The equation of a standing wave produced on a string is given by y = 4 ( x 3 ) (20 t) , where x and y are in cm and t is in seconds. The maximum speed of a particle located at x = 1 cm is v₀ cm/s . The value of v₀ is ________.

Step-by-step solution

The given standing wave equation is y = 4 ( x 3 ) (20 t) . The amplitude of the particle at position x is given by: A(x) = |4 ( x 3 ) | At x = 1 cm , the amplitude is: A = |4 ( 3 ) | = 4 1 2 = 2 cm From the time-dependent part of the wave equation, the angular frequency is = 20 rad/s . The maximum speed of a particle executing simple harmonic motion is given by v_ max = A . v_ max = 2 20 = 40 cm/s Comparing this with v₀ cm/s , we get v₀ = 40 . Answer: 40