Question

The equation of a standing wave produced on a string fixed at both ends is
$y=0.4\sin \left[0.314x\right]\cos \left[600\pi t\right]$ where $x$ and $y$ are in $cm$ and $t$ in $seconds.$ The smallest possible length of the string is

• A. $10cm$
• B. $20cm$
• C. $30cm$
• D. $40cm$

Answer 1

The correct option is A $10cm$

$y=0.4\sin \left(0.314x\right)cos\left(600\pi t\right)$
$k=0.314$ $\omega =600\pi =2\pi f$
$=\frac{2\pi }{\lambda }$ $\Rightarrow f=300$
$\Rightarrow \frac{\pi }{10}=\frac{2\pi }{\lambda }$
$\Rightarrow \lambda =20cm$
$v=f\lambda =300\times 20$
$=6000cm/s$
$=60m/s.$
$f=\frac{\nu }{2l}$
$l=\frac{\nu }{2f}=\frac{60}{2\times 300}=0.10m=10cm$