Question

A transverse wave of frequency $500Hz$ and speed $100m/s$ is travelling in the positive x direction on a long string. At time $t=0s$ the displacements at $x=0.0m$ and at $x=0.25m$ are $0.0m$ and $0.02m$, respectively. The displacement at $x=0.2m$ at $t=5\times {10}^{-4}s$ is?

• A. $-0.04$m
• B. $-0.02$m
• C. $0.04$m
• D. $0.02$m

Answer 1

The correct option is B $-0.02$m

Wavelength is $\lambda =v/f=100/500=0.2m$

Let the wave equation be:

$y=A\sin (kx-\omega t+\varphi )$

Given, $y=0$ at $t=0,x=0$

$0=A\sin \left(\varphi \right)$

$⟹\varphi =0$

Also, $y=0.02$ at $t=0,x=0.25$

$0.02=A\sin \left(\frac{2\pi x}{\lambda }\right)$

$0.02=A\sin \left(\frac{2\pi \times 0.25}{0.2}\right)$

$⟹A=0.02m$

At $t=5\times {10}^{-4}s$ and $x=0.2m$,

$y=0.02\sin (\frac{2\pi \times 0.2}{0.2}-2\pi \times 5\times {10}^{-4}\times 500)$

$=0.02\sin (2\pi -\pi /2)$

$=-0.02m$