A string with a linear mass density of 0.0062 kg / m and a length of 3.00 m is set into the n=10

step 1 The wavelength of the hundredth mode is given by 2l over 100 equals 2 times 3 meter divided by 1

A string with a linear mass density of 0.0062 kg / m and a...

A string with a linear mass density of $0.0062 \mathrm{kg} / \mathrm{m}$ and a length of $3.00 \mathrm{m}$ is set into the $n=100$ mode of resonance. The tension in the string is $20.00 \mathrm{N}$. What is the wavelength and frequency of the wave?

Key Concepts

Resonance in Strings

Resonance occurs when a system is driven at a frequency that matches one of its natural frequencies, leading to large amplitude oscillations. In the context of a string fixed at both ends, resonance leads to standing waves, and the condition for resonance is that the string length equals an integer multiple of half wavelengths.

Frequency and Wavelength Relationship

In any wave system, the frequency (f), wavelength (?), and wave speed (v) are related by the equation v = f?. This fundamental relationship allows one to determine the frequency of the wave if the speed and wavelength are known, or vice versa.

Wave Speed on a String

The wave speed on a string is determined by the tension and the linear mass density of the string. It is given by the equation v = ?(T/?), where T is the tension and ? is the mass per unit length. This relationship shows that increasing the tension increases the wave speed, while increasing the mass density decreases it.

Standing Waves and Harmonics

Standing waves occur when waves traveling in opposite directions interfere to create a pattern that appears fixed in space, with nodes (points of zero amplitude) and antinodes (points of maximum amplitude). The harmonic modes (or resonant modes) are the specific patterns that satisfy the boundary conditions of the system, typically involving integer multiples of half wavelengths fitting into the length of the string.

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