Question

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:

• A. 2 cm
• B. 4 cm
• C. 8 cm
• D. 16 cm

Answer 1

Answer: (A)

Distance between consecutive nodes is half of wavelength.
The standard equation of wave is
$y=a\sin kx\cos \omega t\dots$ (i)
where $a$ is amplitude and co is angular velocity.
Given equation is
$y=5\sin \left(\frac{\pi x}{2}\right)\cos 4\pi t\dots$ (ii)
Comparing Eq. (i) with Eq. (ii),
we get $k=\frac{2\pi }{\lambda }=\frac{\pi }{2}$
$\Rightarrow \lambda =4cm$
Hence, distance between two consecutive nodes is
$N=$ Node $=\frac{\lambda }{2}=\frac{4}{2}=2cm$