One end of a long string of linear mass density `8.0 xx 10^(-3) kgm^(-1)` is connected to an electrically driven tuning fork of frequency 256 Hz. The other passes over a pulley and is tied to a pan containing a mass of 90 kg. The pulley end absorbs all the incoming energy so that reflected waves at this end have negligible amplitude . At t=0, the left end ( fork end) of the string x = 0 has zero transverse displacement ( y = 0 ) and is moving along positive y - direction. The amplitude of the wave is 5.0 cm. Write down the transverse displacement y as a function of x and t that describe the wave on the string.

Explanation

Amplitude A = 0.05m, mass = 90 kg
frequency f = 256 Hz, linear density = `8 xx 10^(-3) kgm^(-1)`
Velocity of the wave `= v = sqrt(( T )/( mu ))`
`v = sqrt((90 xx9.8 )/( 8 xx 10^(-3))) ` `v = 332.04 ms^(-1)`
Wavelength `lambda = ( v)/( f) = ( 332.04)/( 256)`
i.e. `lambda = 1.297m`
Propagation constant `k = ( 2pi)/( lambda ) = ( 2 xx 3.142 )/( 1.297) ` `k = 4.845radm^(-1)`
angular frequency `omega = 2 pi f = 2 xx 3.142 xx 256 = 1608.7 rad s^(-1)`
epoch `phi = 0 `
Standard equation for a wave travelling along `+` ve direction with R = 0, t = 0
`y = A sin ( omega t - kx)`
i.e., ` y = 0.05 sin ( 1609 t - 4.8 x ) m `.