Question

A string vibrates according to equation, y = 5sin$\frac{ \pi x}{3}$cos400πt . Potential energy of the string will be zero at time t equal to
A) $\frac{1}{200}$ and $\frac{1}{400}$
B) $\frac{1}{400}$ and $\frac{1}{800}$
C) $\frac{1}{800}$ and $\frac{3}{800}$
D) $\frac{1}{400}$ and $\frac{3}{400}$

Answer 1

Answer (C).

A string vibrates according to the equation :

 Displacement at x at time t = y(x, t) = 5 Sin (πx/3)  Cos (400πt)

     This is an equation of a standing wave on a rope or string fixed at both ends. We can identify that it is standing wave as x and t are arguments in separate trigonometric functions, and they are multiplied.

    There is no power transmitted along any direction. The particles at any 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 of time KE will be maximum. So total mechanical energy is conserved.

=>   At all x, at the same time we must have  y (x,t) = 0
=>   Cos 400πt = 0
=>   400πt = (2n+1)π/2   ,   n = integer
=>   800 t  =  2n + 1
=>   t = (2n + 1) / 800 sec
=>   t = 1/800, 3/800, 5/800, 7/800 , 9/800 , .... seconds.