A stretched string is 1.97 m long and has a mass of 20.5 g. When the string oscillates at 440 Hz, which is the frequency of the standard A pitch, transverse waves with a wavelength of 15.9 cm travel along the string. Calculate the tension T in the string. T = 100.335
A stretched string is 1.97 m long and has a mass of 20.5 g. When the string oscillates at 440 Hz, which is the frequency of the standard A pitch, transverse waves with a wavelength of 15.9 cm travel along the string. Calculate the tension T in the string. T = 100.335
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Chapter1: Units, Trigonometry. And Vectors
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![### Problem Description
A stretched string is 1.97 meters long and has a mass of 20.5 grams. When the string oscillates at 440 Hz, which is the frequency of the standard A pitch, transverse waves with a wavelength of 15.9 cm travel along the string. Calculate the tension \( T \) in the string.
### Calculation
Given data:
- Length of string, \( L = 1.97 \) m
- Mass of string, \( m = 20.5 \) g = 0.0205 kg
- Frequency of oscillation, \( f = 440 \) Hz
- Wavelength of transverse waves, \( \lambda = 15.9 \) cm = 0.159 m
### Incorrect Calculation Attempt
\[ T = 100.335 \, \text{N} \]
_(Note: The provided calculation has been marked as incorrect.)_
The detailed explanation and calculation for such problems typically involve using the formula for the wave speed \( v \) on a stretched string:
\[ v = \sqrt{\frac{T}{\mu}} \]
where \( \mu \) is the linear mass density given by:
\[ \mu = \frac{m}{L} \]
Additionally, the wave speed can be related to wavelength and frequency by:
\[ v = f \cdot \lambda \]
Combining these equations will allow the calculation of the correct tension \( T \) in the string:
1. Calculate the linear mass density \( \mu \):
\[ \mu = \frac{m}{L} = \frac{0.0205 \, \text{kg}}{1.97 \, \text{m}} \approx 0.0104 \, \text{kg/m} \]
2. Calculate the wave speed \( v \):
\[ v = f \cdot \lambda = 440 \, \text{Hz} \times 0.159 \, \text{m} \approx 69.96 \, \text{m/s} \]
3. Calculate the tension \( T \) using \( v \) and \( \mu \):
\[ T = \mu \cdot v^2 = 0.0104 \, \text{kg/m} \times (69.96 \, \text{m/s})^2 \]
\[ T = 0.0104 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F37e1db01-06d0-4e20-aae1-26f96b6814c3%2Fd15f1973-e88b-4019-bef6-bb62ceabb25b%2F3rge1b_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Description
A stretched string is 1.97 meters long and has a mass of 20.5 grams. When the string oscillates at 440 Hz, which is the frequency of the standard A pitch, transverse waves with a wavelength of 15.9 cm travel along the string. Calculate the tension \( T \) in the string.
### Calculation
Given data:
- Length of string, \( L = 1.97 \) m
- Mass of string, \( m = 20.5 \) g = 0.0205 kg
- Frequency of oscillation, \( f = 440 \) Hz
- Wavelength of transverse waves, \( \lambda = 15.9 \) cm = 0.159 m
### Incorrect Calculation Attempt
\[ T = 100.335 \, \text{N} \]
_(Note: The provided calculation has been marked as incorrect.)_
The detailed explanation and calculation for such problems typically involve using the formula for the wave speed \( v \) on a stretched string:
\[ v = \sqrt{\frac{T}{\mu}} \]
where \( \mu \) is the linear mass density given by:
\[ \mu = \frac{m}{L} \]
Additionally, the wave speed can be related to wavelength and frequency by:
\[ v = f \cdot \lambda \]
Combining these equations will allow the calculation of the correct tension \( T \) in the string:
1. Calculate the linear mass density \( \mu \):
\[ \mu = \frac{m}{L} = \frac{0.0205 \, \text{kg}}{1.97 \, \text{m}} \approx 0.0104 \, \text{kg/m} \]
2. Calculate the wave speed \( v \):
\[ v = f \cdot \lambda = 440 \, \text{Hz} \times 0.159 \, \text{m} \approx 69.96 \, \text{m/s} \]
3. Calculate the tension \( T \) using \( v \) and \( \mu \):
\[ T = \mu \cdot v^2 = 0.0104 \, \text{kg/m} \times (69.96 \, \text{m/s})^2 \]
\[ T = 0.0104 \
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