Question

A string wave equation is given $y=0.002 sin(300t-15x)$ and mass density is $\left(\mu=\dfrac{0.1kg}{m}\right).$ Then find the tension in the string.
\[\begin{align}
  & A.30N \\
 & B.20N \\
 & C.40N \\
 & D.45N \\
\end{align}\]

Answer 1

Hint: We know that the waves are caused due to some disturbances which cause the particles in the medium to move. They are broadly two types of waves namely, the longitudinal and transverse wave. These are represented by the wave equation.

Formula used:
$v=\sqrt{\dfrac{T}{\mu}}$ and$v=\dfrac{\omega}{k}$

Complete step by step answer:
The wave equation of a wave is used to describe the motion of the wave and it is given as $y(x,t)=Asin(kx\pm\omega t+\phi)$where,$x$ is the position of the wave at some given time $t$, $A$ is the amplitude, $k$ is the wavenumber and $\omega$ is the angular frequency of the wave. T
Generally, one wavelength is the phase difference is given as $2\pi rad$. From the wave equation, the wave number $k=\dfrac{2\pi}{\lambda}$ and the angular frequency $\omega=\dfrac{2\pi}{T}$ where $T$ is the time period of the wave and it can also be expressed as $T=\dfrac{1}{f}$, $f$ is the frequency of the wave.
Here, given that $y=0.002 sin(300t-15x)$ and mass density is$\left(\mu=\dfrac{0.1kg}{m}\right).$ Comparing the given equation with the standard wave equation $y(x,t)=Asin(\omega t-kx)$,
we get that, $k=15$ and $\omega=300$
Then we can say that the speed of the wave $v=\dfrac{\omega}{k}$
Substituting we have,
$\implies v=\dfrac{300}{15}=20m/s$
Also the speed of the wave is given as $v=\sqrt{\dfrac{T}{\mu}}$, where $T$ is the tension in the string and $\mu$ is the mass density of the string
Then, we have $T=\mu V^{2}$
Then substituting the values in the appropriate place we have
$\implies T=0.1\times 20^{2}$
$\therefore T=40N$

So, the correct answer is “Option C”.

Note:
A sound wave is both longitudinal and transversal wave which needs a medium to propagate due to rarefaction and compression of the waves. Whereas light waves are transverse waves in nature, which don’t need a medium to propagate.