To demonstrate standing waves, one end of a string is attached to a tuning fork

Question

To demonstrate standing waves, one end of a string is attached to a tuning fork with frequency 180 Hz. The other end of the string passes over a pulley and is connected to a suspended mass M as shown in the figure above. The value of M is such that the standing wave pattern has four "loops." The length of the string from the tuning fork to the point where the string touches the top of the pulley is 1.20 m. The linear density of the string is 2.0 times 10^-4 kg/m, and remains constant throughout the experiment. Determine the mass hanging from the string providing the tension.

Explanation / Answer

f = frequency = 180 Hz

T = tension force in the string = Mg

m = mass density = 2 x 10-4

L = 1.20

since there are 4 loops , fourth mode of vibration is given as

f = (4/2L) sqrt(T/m)

180 = (2/1.20) sqrt(Mg/m)

180 = (2/1.20) sqrt(M(9.8)/0.0002)

M = 0.24 kg

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