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.

Question

Philoid · Accepted Answer

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

y(x,t) = asin(ωt-kx) ………… (i)

Linear mass density, μ = 8.0 × 10^-3 kg m^-1

Frequency of the tuning fork, f = 256 Hz

Amplitude of the wave, a = 5.0 cm = 0.05 m

Mass of the pan, m = 90 kg

Tension in the string, T = mg

⇒ T = 90 kg × 9.8 m s^-2

⇒ T = 882 N

The velocity of the transverse wave (v) is given by the relation

⇒

⇒ v = 332.04 m s^-1

Angular frequency, ω = 2πf

∴ ω = 2 × 3.14 × 256 Hz

⇒ ω = 1608.5 rad/s

Wavelength, λ = v/f

⇒ λ = (332.04 m s^-1)/(256 Hz)

⇒ λ = 1.297 m

Propagation constant, k = 2π/λ

∴ k = (2×3.14)/(1.297m)

⇒ k = 4.84 m^-1

Using these values in equation (i),

y(x,t) = 0.05sin(1608.5t – 4.84x)