A string has a linear density of 8.5 ×10^-3 kg / m and is under a tension of 280 N. The string

step 1 So in this question, we have the third harmonic of a standing wave. First, to find the speed, I

A string has a linear density of 8.5 ×10^-3 kg / m and is...

A string has a linear density of $8.5 \times 10^{-3} \mathrm{~kg} / \mathrm{m}$ and is under a tension of $280 \mathrm{~N}$. The string is $1.8 \mathrm{~m}$ long, is fixed at both ends, and is vibrating in the standing wave pattern shown in the drawing. Determine the (a) speed, (b) wavelength, and (c) frequency of the traveling waves that make up the standing wave.

Key Concepts

Wave Speed on a Stretched String

This concept explains that the speed of a wave traveling along a string is determined by the tension in the string and its linear density. The relationship is given by the formula v = ?(T/?), where v is the wave speed, T is the tension force, and ? is the mass per unit length. It shows that increasing the tension increases wave speed, while increasing the mass per unit length decreases it.

Standing Waves

Standing waves form when two waves of identical frequency and amplitude travel in opposite directions along a medium and interfere, resulting in fixed nodes and oscillating antinodes. In a string fixed at both ends, the boundary conditions force the wave patterns into specific forms, displaying a pattern where only certain wavelengths and frequencies (the harmonics) are allowed.

Harmonic Modes

Harmonic modes refer to the specific standing wave patterns that can exist on a string fixed at both ends. These modes correspond to integer multiples of the fundamental frequency. The fundamental mode (first harmonic) has a wavelength that is twice the length of the string, while higher harmonics have wavelengths given by ?_n = 2L/n, where n is an integer representing the harmonic number.

Frequency-Wavelength Relationship

This relationship is captured by the formula f = v/?, which connects the frequency of the wave, its wavelength, and its speed. It emphasizes that for a given wave speed, a longer wavelength results in a lower frequency and vice versa. This fundamental relationship is essential in analyzing wave motion and properties in any medium.

Transcript

Please give Ace some feedback