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Found in: Page 473

### Fundamentals Of Physics

Book edition 10th Edition
Author(s) David Halliday
Pages 1328 pages
ISBN 9781118230718

# The linear density of a string is ${\mathbf{1}}{\mathbf{.}}{\mathbf{6}}{\mathbf{×}}{{\mathbf{10}}}^{\mathbf{-}\mathbf{4}}{\mathbit{k}}{\mathbit{g}}{\mathbf{/}}{\mathbit{m}}$. A transverse wave on the string is described by the equation ${\mathbit{y}}{\mathbf{=}}\left(0.021m\right){\mathbf{}}{\mathbit{s}}{\mathbit{i}}{\mathbit{n}}\left[\left(2.0{m}^{-1}\right)x+\left(30{s}^{-1}\right)t\right]$. (a)What are the wave speed and (b) What is the tension in the string?

a) The speed of the wave is 15m/s

b) The tension in the string is $3.60×{10}^{-3}\mathrm{N}$.

See the step by step solution

## Step 1: The given data

• The wave equation is given as $y=\left(0.021m\right)\mathrm{sin}\left[\left(2.0{m}^{-1}\right)x+\left(30{s}^{-1}\right)t\right]$
• Linear density of a string, $\left(\mathrm{\mu }\right)=1.6×{10}^{-4}\mathrm{kg}/\mathrm{m}$
• Wavelength, $\left(\mathrm{\lambda }\right)=0.5\mathrm{m}$
• Frequency, $\left(f\right)=30{s}^{-1}$

## Step 2: Understanding the concept of wave equation

The product of wavelength and frequency of the wave is called speed of the wave. the speed of the wave in a stretched string is directly proportional to the square-root of the tension force and inversely proportional to the square-root of linear density of the string.

Formula:

The wave speed of the wave, $v=n×\lambda$ (i)

The velocity of the wave,$v=\sqrt{\frac{T}{\mu }}$ (ii)

## Step 3: a) Calculation of the wave speed

Using equation (i), the wave speed is given as:

$\mathrm{v}=30{\mathrm{s}}^{-1}×0.5\mathrm{m}\phantom{\rule{0ex}{0ex}}=15\mathrm{m}/\mathrm{s}$

Hence, the value of wave speed is 15 m/s

## Step 4: b) Calculation of tension in the string

Using equation (ii), the tension in the string is given as:

$T={v}^{2}\mu \phantom{\rule{0ex}{0ex}}={\left(15\mathrm{m}/\mathrm{s}\right)}^{2}×\left(1.6×{10}^{-4}\mathrm{kg}/\mathrm{m}\right)\phantom{\rule{0ex}{0ex}}=3.60×{10}^{-3}\mathrm{N}$

Hence, the value of the tension in the string is $3.60×{10}^{-3}\mathrm{N}$.