Q. A stretched string is vibrating according to the equation $y=5 \sin \left(\frac{\pi x}{2}\right) \cos 4 \pi t$ where $y$ and $x$ are in $cm$ and $t$ in second. The distance between two consecutive nodes on the string is:
Delhi UMET/DPMTDelhi UMET/DPMT 2004
Solution:
Distance between consecutive nodes is half of wavelength.
The standard equation of wave is
$y=a \sin k x \cos \omega t \ldots$ (i)
where $a$ is amplitude and co is angular velocity.
Given equation is
$y=5 \sin \left(\frac{\pi x}{2}\right) \cos 4 \pi t \ldots$ (ii)
Comparing Eq. (i) with Eq. (ii),
we get $k=\frac{2 \pi}{\lambda}=\frac{\pi}{2}$
$\Rightarrow \lambda=4 cm$
Hence, distance between two consecutive nodes is
$N =$ Node $=\frac{\lambda}{2}=\frac{4}{2}=2 cm$
