### Q.24:- 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 function of *x *and *t *that describes the wave on the string.

**Answer:-**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, *μ *= 8.0 × 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* = *m*g = 90 × 9.8 = 882 N

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

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 × 10^{3}*t* – 4.84 *x*) m.