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 second overtone?

검증된 단계별 안내

First, understand that the second overtone in a stretched string corresponds to the third harmonic. In a string fixed at both ends, the number of antinodes corresponds to the harmonic number.

For the third harmonic, the string will have three antinodes and two nodes (excluding the endpoints). The wavelength \( \lambda \) of the third harmonic is given by \( \lambda = \frac{2L}{3} \), where \( L \) is the length of the string.

Substitute the given length of the rope \( L = 1.50 \text{ m} \) into the wavelength formula to find the wavelength of the third harmonic.

The frequency \( f \) of a wave is related to its speed \( v \) and wavelength \( \lambda \) by the equation \( f = \frac{v}{\lambda} \). Use the wave speed \( v = 62.0 \text{ m/s} \) and the wavelength calculated in the previous step to find the frequency of the third harmonic.

By following these steps, you will determine both the wavelength and frequency of the second overtone (third harmonic) on the rope.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Wave Speed

Wave speed is the rate at which a wave propagates through a medium, calculated as the product of frequency and wavelength. In this scenario, the wave speed is given as 62.0 m/s, which is crucial for determining the wavelength and frequency of the wave on the rope.

Standing Waves and Overtones

Standing waves occur when waves reflect back and forth between fixed points, creating nodes and antinodes. Overtones are higher frequency modes of vibration. The second overtone corresponds to the third harmonic, where the rope has three segments vibrating, affecting the wavelength and frequency calculations.

Wave Equation for Harmonics

The wave equation for harmonics relates the length of the medium to the wavelength of the standing wave. For a rope fixed at both ends, the wavelength of the nth harmonic is given by λ = 2L/n. Understanding this equation helps calculate the wavelength and frequency of the second overtone, which is the third harmonic.