is found that standing waves can be set up on this string when it is driven by an oscillator at a frequency of 28 cycles/second. When the oscillator frequency is slowly increased, another standing wave pattern appears at 35 cycles/second. a) What is the fundamental frequency of the string? b) What is the tension in the string?
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![### Problem Description
A string of mass 35 g and length 3.0 m is clamped at the ends. It is found that standing waves can be set up on this string when it is driven by an oscillator at a frequency of 28 cycles/second. When the oscillator frequency is slowly increased, another standing wave pattern appears at 35 cycles/second.
#### Questions:
a) What is the fundamental frequency of the string?
b) What is the tension in the string?
### Explanation:
To analyze and solve the given problems, we need to understand the concept of standing waves on a string:
1. **Fundamental Frequency and Harmonics**:
- The fundamental frequency (f₁) is the lowest frequency at which a string can vibrate and form a standing wave.
- Harmonics are integer multiples of the fundamental frequency.
Given in the problem:
- First standing wave pattern at 28 cycles/second (Hz).
- Another standing wave pattern at 35 cycles/second (Hz).
These frequencies correspond to harmonic frequencies. Let's denote them as \( f_n \) and \( f_{n+1} \):
\[ f_n = 28 \, \text{Hz} \]
\[ f_{n+1} = 35 \, \text{Hz} \]
Harmonic frequencies are given by \( f_n = n \cdot f_1 \) where \( f_1 \) is the fundamental frequency.
2. **Finding Fundamental Frequency**:
To find the fundamental frequency, calculate the greatest common divisor (gcd) of the harmonic frequencies:
\[ \text{gcd}(28 \, \text{Hz}, 35 \, \text{Hz}) = 7 \, \text{Hz} \]
Therefore, the fundamental frequency \( f_1 \) is \( 7 \, \text{Hz} \).
3. **Finding the Tension in the String**:
- The wave speed \( v \) on a string is related to the tension \( F \) and the mass per unit length \( \mu \) by the equation:
\[ v = \sqrt{\frac{F}{\mu}} \]
where
\[ \mu = \frac{\text{mass}}{\text{length}} = \frac{35 \, \text{g}}{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24b1d657-8a50-4e7c-90a8-6e71b5a64daf%2F7953d7f4-9d6a-48cc-a68f-4250920aa85f%2F1h5vo48l_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Description
A string of mass 35 g and length 3.0 m is clamped at the ends. It is found that standing waves can be set up on this string when it is driven by an oscillator at a frequency of 28 cycles/second. When the oscillator frequency is slowly increased, another standing wave pattern appears at 35 cycles/second.
#### Questions:
a) What is the fundamental frequency of the string?
b) What is the tension in the string?
### Explanation:
To analyze and solve the given problems, we need to understand the concept of standing waves on a string:
1. **Fundamental Frequency and Harmonics**:
- The fundamental frequency (f₁) is the lowest frequency at which a string can vibrate and form a standing wave.
- Harmonics are integer multiples of the fundamental frequency.
Given in the problem:
- First standing wave pattern at 28 cycles/second (Hz).
- Another standing wave pattern at 35 cycles/second (Hz).
These frequencies correspond to harmonic frequencies. Let's denote them as \( f_n \) and \( f_{n+1} \):
\[ f_n = 28 \, \text{Hz} \]
\[ f_{n+1} = 35 \, \text{Hz} \]
Harmonic frequencies are given by \( f_n = n \cdot f_1 \) where \( f_1 \) is the fundamental frequency.
2. **Finding Fundamental Frequency**:
To find the fundamental frequency, calculate the greatest common divisor (gcd) of the harmonic frequencies:
\[ \text{gcd}(28 \, \text{Hz}, 35 \, \text{Hz}) = 7 \, \text{Hz} \]
Therefore, the fundamental frequency \( f_1 \) is \( 7 \, \text{Hz} \).
3. **Finding the Tension in the String**:
- The wave speed \( v \) on a string is related to the tension \( F \) and the mass per unit length \( \mu \) by the equation:
\[ v = \sqrt{\frac{F}{\mu}} \]
where
\[ \mu = \frac{\text{mass}}{\text{length}} = \frac{35 \, \text{g}}{
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