One end of a long string of linear mass density 8.0×10-3 kgm-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 a function of x and t that describes the wave on the string.
Theequationofatravellingwavepropagatingalongthepositive
x-directionisgivenby:
yx,t=asinωtkx....i
Itisgiveninthequestionthat:
Linearmassdensity,μ=8.0×10-3kgm-1
Frequencyofthetuningfork,ν=256Hz
Amplitudeofthewave,a=5.0cm=0.05m....ii
Massofthepan,m=90kg
Tensioninthestring,T=mg=90×9.8=882N
Thevelocityofthetransversewavev,
v=Tμ=8828.0×103=332m/s
Angularfrequency,
ω=2πν=2×3.14×256=1608.5=1.6×103rad/s...iii
Wavelengthisgivenby,λ=vv=332256m
Propagationconstantisgivenby,
k=2πλ=2×3.14332256=4.84m1...iv
Substitutingthevaluesfromequationsii,iii,andivinequationi,
yx,t=0.05sin1.6×103t4.84xm