Answer :
Answer: the speed of the transverse waves on the guitar string is 62 m/s62m/s.
Explanation: To find the speed of the transverse waves on the guitar string, we can use the relationship between the frequency, wavelength, and speed of the wave.Given:The length of the guitar string L=20L=20 cm = 0.20 mThe frequency of vibration f=620f=620 HzThe string resonates in four segments (4th harmonic)The string resonating in four segments means it is vibrating in its fourth harmonic. In the nth harmonic, the wavelength λnλn is related to the length of the string LL by:L=nλn2L=2nλnFor the fourth harmonic (n=4n=4):L=4λ42=2λ4L=24λ4=2λ4Solving for λ4λ4:λ4=L2=0.20 m2=0.10 mλ4=2L=20.20m=0.10mThe speed vv of the wave on the string can be found using the wave equation:v=f⋅λv=f⋅λSubstituting the given frequency and the calculated wavelength:v=620 Hz⋅0.10 m=62 m/s
Explanation:
To determine the speed of transverse waves on the guitar string, we need to use the relationship between the frequency, the wavelength, and the speed of the wave. The formula for the speed \( v \) of a wave on a string is:
\[ v = f \lambda \]
where:
- \( f \) is the frequency of the wave,
- \( \lambda \) is the wavelength.
First, let's determine the wavelength of the vibrating string.
The string is resonating in four segments, meaning it has four antinodes and three nodes (excluding the endpoints). This indicates that the string is vibrating in its fourth harmonic. The wavelength of the nth harmonic on a string fixed at both ends is given by:
\[ \lambda_n = \frac{2L}{n} \]
where:
- \( L \) is the length of the string,
- \( n \) is the harmonic number.
In this case:
- The length of the string \( L = 20 \) cm = 0.20 m,
- The harmonic number \( n = 4 \).
So, the wavelength for the fourth harmonic is:
\[ \lambda_4 = \frac{2L}{4} = \frac{2 \times 0.20\, \text{m}}{4} = \frac{0.40\, \text{m}}{4} = 0.10\, \text{m} \]
Now that we have the wavelength, we can calculate the speed of the transverse waves using the given frequency \( f = 620 \) Hz.
\[ v = f \lambda \]
\[ v = 620\, \text{Hz} \times 0.10\, \text{m} \]
\[ v = 62\, \text{m/s} \]
Therefore, the speed of the transverse waves on the string is \( 62 \, \text{m/s} \).
