A 3.5 m string is stretched between a pulley and a wave generator consisting of

Question

A 3.5 m string is stretched between a pulley and a wave generator consisting of a plate oscillating up and down with small amplitude and frequency 150 Hz. When a mass of 300 g is attached to the string, the following resonance pattern is achieved with three antinodes.

a) What is the wavelength of this standing wave?

b) What is the wave velocity along the string?

c) What is the linear mass density of the string?

d) How large a load would ne needed to generate a standing wave with two antinodes? With one antinode?

Explanation / Answer

Here Pulley end is fixed i.e. a node is formed.Generator end is free i.e. antinode is formed.

As three antinodes are formed it is a 2nd overtone.

=> f= (2n+1)v/(4L) where n=2 => f= 5v/(4L)

b) v = 4fL/5 = 4 x 150 x 3.5 / 5 = 420 m/s

a) lambda = v/f = 420/150 = 2.8 m

c) velocity in a string , v = sqare root (T/u) where T is tension and u is linear mass density

T= mg = 0.3 x 9.8 = 2.94 N

=>u = T/(v^2) = 2.94 / (420^2)= (5/3) x 10^(-5) kg/m = 1.6667 x 10^(-5) kg/m

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