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 function of x and t that describes the wave on the string.

Answer 1

The wave is traveling along positive x-axis direction . As the turning- fork is attached at the left end . if is sending the disturbance along right (left →right ) which corresponds to positive x-axis.
Y(x, t) = asin(wt - kx )
a = 5 cm = 0.05 m
Frequency ( f) = 256 Hz
Tension in string = mg = 90 × 9.8 = 882N
We know,
Velocity of the transverse wave
V = √{T/u}
Where u is mass density
V = √{882/8 × 10^-3 }
= 332 m/s

Angular frequency ( w) = 2πf
= 2 × 22/7 × 256
= 1607.6 rad/s

Wavelength = speed of wave / frequency = 332/256

Propagation constant ( K) = 2π/wavelength
= 2 × 3.14/(332/256)
= 4.84 m

Hence, the required equation of the wave .
Y(x, t) = 0.05 sin(1.6 × 10³t - 4.84x)