Transverse waves are traveling on a long string that is...

Transverse waves are traveling on a long string that is under a tension of $4.00 \mathrm{~N}$. The equation describing these waves is $$y(x, t)=(1.25 \mathrm{~cm}) \sin \left[\left(415 \mathrm{~s}^{-1}\right) t-\left(44.9 \mathrm{~m}^{-1}\right) x\right]$$ (a) Find the speed of the wave using the equation. (b) Find the lincar mass density of this string.

Transverse waves are traveling on a long string that is under a tension of $4.00 \mathrm{~N}$. The equation describing these waves is
$$y(x, t)=(1.25 \mathrm{~cm}) \sin \left[\left(415 \mathrm{~s}^{-1}\right) t-\left(44.9 \mathrm{~m}^{-1}\right) x\right]$$
(a) Find the speed of the wave using the equation. (b) Find the lincar mass density of this string.

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Key Concepts

Wave Speed (v)

This concept describes the propagation speed of a traveling wave, which can be determined by the relation v = ?/k, where ? is the angular frequency and k is the wave number. It explains how quickly the disturbance moves along the medium.

Wave Equation Representation

A wave equation of the form y(x, t) = A sin(kx - ?t) represents a traveling wave moving in the positive x direction, where A is the amplitude, k is the wave number indicating spatial periodicity, and ? is the angular frequency indicating time periodicity. This mathematical form encapsulates both the spatial and temporal variations of the wave.

Tension and Linear Mass Density Relationship

In the context of waves on a string, the wave speed is connected to the properties of the medium by the equation v = ?(T/?), where T is the tension in the string and ? is the linear mass density. This relationship shows that higher tension increases wave speed, while greater mass density reduces it.

Linear Mass Density Calculation

Using the relation v = ?(T/?), one can calculate the linear mass density of a string by rearranging the formula to ? = T/v². This concept is important as it connects measurable quantities of the string (tension and wave speed) to its intrinsic mass distribution per unit length.

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