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Chapter 11: Problem 8

When the tension in a cord is \(75 \mathrm{N}\), the wave speed is $140 \mathrm{m} / \mathrm{s} .$ What is the linear mass density of the cord?

Short Answer

Expert verified
Answer: The linear mass density of the cord is approximately 0.0038 kg/m.

Step by step solution

01

Rearrange the formula for the linear mass density

In order to find the linear mass density (\(\mu\)), we'll first rearrange the formula \(v = \sqrt{\frac{T}{\mu}}\). Square both sides of the equation to get rid of the square root: \(v^2 = \frac{T}{\mu}\) Now, multiply both sides by \(\mu\): \(\mu v^2 = T\) Finally, divide both sides by \(v^2\) to solve for \(\mu\): \(\mu = \frac{T}{v^2}\)
02

Substitute the given values

Now that we have the formula \(\mu = \frac{T}{v^2}\), we can plug in the given values for tension (\(T=75\,\mathrm{N}\)) and wave speed (\(v=140\,\mathrm{m/s}\)): \(\mu = \frac{75\,\mathrm{N}}{(140\,\mathrm{m/s})^2}\)
03

Calculate the linear mass density

Finally, we calculate the value of the linear mass density: \(\mu = \frac{75\,\mathrm{N}}{19600\,\mathrm{m^2/s^2}}\) \(\mu = 0.0038265\,\mathrm{kg/m}\) So the linear mass density of the cord is approximately \(0.0038\,\mathrm{kg/m}\).

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Most popular questions from this chapter

The speed of sound in air at room temperature is \(340 \mathrm{m} / \mathrm{s}\) (a) What is the frequency of a sound wave in air with wavelength $1.0 \mathrm{m} ?$ (b) What is the frequency of a radio wave with the same wavelength? (Radio waves are electromagnetic waves that travel at $3.0 \times 10^{8} \mathrm{m} / \mathrm{s}$ in air or in vacuum.)
(a) Sketch graphs of \(y\) versus \(x\) for the function $$ y(x, t)=(0.80 \mathrm{mm}) \sin (k x-\omega t) $$ for the times \(t=0,0.96 \mathrm{s},\) and \(1.92 \mathrm{s} .\) Make all three graphs of the same axes, using a solid line for the first, a dashed line for the second, and a dotted line for the third. Use the values \(k=\pi /(5.0 \mathrm{cm})\) and $\omega=(\pi / 6.0) \mathrm{rad} / \mathrm{s}$ (b) Repeat part (a) for the function $$ y(x, t)=(0.50 \mathrm{mm}) \sin (k x+\omega t) $$ (c) Which function represents a wave traveling in the \(-x\) direction and which represents a wave traveling in the \(+x\) -direction?
A sound wave with intensity \(25 \mathrm{mW} / \mathrm{m}^{2}\) interferes destructively with a sound wave that has an intensity of $28 \mathrm{mW} / \mathrm{m}^{2} .$ What is the intensity of the superposition of the two?
A wave on a string has equation $$ y(x, t)=(4.0 \mathrm{mm}) \sin (\omega t-k x) $$ where \(\omega=6.0 \times 10^{2} \mathrm{rad} / \mathrm{s}\) and $k=6.0 \mathrm{rad} / \mathrm{m} .$ (a) What is the amplitude of the wave? (b) What is the wavelength? (c) What is the period? (d) What is the wave speed? (e) In which direction does the wave travel?
A uniform string of length \(10.0 \mathrm{m}\) and weight \(0.25 \mathrm{N}\) is attached to the ceiling. A weight of \(1.00 \mathrm{kN}\) hangs from its lower end. The lower end of the string is suddenly displaced horizontally. How long does it take the resulting wave pulse to travel to the upper end? [Hint: Is the weight of the string negligible in comparison with that of the hanging mass?]
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