In the arrangement shown below, an object can be hung from a string (with linear

Question

In the arrangement shown below, an object can be hung from a string (with linear mass density mu = 0.002 00 kg/m) that passes over a light pulley. The string is connected to a vibrator (of constant frequency f), and the length of the string between point P and the pulley is L = 2.10 m. When the mass m of the object is either 25.0 kg or 36.0 kg, standing waves are observed; no standing waves are observed with any mass between these values, however. What is the frequency of the vibrator? {Note: The greater the tension in the string, the smaller the number of nodes in the standing wave.) What is the largest object mass for which standing waves could be observed?

Explanation / Answer

Let n be the number of nodes in the standing wave resulting from the 36.0-kg mass.

Then n + 1 is the number of nodes for the standing wave resulting from the 25.0-kg mass.

For standing waves, = 2L/n

and the frequency f = v/

f = [(n+1)/2L][(Tn+1/)]

[n+1/n] = (Tn/Tn+1)

= (36/25)

= 6/5

5n + 5 = 6n

n = 5

f = [5 / 2(2.10)][(36*9.8)/0.00200]

= 500 Hz

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The largest mass will correspond to a standing wave of 1 loop

500 =  [1 / 2(2.10)][(m * 9.8)/0.00200]

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