A cord of length 1.5 m is fixed at both ends. Its mass per unit length is 1.2 g / m and the tens

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A cord of length 1.5 m is fixed at both ends. Its mass per...

A cord of length $1.5 \mathrm{m}$ is fixed at both ends. Its mass per unit length is $1.2 \mathrm{g} / \mathrm{m}$ and the tension is $12 \mathrm{N}$. (a) What is the frequency of the fundamental oscillation? (b) What tension is required to make the $n=3$ mode have a frequency of $0.50 \mathrm{kHz}$ ?

Key Concepts

Standing Waves on a Fixed String

When a string is fixed at both ends, only certain wavelengths, and hence frequencies, are allowed. These specific patterns of vibration, known as standing waves, occur because the reflections at the boundaries interfere with the incoming waves, producing nodes at the fixed ends. Understanding these boundary conditions is fundamental to analyzing the vibrational modes of the string.

Fundamental Frequency and Harmonics

The fundamental frequency is the lowest natural frequency at which a string vibrates, corresponding to the simplest mode with just one antinode in the center. Higher modes, or harmonics, have frequencies that are integer multiples of the fundamental frequency. This concept is crucial because it links the mode number (n) to the actual frequency observed in the vibration.

Wave Speed on a String and Its Relation to Tension and Mass Density

The speed of a wave on a string is determined by the tension in the string and its mass per unit length. This relationship is given by the formula v = ?(T/?), where T is the tension and ? is the linear mass density. This concept is essential for calculating how fast disturbances travel along the string, which in turn affects the frequencies of the standing waves.

Frequency Formula for Vibrating Strings

The frequencies of the standing wave modes on a string fixed at both ends are given by the formula f_n = (n/2L) ?(T/?), where n is the mode number, L is the length of the string, T is the tension, and ? is the linear mass density. This equation ties together the physical properties of the string with the observable frequencies, making it a central concept in the study of wave mechanics on strings.

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