Question

The vibrations of a string of length $60cm$ fixed at both ends are represented by the equation
$y=4\sin \left(\frac{\pi x}{15}\right)\cos \left(96\pi t\right)$, where $x$ and $y$ are in $cm$ and $t$ in 𝑠𝑒𝑐𝑜𝑛𝑑. The maximum displacement (in $cm$) at $x=\frac{5}{2}cm$ is

Answer 1

Given, length of the string $=60cm$,
Equation of vibration $y=4\sin \left(\frac{\pi x}{15}\right)\cos \left(96\pi t\right)$
For $x=\frac{5}{2}$,
$y=4\sin \left(\frac{\frac{5\pi }{2}}{15}\right)\cos \left(96\pi t\right)$ = $2cos\left(96\pi t\right)$
So, $y$ will be maximum when $\cos \left(96\pi t\right)=1$
$y_{max}=2cm$ at $x=\frac{5}{2}cm$.
Final answer: ($2)$