A travelling wave is produced on a long horizontal string by vibrating an end up and down sinusoidally. The amplitude of vibration is 1.0cm and the displacement becomes zero 200 times per second. The linear mass denstiy of the string is` 0.10 kg m^(-1)` and it is kept under a tension of 90 N. (a) Find the speed and the wavelength of the wave. (b) Assume that the wave moves in the positeve x-directin and at t = 0 the end x= 0 is at its positive extreme position. Write the wave equation. (c ) Find the velocity and acceleration of the particle at x = 50 cm at time t = 10ms.

Answer

30 m/s , 30 cm

Explanation

As displacement becomes zero zoo times per second frequency is 100 Hz
`v =sqrt((T)/(mu)) = sqrt((90)/(0.1)) = 30m//s`
`lambda = (v)/(n) = 0.3m`
Wave equation is
`y = Asin(omegat - kx + phi)`
As at t = 0, particle starts from positive extreme position we use `phi = (pi)/(2)`
`y = 0.01 cos (200pit - (2pi)/(0.3)x)`
`y = 0.01 cos (100t - (x)/(0.3))`
Velocity and acceleration of medium particles is given as
`v = (partialy)/(partialt) = - 0.01 xx 200 pi sin2pi (100t - (x)/(0.3))`
`equiv2pi sin 2pi(1-(5)/(3)) = - 2pi sin((4pi)/(3))`
` = -sqrt(3pi) = - 5.45m//s`
`a = (partialv)/(partialt) = -0.01 xx (200pi)^(2) cos2pi (100t- (x)/(0.3))`
` = - 4000 cos((4pi)/(3)) = 2000m//s^(2)`