Question

The equation of a standing wave is $y=4\sin \left(\frac{\pi x}{5}\right)\cos \left(100\pi t\right)$, where $y$ and $x$ are in $\text{cm}$ and $t$ in $\text{s}$. The wave is formed using a string of length $20\text{cm}$. The second and $4^{th}$ antinodes are located at positions (in $\text{cm}$)

• A. $2.5,7.5$
• B. $7.5,11.5$
• C. $7.5,17.5$
• D. $7.5,12.5$

Answer 1

The correct option is C $7.5,17.5$


Given equation is

$y=4\sin \left(\frac{\pi x}{5}\right)\cos \left(100\pi t\right)$

$k=\frac{\pi }{5};\omega =100\pi$

So, $k=\frac{2\pi }{\lambda }=\frac{\pi }{5}\Rightarrow \lambda =10\text{cm}$

The distance of the anti nodes = $\frac{\lambda }{4},\frac{3\lambda }{4},...$

So,
Position of second antinode $d_2=\frac{3\lambda }{4}=7.5\text{cm}$

And, position of fourth antinode $d_4=\frac{7\lambda }{4}=17.5\text{cm}$

Hence, option $\left(c\right)$ is the correct answer.