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 fundamental?

Verified step by step guidance

1. First, understand that the fundamental frequency of a rope stretched between two supports is the lowest frequency at which the rope can vibrate. It corresponds to the situation where there is one antinode in the middle and nodes at the ends.
2. The wavelength of the fundamental frequency is twice the length of the rope. Therefore, calculate the wavelength using the formula: $\lambda =2L$, where $L$ is the length of the rope.
3. Substitute the given length of the rope into the formula: $\lambda =2\times 1.50\text{m}$.
4. Next, use the wave speed formula to find the frequency of the fundamental: $v=f\lambda$, where $v$ is the speed of the wave, $f$ is the frequency, and $\lambda$ is the wavelength.
5. Rearrange the formula to solve for frequency: $f=\frac{v}{\lambda }$. Substitute the values for wave speed and wavelength to find the frequency.

Key Concepts

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 speed of transverse waves on the rope is given as 62.0 m/s, which is crucial for determining the fundamental frequency and wavelength.

Fundamental Frequency

The fundamental frequency is the lowest frequency at which a system vibrates. For a rope fixed at both ends, the fundamental frequency corresponds to the simplest standing wave pattern, with one half-wavelength fitting between the supports. This concept helps in calculating the frequency using the given wave speed and rope length.

Wavelength

Wavelength is the distance between consecutive points of a wave in phase, such as crest to crest. For the fundamental mode on a rope, the wavelength is twice the length of the rope. Understanding this relationship allows us to calculate the wavelength using the rope's length and the wave speed.