Question

A transverse sinusoidal wave moves along a string in the positive x - direction at a speed of $10cm{s}^{-1}$. The wavelength of the wave is 0.5 m and its amplitude is 10 cm. At a particular time t, the snap - shot of the wave is shown in figure. The velocity of point P when its displacement is 5 cm is

Answer 1

In a transverse wave, the particle displacement and particle velocity are perpendicular to the direction of propagation of the wave. Hence choices (c) and (d) are wrong. The particle displacements is given by
$y=asin\left[\frac{2\pi }{\lambda }\right(vt-x\left)\right]$ (1)
Putting y = 5 cm and a = 10 cm in Eq. (1), we get
$sin\left[\frac{2\pi }{\lambda }\right(vt-x\left)\right]=\frac{1}{2}$
$\Rightarrow \frac{2\pi }{\lambda }(vt-x)=\frac{\pi }{6}$
Particle velocity is
$V=\frac{dy}{dt}=a\left(\frac{2\pi }{\lambda }v\right)cos\left[\frac{2\pi }{\lambda }\right(vt-x\left)\right]$ (2)
Given a = 0.1 m, $v=0.1m{s}^{-1}$, a = 0.1 m and $\lambda =0.5m$. Also $\frac{2\pi }{\lambda }(vt-x)=\frac{\pi }{6}$. Putting these values in Eq. (2), we get
$V=\frac{\sqrt{3}\pi }{50}m{s}^{-1}$ along the y - axis.