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 ?= 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. (5 pnts) 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.) (b) (5 pnts) What is the largest object mass for which standing waves could be observed?

Explanation / Answer

The wave has 3 complete cycles in the length of the string. We want to solve for tension T given length L, mass density m/L and frequency f. There's a potential pitfall here. The standard equations (ref.) assume the string vibrates at its fundamental frequency, such the string length is 1/2 wavelength. Obviously we have to tailor our dimensions to make those equations applicable. What we do is adjust L = 1/2 wavelength = 1/3 m. First we can easily find wave velocity v. v = f*wavelength = 120*2/3 = 80 m/s. And permuting the 2nd eq. in the ref., T = v^2*m/L = 12.8 N m(hanging) = T/g = 1.306 kg.

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