The displacement of a string is given by y(x, t)=0.06 (2 / 3) (12… - Vidyadip

MCQ

The displacement of a string is given by $\text{y}(\text{x, t})=0.06\sin(2\pi\text{x}/ 3)\cos(120\pi\text{t})$ where $x$ and $y$ are in $m$ and $t$ in $s.$ The length of the string is $1.5m$ and its mass is $3.0\times10^{-2}\text{kg}$

A
It represents a progressive wave of frequency $60Hz.$
B
It represents a stationary wave of frequency $60Hz.$
C
It is the result of superposition of two waves of wavelength $3m,$ frequency $60Hz$ each travelling with a speed of $180\ m/ s$ in opposite direction.
✓
Both $B$ and $C$

✓

Answer

Correct option: D.

Both $B$ and $C$

We know thet standard equation of stationary wave is $\text{y}(\text{x, t})=\text{a}\sin(\text{kx})\cos(\omega\text{t})$ and given equation is $\text{y}(\text{x, t})=0.06\sin\Big(\frac{2\pi}{3}\Big)\cos[9120\pi)\text{t})$

Comparring both equation $\omega=120\pi$

$2\pi\text{v}=120\pi$ or $\text{v}=\frac{120}{2}=60\text{Hz}$
Verifies the option $(b).$

$\frac{2\pi}{\lambda}=\text{k}$ from $\text{k}=\frac{2\pi}{3}$

$\therefore\frac{2\pi}{\lambda}=\frac{2\pi}{3}$
$\Rightarrow\lambda=3\text{m},\ \text{v}=60\text{Hz}$
speed $\text{v}=\text{v}\lambda=60\times3=180\text{m/ s}.$
Hence, verifies the option $(D).$

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