Question

A transverse wave is travelling along a string from left to right. The fig. represents the shape of the string (snap-shot) at a given instant. At this instant Which points have maximum magnitude of velocity.

Write sum of those points in the answer.

Answer 1

Equation of travelling wave is given by:
$y=A\sin (kx-\omega t)$
Differentiating with respect to time.
$v\left(ycomponent\right)=-A\omega \cos (kx-\omega t)$
The snapshot is taken at an instant ($t$ is fixed), without any loss of generality let us take $t=0$
Then
$v=-A\omega \cos \left(kx\right)$
Now we analyze this function for maxima and minima.
$V$ takes maximum value at $kx=0$;
$V$ takes minimum values at $kx=\pi$
Both of them have the same magnitude but in opposite directions.
$kx=\pi$ implies $x=\frac{\lambda }{2}$
These are the two points where magnitude of velocity is maximum.
These are points $1,5$.
Hence, sum of these two points, $sum=1+5$
$\Rightarrow sum=6$