Question

A string oscillates according to the equation $y^{\prime }=\left(0.50cm\right)\sin \left[\left(\frac{\pi }{3}c{m}^{-1}\right)x\right]\cos \left[\right(40\pi s^{-1}\left)t\right]$. What is the transverse speed of a particle of the string at the position $x=1.5$cm when $t=\frac{9}{8}s$?

Answer 1

For $x=1.5cm=\frac{3}{2}$

$y=0.5\sin [\frac{\pi }{3}\times \frac{3}{2}]\cos 40\pi t$

$⟹y=0.5\sin \left[\frac{\pi }{2}\right]\cos 40\pi t$

$⟹y=0.5\sin \cos 40\pi t$

Transverse speed=$v=\frac{dy}{dt}=-20\sin 40\pi t=-20\sin [40\pi \times \frac{9}{8}]$

$v=\frac{dy}{dt}=-20\sin 40\pi t=-20\sin \left[45\pi \right]$

$⟹v=0$