Question

A string vibrates according to equation $y=\sin \frac{\pi x}{3}\cos 40\pi t$. The potential energy of the string will be zero at times

• A. $\frac{1}{20}s$ and $\frac{1}{40}s$
• B. $\frac{1}{40}s$ and $\frac{1}{80}s$
• C. $\frac{1}{80}s$ and $\frac{3}{80}s$
• D. $\frac{1}{40}s$ and $\frac{3}{40}s$

Answer 1

The correct option is C $\frac{1}{80}s$ and $\frac{3}{80}s$

$y=sin\frac{\pi x}{3}\cos 40\pi t$

The potential energy depends on the displacement of particles from the equilibrium position. So P.E is zero only when displacement is $0$.

So let $y=0$

$\sin \left(\frac{\pi x}{3}\right)\cos \left(40\pi t\right)=0$

$\Rightarrow \cos \left(40\pi t\right)=0$

$40\pi t=(2n+1)\frac{\pi }{2}$ (where $n=$ integer)

$80t=2n+1$

$[t=\frac{2n+1}{80}sec]$

$\Rightarrow t=\frac{1}{80},\frac{3}{80},\frac{5}{80},............seconds$

Hence Option (C) is correct.