Question

A stretched string is vibrating according to the equation $y=5\sin \left(\frac{\pi x}{3}\right)\cos 400\pi t$ , where $y$ and a are in cm and $t$ is second. Potential energy will be zero at time t:-

• A. 2 s
  
• B. 4 s
  
• C. 8 s
  
• D. 16 s
  

Answer 1

The correct option is C 8 s

A string vibrates according to the equation displacement at $x$ at time $t=y(x,t)=5\sin \left(\frac{\pi x}{3}\right)\cos \left(4\cos \pi t\right).$ This is equation of a standing have of a rope or string fixed at both ends. There is no power transmitted along any direction. The particle at x vibrate in y direction.

The potential energy depends on the displacement of particles from the equilibrium position. So PE is zero only when displacement is zero. At that point kinetic energy will be maximum. So total mechanical energy is conserved.

At all x at the same time he must have $y(n,t)=0$

$\Rightarrow \cos 400{\pi }^{+}=0$

$\Rightarrow 400{\pi }^{+}=\frac{(2x+1)\pi }{2}$ $n=$ integer

$\Rightarrow 800t=2x+1$

$\Rightarrow t=\frac{(2x+1)}{800}$ sec

$\Rightarrow t=\frac{1}{800},\frac{3}{800},\frac{5}{800},\frac{7}{800},\frac{9}{800}------seconds$

Hence, solved.