Question

A transverse wave is travelling along a string from left to right. The adjoining figure represents the shape of the string at a given instant. At this instant, among the following statements, choose the incorrect one.

• A. Points $D,E$ and $F$ have upward (positive) velocity.
• B. Points $A,B$ and $H$ have downward (negative) velocity.
• C. Point $C$ and $G$ have zero velocity.
• D. Points $A$ and $E$ have minimum velocity.

Answer 1

The correct option is D Points $A$ and $E$ have minimum velocity.

Particle velocity is,
$\frac{dy}{dt}=-v\left(\frac{dy}{dx}\right)$
$v_p=-\text{wave velocity}\times \text{slope of the wave}$
Option$\left(a\right)$:
For upward velocity, $v_p=+ve$, so slope must be negative. Slope is negative at the points $D,E$ and $F$.
Hence points $D,E$ and $F$ will have upward velocity.


Option$\left(b\right)$:
For downward velocity, $v_p=-ve$, so slope must be positive. Slope of the wave is positive at the points $A$ and $B,H$
Hence points $A,B,H$ will have downward velocity.

Option$\left(c\right)$:
For zero velocity, $v_p=0$, so slope of wave must be zero. Tangent drawn at points $C,G$ is parallel to the $x-$axis, hence $\tan \theta =\text{slope}=0$ at points $C,G$

Option$\left(d\right)$:
For maximum magnitude of velocity, $\left|\text{slope}\right|=\text{maximum}$
At points $A$ and $E$ the angle$\left(\theta \right)$ made by tangent from horizontal is maximum, i.e magnitude of $\tan \theta$ is maximum at $A$ and $E$.
$\therefore$Option$\left(d\right)$ is incorrect.