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Q17P

Expert-verifiedFound in: Page 473

Book edition
10th Edition

Author(s)
David Halliday

Pages
1328 pages

ISBN
9781118230718

**The linear density of a string is ${\mathbf{1}}{\mathbf{.}}{\mathbf{6}}{\mathbf{\times}}{{\mathbf{10}}}^{\mathbf{-}\mathbf{4}}{\mathit{k}}{\mathit{g}}{\mathbf{/}}{\mathit{m}}$****. A transverse wave on the string is described by the equation ${\mathit{y}}{\mathbf{=}}{\left(0.021m\right)}{\mathbf{}}{\mathit{s}}{\mathit{i}}{\mathit{n}}{\left[\left(2.0{m}^{-1}\right)x+\left(30{s}^{-1}\right)t\right]}$****. (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 $3.60\times {10}^{-3}\mathrm{N}$.

- 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, $\left(\mathrm{\mu}\right)=1.6\times {10}^{-4}\mathrm{kg}/\mathrm{m}$
- Wavelength, $\left(\mathrm{\lambda}\right)=0.5\mathrm{m}$
- Frequency, $\left(f\right)=30{s}^{-1}$

**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\times \lambda $ (i)

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

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

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

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

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

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

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

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