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 zero velocity

Answer 1

Equation of travelling wave is given by:
$y=Asin(kx-\omega t)$
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
$v=-A\omega cos\left(kx\right)$
For $v$ to be zero $kx$ must be equal to either $\frac{\pi }{2}$ or $\frac{3\pi }{2}$
In other words $x$ must be equal to $\frac{\lambda }{4}$ or $\frac{3\lambda }{4}$
$3$ and $7$ satisfy this.