A 1.50-m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 62.0 m/s.What are the wavelength and frequency of the fourth harmonic?

Verified step by step guidance

1. First, understand that the harmonics of a stretched rope are related to the standing wave patterns that can form on the rope. The fundamental frequency (first harmonic) has one antinode, the second harmonic has two antinodes, and so on.
2. For the fourth harmonic, the rope will have four antinodes. The formula for the wavelength of the nth harmonic on a rope of length L is: $\lambda =\frac{2L}{n}$, where n is the harmonic number.
3. Substitute the given values into the wavelength formula: $\lambda =\frac{2\times 1.50}{4}$. This will give you the wavelength of the fourth harmonic.
4. Next, use the wave speed formula to find the frequency. The formula is: $v=f\times \lambda$, where v is the wave speed, f is the frequency, and $\lambda$ is the wavelength.
5. Rearrange the wave speed formula to solve for frequency: $f=\frac{v}{\lambda }$. Substitute the values for v (62.0 m/s) and the calculated wavelength to find the frequency of the fourth harmonic.

Key Concepts

Wave Speed

Wave speed is the rate at which a wave propagates through a medium. It is determined by the tension and mass per unit length of the medium, such as a rope. In this scenario, the wave speed is given as 62.0 m/s, which is crucial for calculating the wavelength and frequency of harmonics.

Harmonics

Harmonics are the natural frequencies at which a system, like a stretched rope, vibrates. Each harmonic corresponds to a specific mode of vibration. The fourth harmonic means the rope vibrates in a pattern with four segments, affecting the wavelength and frequency calculations.

Wavelength and Frequency Relationship

The wavelength and frequency of a wave are inversely related through the wave speed: v = fλ, where v is the wave speed, f is the frequency, and λ is the wavelength. For the fourth harmonic, the wavelength is determined by the rope's length and harmonic number, while the frequency is derived from the wave speed and wavelength.