Question

One end of a long string of linear mass density 8.0 × 10^–3 kg m^–1 is connected to an electrically driven tuning fork of frequency 256 Hz. The other end 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 the function of x and t that describes the wave on the string.

Answer 1

The equation of a travelling wave propagating along the positive y-direction is given by the displacement equation:

y (x, t) = a sin (wt – kx) … (i)

Linear mass density,  `mu = 8.0 xx 10^(-3) kg m^(-1)`

Frequency of the tuning fork, ν = 256 Hz

Amplitude of the wave, a = 5.0 cm = 0.05 m … (ii)

Mass of the pan, m = 90 kg

Tension in the string, T = mg = 90 × 9.8 = 882 N

The velocity of the transverse wave v, is given by the relation:

`v = sqrt(T/mu)`

`= sqrt(882/(8.0xx10^(-3))) = 332 "m/s"`

Angular frequency , `omega = 2piv`

`= 2xx 3.14xx 256`

`= 1608.5 = 1.6 xx 10^(3) "rad/s"`  ....(iii)

Wavelength, lambda = `v/v = 332/256 m`

:. Propagation constant, `k = (2pi)/lambda`

`= (2xx3.14)/(332/256) = 4.84 m^(-1)  ....(iv)`

Substituting the values from equations (ii), (iii), and (iv) in equation (i), we get the displacement equation:

y (x, t) = 0.05 sin (1.6 × 10³t – 4.84 x) m

Answer 2

Here, mass per unit length, g = linear mass density = 8 x 10^-3 kg m^-1;
Tension in the string, T = 90 kg = 90 x 9.8 N= 882 N;

Frequency v = 256 Hz and amplitude A = 5.0 cm = 0.05 m

As the wave propagating along the string isa transverse travelling wave, the velocity of the wave

v = `sqrt(T/mu) = sqrt(882/8xx10^(-3)) ms^(-1) = 3.32 xx 10^2 ms^(-1)`

Now `omega = 2piv = 2 xx 3.142 xx 256 = 1.61 xx 10^2 ms^(-1)` 

Also `v = vlambda` or `lambda = v/v`  = `(3.32 xx 10^2)/256 m`

Propagation constant, `k = (2pi)/lambda = (2xx3.142xx256)/(3.32xx10^2)`

`= 4.84 m^(-1)`

:. The equation of of the wave is

`v(x,t) = A sin(omegat - kx)`

`= 0.05 sin (1.61 xx 10^3 t - 4.84 x)`

Here x,y are in meter  and t is in second