(II) One end of a horizontal string of linear density 6.6 ×10^-4 kg / m is attached to a small-a

step 1 In this cross string, we have a string that is hanging over a pulley and has a mesh attached to

(II) One end of a horizontal string of linear density 6.6...

(II) One end of a horizontal string of linear density $6.6 \times 10^{-4} \mathrm{kg} / \mathrm{m}$ is attached to a small-amplitude mechanical $120-\mathrm{Hz}$ oscillator. The string passes over a pulley, a distance $\ell=1.50 \mathrm{m}$ away, and weights are hung from this end, Fig. $37 .$ What mass $\mathrm{m}$ must be hung from this end of the string to produce $(a)$ one loop, $(b)$ two loops, and $(c)$ five loops of a standing wave? Assume the string at the oscillator is a node, which is nearly true.

Key Concepts

Wave Speed on a Tensioned String

The speed at which waves travel along a string depends on the tension in the string and its linear mass density, following the relationship v = ?(T/?). This concept is crucial for understanding how variations in tension (for example, due to a hanging mass) affect the propagation of waves and the formation of standing waves.

Harmonics and Resonance

Harmonics refer to the distinct vibrational modes that occur at integer multiples of the fundamental frequency. In a string with fixed boundary conditions, the number of segments (or loops) corresponds to the harmonic number, and each mode forms a unique standing wave pattern that resonates at a specific frequency.

Standing Waves

Standing waves occur when waves traveling in opposite directions interfere, creating fixed points of no displacement (nodes) and points of maximum displacement (antinodes). This phenomenon is fundamental in systems with fixed boundaries, as it leads to discrete resonant frequencies corresponding to the vibrational modes of the medium.

Tension Due to a Suspended Mass

When a mass is hung from a string, the gravitational force provides the tension in the string, which is given by the product of the mass and the acceleration due to gravity (mg). This tension is a key factor in determining the wave speed and, consequently, the resonant frequencies and standing wave patterns on the string.

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