Question

The equation of a standing wave in a stretched string is given by $y=5\sin \left(\frac{\pi x}{3}\right)\cos \left(40\pi t\right)$, where $x$ and $y$ are in $\text{cm}$ and $t$ is in $\text{sec}$. The separation between two consecutive nodes is (in $\text{cm}$)

• A. $1.5$
• B. $3$
• C. $6$
• D. $4$

Answer 1

The correct option is B $3$

Given that, equation of standing wave is
$y=5\sin \left(\frac{\pi x}{3}\right)\cos \left(40\pi t\right)$

Comparing this with
$y=2A\sin \left(\frac{2\pi x}{\lambda }\right)\cos \left(\frac{2\pi t}{T}\right)$

we get, $\lambda =6\text{cm};T=\frac{1}{20}\text{sec}$

Position of a node on the string is given by $x=\frac{n\lambda }{2}$, where $n=0,1,2,3,\mathrm{4.....}$

Seperation between two consecutive nodes is given by
$\Delta x=|\frac{n\lambda }{2}-\frac{(n-1)\lambda }{2}|=\frac{\lambda }{2}$

From the given data,
$\Delta x=\frac{6}{2}=3\text{cm}$

Alternate method :

At the nodes, the value of $x$ is such that $y=0$.
To make sin function zero,
$\frac{\pi x_n}{3}=n\pi$
For $n=0,x_0=0\text{cm}$
For $n=1,x_1=3\text{cm}$
$\therefore$ seperation between two consecutive nodes is $\Delta x=|x_1-x_0|=3\text{cm}$

Hence, option (b) is the correct answer.