A 50-cm -long wire with a mass of 1.0 g and a tension of 440...

A $50-\mathrm{cm}$ -long wire with a mass of $1.0 \mathrm{g}$ and a tension of $440 \mathrm{N}$ passes across the open end of an open-closed tube of air. The wire, which is fixed at both ends, is bowed at the center so as to vibrate at its fundamental frequency and generate a sound wave. Then the tube length is adjusted until the fundamental frequency of the tube is heard. What is the length of the tube? Assume $v_{\text {sound }}=340 \mathrm{m} / \mathrm{s}$.

Key Concepts

Wave Propagation on a String

This concept involves understanding how waves travel along a string that is under tension. The wave speed on a string is determined by the tension in the string and its mass per unit length (linear density), according to the equation v = ?(T/?). This is a foundational concept for analyzing vibrations and resonant frequencies in mechanical systems.

Fundamental Frequency of a Vibrating String

For a string with fixed ends, the fundamental frequency is given by f = v/(2L) where v is the wave speed and L is the length of the string. This formula is crucial because it directly relates the physical properties of the string, such as tension and mass per unit length, to the sound produced by its vibration.

Standing Waves in Air Columns

In acoustics, air columns within tubes can support standing waves. For an open-closed tube, the fundamental frequency is established by the condition that the column has a node at the closed end and an antinode at the open end. The fundamental frequency in such a tube is given by f = v/(4L), where v is the speed of sound in air and L is the length of the tube.

Resonance Matching Between Systems

This concept involves adjusting one system so that its frequency matches the resonance of another system. In problems where a vibrating string generates sound that is coupled with an air column’s resonance, the frequency of the string is matched with the fundamental frequency of the air column. This matching is critical for achieving effective energy transfer and resonance.

Mass per Unit Length (Linear Density)

The mass per unit length is calculated by dividing the total mass of the string by its total length. This value (?) is a key parameter in determining the wave speed along the string, which in turn affects its vibrational frequency. Understanding how to compute linear density from mass and length is a fundamental skill in wave mechanics.

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