Question

A standing wave set up in a medium is $y=4\cos \left(\frac{\pi x}{3}\right)\sin 40\pi t$ where $x,y$ are in $\text{cm}$ and $t$ in $\text{sec}$. The velocity of particle at $x=6\text{cm}$ at $t=\frac{1}{8}\text{sec}$ is

• A. $120\pi \text{cm/s}$
• B. $-160\pi \text{cm/s}$
• C. $-80\pi \text{cm/s}$
• D. $40\pi \text{cm/s}$

Answer 1

The correct option is B $-160\pi \text{cm/s}$

Given that,
$x=6\text{cm};t=\frac{1}{8}\text{sec}$
Equation of standing wave is
$y=4\cos \left(\frac{\pi x}{3}\right)\sin 40\pi t$
By differentiating we get velocity of particle as,
$V=\frac{dy}{dt}=4\cos \left(\frac{\pi \times 6}{3}\right)\left[\right(40\pi \left)\cos 40\pi \right(\frac{1}{8})$]
$=160\pi \cos \left(2\pi \right)\left(\cos 5\pi \right)$
$=160\pi \left(1\right)(-1)$
$V=-160\pi \text{cm/s}$
Hence, the correct answer is (d).