Question

A wave travelling on a string at a speed of $10m{s}^{-1}$ causes each particle of the string to oscillate with a time period of 20 ms. (a) What is the wavelength of the wave ? (b) If the displacement of a particle is 1.5 mm at a certain instant, what will be the displacement of a particle 10 cm away from it at the same instant ?

Answer 1

Wave speed = v = 10 m/sec

Time period = T = 20 ms

$=20\times {10}^{-3}=2\times {10}^{-2}sec$

(a) Wavelength,

$\lambda =vt=10\times 2\times {10}^{-2}$

$=0.02m=20cm$

(b) Wavelength,

$\lambda =20cm$

$\therefore$ Phase difference $=\left(\frac{2\pi }{\lambda }\right).x$

$=\left(\frac{2\pi }{20}\right)\times 10=\pi rad$

$y_1=asin(\omega t-kx)$

$\Rightarrow$ $1.5=asin(\omega t-kx)$

So, the displacement of the particle at a distance x = 10 cm

$[\varphi =\frac{2\pi x}{\lambda }=2\pi \times \frac{10}{20}=\pi ]$ is given by

$y_2=asin(\omega t-kx+\pi )$

$=asin(\omega t-kx)=1.5mm$

$\therefore$ Displacement = 1.5 mm