A 1.5m string is stretched between a pulley and a wave generator consisting of a

Question

A 1.5m string is stretched between a pulley and a wave generator consisting of a plate oscillating up and down with a small amplitude and frequency of 120 Hz. When a mass of 216g 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? (find tension first)

d) How large a mass would have to be used to generate a standing wave with two antinodes? With one antinode?

Explanation / Answer

Here ,

frequency , f = 120 Hz

tension in the string , T = m * g

T = 0.216 * 9.8

T = 2.12 N

a) for three antinodes

L = 3 * wavelength/2

1.5 = 3 * wavelength/2

wavelength = 1 m

the wavelength of standing wave is 1 m

b)

wave velocity= wavelength * frequency

wave velocity = 1 * 120 m/s

wave velocity is 120 m/s

c)

tension in the string , T = m * g

T = 0.216 * 9.8

T = 2.12 N

as velocity of wave = sqrt(T/u)

120 = sqrt(2.12/u)

u = 1.47 *10^-4 Kg/m

the linear mass density of string is 1.47 *10^-4 Kg/m

d)

for two antinode ,

wavelength = L

wavelength = 1.5 m

Now , let mass is m

speed of wave = sqrt(m*g/u)

1.5 * 120 = sqrt(m * 9.8/(1.47 *10^-4 ))

sovling for m

m = 0.486 Kg

the mass hanged is 0.486 Kg

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