It is observed that a pulse requires 0.2 sec to travel from one end to the other

Question

It is observed that a pulse requires 0.2 sec to travel from one end to the other of a long string. The tension in the string is provided by passing the string over a pulley to a weight which has 50 times the mass of the string.
(a) What is the length of the string?
(b) What is the equation of the third normal mode?
(c) If a traveling wave is set up in it with the following equation:
Where x and y are in meters, t in seconds, and v is the wave velocity (which you can calculate). Find the transverse displacement and velocity of the string at the point x=4 m at the time t=0.1 sec.

Explanation / Answer

Given Time taken to travel for pulse , t = 0.2 s Tension in the string , T = weight of the object (w )= 50 (mass of the string) a) From the relation     v = T / (m /L)       = 50 m / (m /L)     v = 50 L ---1 But velocity, v = L / 0.2 s   Taking e q1 L / 0.2 s = 50 L    L^2 /0. 04 = 50 L     L = 2 units b) The equation of third normal mode is      f3 = 3 v / 2L v is the wave speed and L is the length of the string c) The wave equation is    y(x, t) = A sin (kx - t)     You didnot give the equation in part c    

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