Type your question hereIn the figure, a string, tied to a sinusoidal oscillator

Question

Type your question hereIn the figure, a string, tied to a sinusoidal oscillator at P and running over a support at Q, is stretched by a block of mass m. Separation L = 0.8 m, linear density ? = 0.9 g/m, and the oscillator frequency f = 140 Hz. The amplitude of the motion at P is small enough for that point to be considered a node. A node also exists at Q.

(a) What mass m allows the oscillator to set up the fourth harmonic on the string?
(b) What standing wave mode, if any, can be set up if m = 1 kg (Give 0 if the mass cannot set up a standing wave)?

Explanation / Answer

For the string to be vibrating in its fourth harmonic, the wavelength of the wave must be half of the string length so that two full waves can fit on the piece of string simultaneously.


lambda = {L}/{2} = 0.4m



We can then use this to calculate the velocity of a wave in the string:



v = lambda* f = 0.4(140) = 56ms^{ - 1}


Which allows the strings tension to be calculated:


T = v^2 linear density = 3.484 N



The mass of the block is thereforem=T/9.8= 0.3555 kg.

b)Knowing the mass of the block (thus the tension), the velocity of a wave in the string can be calculated:

v = sqrt {rac{T}{mu }} = sqrt {{9.81}/{{0.0009}}} pprox 78.3ms^{ - 1}




The wavelength of a wave with frequency of 140Hz can then be calculated:



lambda = {v}/{f} = rac{{0.0009}/{{120}} = 0.6525m

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