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Question

A tuning fork of frequency 440 Hz is attached to a long string of linear mass density 0⋅01 kg m−1 kept under a tension of 49 N. The fork produces transverse waves of amplitude 0⋅50 mm on the string. (a) Find the wave speed and the wavelength of the waves. (b) Find the maximum speed and acceleration of a particle of the string. (c) At what average rate is the tuning fork transmitting energy to the string?

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Solution

Given,
Frequency of the tuning fork, f = 440 Hz
Linear mass density, m = 0.01 kgm−1
Applied tension, T = 49 N
Amplitude of the transverse wave produce by the fork = 0.50 mm
Let the wavelength of the wave be λ.

(a) The speed of the transverse wave is given by
ν=Tm
v=490.01=70 m/s
Also, ν=fλ λ=fv=70440=16 cm

(b) Maximum speed (vmax) and maximum acceleration (amax):
We have: y=A sin ωt-kx
ν=dydt=Aω cos ωt-kxNow,νmax=dydt=Aω =0.50×10-3×2π×440 =1.3816 m/s.And, a=d2ydt2a=-Aω2 sin ωt-kxamax=-Aω2 =0.50×10-3×4π2 4402 =3.8 km/s2

(c) Average rate (p) is given by
p=2π2νA2f2 =2×10×0.01×70×0.5×10-32×4402 =0.67 W

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