Question

The vibrations of a string of length $20\text{cm}$ fixed at both ends are represented by the equation $y=2\sin \left(\frac{\pi x}{5}\right)\cos \left(12\pi t\right)$, where $x$ and $y$ are in $\text{cm}$ and $t$ in $\text{sec}$. How many nodes are located along the string?

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

The correct option is C $5$

Given equation for string
$y=2\sin \left(\frac{\pi x}{5}\right)\cos \left(12\pi t\right)$

Comparing this with

$y=2A\sin \left(\frac{2\pi x}{\lambda }\right)\cos \left(\frac{2\pi t}{T}\right)$
we get, $\lambda =10\text{cm}$

Position of a node on the string is given by $x=\frac{n\lambda }{2}$ , where $n=0,1,2,\mathrm{3.....}$
From the given data,
$x=5n\Rightarrow x=0,5,10,15,20$
So for a $20\text{cm}$ long string, $5$ nodal points are formed.

Thus, option (c) is the correct answer.