Question

The transverse displacement of a string clamped at both ends is given by $y(x,t)=0.06\sin (\dfrac{2\pi x}{3})\cos 60\pi t$ .where x and y are in m and t in s, the length of the string is $1.5m$ and its mass is $3\times {{10}^{-2}}k{{g}^{2}}$ .The tension developed in string is ?
A) $81N$
B) $162N$
C) $90N$
D)$180N$

Answer 1

Hint: Travelling wave creates a disturbance in the transmission line and it occurs for a short duration of time which is in microseconds and travelling wave is a temporary wave .Travelling wave is important and it helps in knowing the voltage and current in the transmission line. Travelling waves can be represented mathematically.

Complete step-by-step solution:
The travelling wave corresponding to a solution of the linearized Euler equations
Travelling wave is given by
 $y(x,t)=A\sin (\omega t\pm kx+\varphi )$ $\cdots \cdots (1)$
From the data
 $y(x,t)=0.06\sin (\dfrac{2\pi x}{t})\cos 60\pi t$ $\cdots \cdots (2)$
Compare equation (1) and (2) we get
A=0.06, \[\omega =60\pi \], $k=\dfrac{2\pi }{3}$
Wavelength, $\lambda =\dfrac{2\pi }{k}$
On substituting
$\lambda =\dfrac{2\pi }{(\dfrac{2\pi }{3})}$
$\lambda =3m$
Frequency $f=\dfrac{\omega }{2\pi }$
On substituting value of $\omega $
$f=\dfrac{60\pi }{2\pi }$
$f=30Hz$
Speed,$v=f\lambda $
On substituting value of $f$ and$\lambda $
$v=90m{{s}^{-1}}$
The speed of a wave in a string is given by:
$\begin{align}
  & v=\sqrt{\dfrac{T}{\mu }} \\
 & T=\dfrac{{{v}^{2}}m}{l} \\
\end{align}$
$T=\dfrac{{{(90)}^{2}}\times 3\times 1{{0}^{-2}}}{1.5}$
$T=162N$
So the correct option is B.

Note: Students travelling wave which helps in designing the insulator and also protective equipment a travelling wave is commonly represented in step and infinite rectangular wave. Surge which occurs due to movement of charges along the conductor is also a type of travelling wave. Speed of waves depends on the tension and linear mass density.