Question

The wave-function for a certain standing wave on a string fixed at both ends is $y(x,t)=0.5\sin \left(0.025\pi x\right)\cos 500t$ where $x$ and $y$ are in centimeters and t is in seconds. The shortest possible length of the string is:

• A. $126cm$
• B. $160cm$
• C. $40cm$
• D. $80cm$

Answer 1

The correct option is C $40cm$

For the shortest possible length, it should be allowing fundamental frequency resonance.

In fundamental frequency

$L=\frac{\lambda }{2}(\because K=\frac{2\pi }{\lambda })$

$=\frac{2\pi /K}{2}=\frac{\pi }{K}$

from $y=0.5\sin \left(0.025\pi x\right)\cos \left(500t\right)$

$K=0.025\pi (oncomparingwithy=A\sin \left(Kx\right)\cos \left(wt\right))$

$\therefore L=\frac{\pi }{0.025\pi }=\frac{1000}{25}$

$[L=40cm]$

Hence Option (C) is correct.