A string wave equation is given by $y = 0.002 \sin(300t - 15x)$ and mass density is ($\mu = 0.1\text{…
A string wave equation is given by $y = 0.002 \sin(300t - 15x)$ and mass density is ($\mu = 0.1\text{ kg/m}$). Then, find the tension force in the string. [AIIMS 2019]
30 N
20 N
40 N
45 N
Solution
Given, wave equation, $y = 0.002 \sin(300t - 15x)$
Mass density, $\alpha = 0.1\text{ kg/m}$
On comparing the given wave equation with
$y = A \sin(\omega t - kx)$, we get
$\omega = 300, k = 15$
$\therefore$ Velocity of wave, $v = \frac{\omega}{k} = \frac{300}{15} = 20\text{ m/s}$
Wave speed in string is also given by
$v = \sqrt{\frac{T}{\alpha}}$
where, $T = \text{tension in the string}$
and $\alpha = \text{mass per unit length of string.}$
$\therefore 20 = \sqrt{\frac{T}{0.1}}$
$\Rightarrow 400 = \frac{T}{0.1}$
$\Rightarrow T = 40\text{ N}$