A 22 -cm-long, 1.0 -mm-diameter copper wire is joined...

A 22 -cm-long, 1.0 -mm-diameter copper wire is joined smoothly to a 60 -cm-long, 1.0 -mm-diameter aluminum wire. The resulting wire is stretched with $20 \mathrm{N}$ of tension between fixed supports $82 \mathrm{cm}$ apart. The densities of copper and aluminum are $8920 \mathrm{kg} / \mathrm{m}^{3}$ and $2700 \mathrm{kg} / \mathrm{m}^{3},$ respectively. a. What is the lowest-frequency standing wave for which there is a node at the junction between the two metals? b. At that frequency, how many antinodes are on the aluminum wire?

A 22 -cm-long, 1.0 -mm-diameter copper wire is joined smoothly to a 60 -cm-long, 1.0 -mm-diameter aluminum wire. The resulting wire is stretched with $20 \mathrm{N}$ of tension between fixed supports $82 \mathrm{cm}$ apart. The densities of copper and aluminum are $8920 \mathrm{kg} / \mathrm{m}^{3}$ and $2700 \mathrm{kg} / \mathrm{m}^{3},$ respectively.
a. What is the lowest-frequency standing wave for which there is a node at the junction between the two metals?
b. At that frequency, how many antinodes are on the aluminum wire?

Key Concepts

Composite String

A composite string or wire consists of segments made from different materials, each potentially having different densities and mechanical properties. This variation can affect the local wave speeds and the overall standing wave patterns, necessitating special consideration of the interface where the segments join, as continuity conditions must be satisfied at that junction.

Mode Shapes

Mode shapes describe the spatial distribution of nodes and antinodes in a vibrating system under resonance conditions. Each mode corresponds to a different harmonic, characterized by an integer number of half-wavelengths fitting within the system's length. These discrete modes determine the possible frequencies at which the system can oscillate.

Boundary Conditions

Boundary conditions specify the behavior of a system at its endpoints or any discontinuities, such as a junction between two different materials. In the context of standing waves, fixed boundaries require the displacement to be zero (nodes), and in composite strings, junctions often impose continuity of displacement and force, leading to additional nodes or specific standing wave patterns.

Wave Speed on a String

The speed of a transverse wave on a string is determined by the tension in the string and its linear mass density, following the equation v = ?(T/µ), where T is the tension and µ is the mass per unit length. This relationship is crucial for predicting how quickly disturbances travel along the string and for determining the frequencies of standing wave patterns.

Standing Waves

Standing waves are interference patterns established by the superposition of two waves traveling in opposite directions along the same medium. They are characterized by fixed points called nodes, where there is no displacement, and antinodes, where the displacement is maximal. Standing waves appear in systems where the boundaries reflect the waves, creating conditions for resonance at specific frequencies.

Linear Mass Density

Linear mass density is the mass per unit length of a string or wire and is calculated by dividing the total mass by its length. It plays a critical role in determining the wave speed, as heavier strings (or those with higher density) tend to support slower wave propagation, affecting the resonance conditions for standing waves.

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