A stretched string fixed at each end has a mass of 34.0 g and a length of 8.20 m. The tension in the string is 52.0 N.
(a) Determine the positions of the nodes and antinodes for the third harmonic. (Enter your answers from smallest to largest distance from one end of the string.)
nodes:
1.

antinodes:


(b) What is the vibration frequency for this harmonic?
  

To solve this problem, we'll first need to calculate the wavelength, node, and antinode positions for the third harmonic. Then, we'll calculate the vibration frequency of the string for this harmonic.

### (a) Positions of Nodes and Antinodes for the Third Harmonic

For a string fixed at both ends, the harmonic frequencies are given by:
\[
f_n = \frac{n v}{2L}
\]
where:
- \( n \) is the harmonic number (3 for the third harmonic),
- \( v \) is the wave speed on the string,
- \( L \) is the length of the string.

The wave speed \( v \) can be calculated using the formula:
\[
v = \sqrt{\frac{T}{\mu}}
\]
where:
- \( T = 52.0 \, \text{N} \) (tension),
- \( \mu \) is the linear mass density, calculated as \( \mu = \frac{m}{L} \), where \( m \) is the mass of the string (in kg).

For the third harmonic:
- The wavelength \( \lambda_3 = \frac{2L}{3} \).

**Node Positions:**
Nodes occur at multiples of \(\lambda_3/2\). For the third harmonic, the nodes occur at:
\[
\text{Node positions} = 0, \, \frac{L}{3}, \, \frac{2L}{3}, \, L
\]

**Antinode Positions:**
Antinodes occur at the midpoints between nodes. For the third harmonic, the antinodes occur at:
\[
\text{Antinode positions} = \frac{L}{6}, \, \frac{L}{2}, \, \frac{5L}{6}
\]

### (b) Vibration Frequency for the Third Harmonic

Using the formula for the frequency \( f_3 = \frac{3v}{2L} \), where \( v \) is the wave speed calculated above.

Let's calculate these values.### (a) Positions of Nodes and Antinodes for the Third Harmonic

**Nodes** (distances from one end of the string):
1. \( 0 \, \text{m} \)
2. \( 2.73 \, \text{m} \)
3. \( 5.47 \, \text{m} \)
4. \( 8.20 \, \text{m} \)

**Antinodes** (distances from one end of the string):
1. \( 1.37 \, \text{m} \)
2. \( 4.10 \, \text{m} \)
3. \( 6.83 \, \text{m} \)

### (b) Vibration Frequency for the Third Harmonic
The vibration frequency for the third harmonic is approximately \( 20.49 \, \text{Hz} \).

Do you have any questions or need further details?

Here are some related questions:
1. How would the frequency change if the tension in the string was doubled?
2. What would be the frequency of the first harmonic for this string?
3. Can you calculate the frequency for the fourth harmonic?
4. How would the node and antinode positions change for the second harmonic?
5. What is the relationship between the wavelength and the length of the string for any harmonic?

**Tip:** Understanding harmonics and standing waves on a string is crucial for topics in sound and wave physics.

Explore the positions of nodes and antinodes for the third harmonic of a stretched string fixed at both ends. Calculate the vibration frequency using wave physics principles. Learn how tension, mass, and length affect standing wave patterns. Ideal for college-level physics students.

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