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Chapter 11: Problem 51

A guitar's E-string has length \(65 \mathrm{cm}\) and is stretched to a tension of \(82 \mathrm{N}\). It vibrates at a fundamental frequency of $329.63 \mathrm{Hz}$ Determine the mass per unit length of the string.

Short Answer

Expert verified
Answer: The mass per unit length of the guitar's E-string is approximately \(7.68\times10^{-4}\,\text{kg/m}\).

Step by step solution

01

Convert given information to SI units

We are given the string length \(L = 65\,\text{cm}\). First, we need to convert it to meters: $$ L = 65\,\text{cm} \times \frac{1\,\text{m}}{100\,\text{cm}} = 0.65\,\text{m}. $$ The tension, \(T\), is given in SI units, so \(T = 82\,\text{N}\).
02

Rearrange the formula for mass per unit length

To find the mass per unit length, we will rearrange the fundamental frequency formula to solve for \(μ\): $$ μ = \frac{T}{(2Lf)^2}. $$
03

Plug in given values and calculate

Now, we can plug in the given values for string length (\(L = 0.65\,\text{m}\)), tension (\(T = 82\,\text{N}\)), and fundamental frequency (\(f = 329.63\,\text{Hz}\)) to find the mass per unit length: $$ μ = \frac{82\,\text{N}}{(2\times0.65\,\text{m}\times329.63\,\text{Hz})^2} = \frac{82}{(1.3\times329.63)^2}\,\text{kg/m}. $$ After doing the math, we get: $$ μ \approx 7.68\times10^{-4}\,\text{kg/m}. $$ The mass per unit length of the guitar's E-string is approximately \(7.68\times10^{-4}\,\text{kg/m}\).

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Most popular questions from this chapter

A transverse wave on a string is described by the equation $y(x, t)=(2.20 \mathrm{cm}) \sin [(130 \mathrm{rad} / \mathrm{s}) t+(15 \mathrm{rad} / \mathrm{m}) x].$ (a) What is the maximum transverse speed of a point on the string? (b) What is the maximum transverse acceleration of a point on the string? (c) How fast does the wave move along the string? (d) Why is your answer to (c) different from the answer to (a)?
A longitudinal wave has a wavelength of \(10 \mathrm{cm}\) and an amplitude of \(5.0 \mathrm{cm}\) and travels in the \(y\) -direction. The wave speed in this medium is \(80 \mathrm{cm} / \mathrm{s},\) (a) Describe the motion of a particle in the medium as the wave travels through the medium. (b) How would your answer differ if the wave were transverse instead?
(a) Plot a graph for $$ y(x, t)=(4.0 \mathrm{cm}) \sin [(378 \mathrm{rad} / \mathrm{s}) t-(314 \mathrm{rad} / \mathrm{cm}) x] $$ versus \(x\) at \(t=0\) and at \(t=\frac{1}{480} \mathrm{s}\). From the plots determine the amplitude, wavelength, and speed of the wave. (b) For the same function, plot a graph of \(y(x, t)\) versus \(t\) at \(x=0\) and find the period of the vibration. Show that \(\lambda=v T.\)
A metal guitar string has a linear mass density of $\mu=3.20 \mathrm{g} / \mathrm{m} .$ What is the speed of transverse waves on this string when its tension is \(90.0 \mathrm{N} ?\)
What is the speed of the wave represented by \(y(x, t)=\) $A \sin (k x-\omega t),\( where \)k=6.0 \mathrm{rad} / \mathrm{cm}\( and \)\omega=5.0 \mathrm{rad} / \mathrm{s} ?$
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