Question

The equation of a transverse wave traveling on a rope is given by $y=10\sin \pi (0.01x-2.000t)$ where $y$ and $x$ are in cm and $t$ in seconds. The maximum transverse speed of a particle in the rope is about

• A. $62.8cm/s$
• B. $75cm/s$
• C. $100cm/s$
• D. $121cm/s$

Answer 1

The correct option is A $62.8cm/s$

Given a string wave of the form:

$y=A_o\sin (kx-\omega t)$

The transverse speed is the rate change of the y coordinate of the particle on the rope

so,

$v_y=\frac{\partial y}{\partial t}$

$v_y=\frac{\partial }{\partial t}\left(A_o\sin \pi (kx-\omega t)\right)$

$v_y=-A_o\omega \pi \cos \pi (kx-\omega t)$

Now for maximum transverse speed $\cos \pi (kx-\omega t)=1$

So, $v_{y\left(max\right)}=A_o\pi \omega$

Putting the values

$v_{y\left(max\right)}=10\times \pi \times 2.000=62.8cm/s$

So A is the right answer.