Question

A standing wave $y=\text{A sin}\left(\frac{20\pi x}{3}\right)\text{cos (1000}\pi \text{t)}$ is set up in a taut string, where $x$ and $y$ are in metre and $t$ is in seconds. The distance between first two successive points oscillating with the amplitude $\frac{A}{2}$, from the fixed end will be equal to

• A. $10\text{cm}$
• B. $15\text{cm}$
• C. $2.5\text{cm}$
• D. $4\text{cm}$

Answer 1

The correct option is A $10\text{cm}$

For the standing wave represented by
$y=\text{A sin}\left(\frac{20\pi x}{3}\right)\text{cos}\left(1000\pi t\right)$


A particle on string in a standing wave execute SHM with amplitude of
$A_x=\text{A sin}\left(\frac{20\pi x}{3}\right)$
$\Rightarrow$ Putting $A_x=\frac{A}{2}$
$\Rightarrow \text{sin}\left(\frac{20\pi x}{3}\right)=\frac{1}{2}$
$\Rightarrow \left(\frac{20\pi x}{3}\right)=\frac{\pi }{6},\frac{5\pi }{6}$
$\Rightarrow x=\frac{1}{40},\frac{1}{8}\text{m}$
So, distance between two consecutive points having amplitude $\frac{A}{2}$ is $\frac{1}{8}-\frac{1}{40}=\frac{1}{10}\text{m}=10\text{cm}$