Question

The equation for the vibration of a string fixed at both ends vibrating in its third harmonic is given by $y=2\text{(cm)}\sin \left[\right(0.6{\text{cm}}^{-1}x\left]\cos \right[\left(500\pi {\text{s}}^{-1}\right)t].$
What is its length ?

• A. $24.6\text{cm}$
• B. $12.5\text{cm}$
• C. $20.6\text{cm}$
• D. $15.7\text{cm}$

Answer 1

The correct option is D $15.7\text{cm}$

Given :
$y=2\text{(cm)}\sin \left[\right(0.6{\text{cm}}^{-1}x\left]\cos \right[\left(500\pi {\text{s}}^{-1}\right)t]$
on comparing with standard equation
$\omega =500\pi =2\pi f$
$A=2\text{(cm)}$
$k=0.6=\frac{2\pi }{\lambda }$
$\lambda =\left(\frac{2\pi }{0.6}\right)\text{(cm)}$
Now string length is given by $=\frac{3\lambda }{2}=\frac{3}{2}\left(\frac{2\pi }{0.6}\right)=15.7\text{(cm)}$.