A guitar string is 96 cm long and has a mass of 4.3 g. The distance from the bridge to the support post is 70.08 cm, and the string is under a tension of 477 N. What is the frequency of the 10th overtone? Answer: 63°F Mo O 10
A guitar string is 96 cm long and has a mass of 4.3 g. The distance from the bridge to the support post is 70.08 cm, and the string is under a tension of 477 N. What is the frequency of the 10th overtone? Answer: 63°F Mo O 10
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![### Physics of Guitar Strings
**Problem Statement:**
A guitar string is 96 cm long and has a mass of 4.3 g. The distance from the bridge to the support post is 70.08 cm, and the string is under a tension of 477 N. What is the frequency of the 10th overtone?
**Answer:**
\[ \_\_\_\_\_\_\_\_ \]
___
**Detailed Explanation:**
This problem involves understanding the physical properties of a vibrating string and applying the principles of wave mechanics to determine the frequency of a given overtone. Here’s a step-by-step guide to approach the problem:
1. **Given Data:**
- Length of the string, \( L = 96 \) cm = \( 0.96 \) m
- Mass of the string, \( m = 4.3 \) g = \( 0.0043 \) kg
- Distance between the bridge and support post (effective length), \( l = 70.08 \) cm = \( 0.7008 \) m
- Tension in the string, \( T = 477 \) N
2. **Fundamentals:**
- The frequency of the nth overtone (or the (n+1)th harmonic) is given by:
\[
f_n = \left( n + 1 \right) \frac{v}{2l}
\]
where \( v \) is the wave speed on the string.
- The wave speed \( v \) can be calculated using the formula:
\[
v = \sqrt{\frac{T}{\mu}}
\]
where \( \mu \) (linear mass density) is given by:
\[
\mu = \frac{m}{L}
\]
3. **Calculations:**
Calculate the linear mass density \( \mu \):
\[
\mu = \frac{0.0043 \text{ kg}}{0.96 \text{ m}} = 0.00448 \text{ kg/m}
\]
Calculate the wave speed \( v \):
\[
v = \sqrt{\frac{477 \text{ N}}{0.00448 \text{ kg/m}}} = 327.50 \text{ m/s}
\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff5b22acd-e686-40ab-bd08-7f88017cfb76%2Fa7758793-2c92-4645-9e0e-cb8ee1093ad7%2F71s01v_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Physics of Guitar Strings
**Problem Statement:**
A guitar string is 96 cm long and has a mass of 4.3 g. The distance from the bridge to the support post is 70.08 cm, and the string is under a tension of 477 N. What is the frequency of the 10th overtone?
**Answer:**
\[ \_\_\_\_\_\_\_\_ \]
___
**Detailed Explanation:**
This problem involves understanding the physical properties of a vibrating string and applying the principles of wave mechanics to determine the frequency of a given overtone. Here’s a step-by-step guide to approach the problem:
1. **Given Data:**
- Length of the string, \( L = 96 \) cm = \( 0.96 \) m
- Mass of the string, \( m = 4.3 \) g = \( 0.0043 \) kg
- Distance between the bridge and support post (effective length), \( l = 70.08 \) cm = \( 0.7008 \) m
- Tension in the string, \( T = 477 \) N
2. **Fundamentals:**
- The frequency of the nth overtone (or the (n+1)th harmonic) is given by:
\[
f_n = \left( n + 1 \right) \frac{v}{2l}
\]
where \( v \) is the wave speed on the string.
- The wave speed \( v \) can be calculated using the formula:
\[
v = \sqrt{\frac{T}{\mu}}
\]
where \( \mu \) (linear mass density) is given by:
\[
\mu = \frac{m}{L}
\]
3. **Calculations:**
Calculate the linear mass density \( \mu \):
\[
\mu = \frac{0.0043 \text{ kg}}{0.96 \text{ m}} = 0.00448 \text{ kg/m}
\]
Calculate the wave speed \( v \):
\[
v = \sqrt{\frac{477 \text{ N}}{0.00448 \text{ kg/m}}} = 327.50 \text{ m/s}
\
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