In the arrangement shown below, an object can be hung from a string(with a linea
ID: 2245165 • Letter: I
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In the arrangement shown below, an object can be hung from a string(with a linear mass density In the arrangement shown below, an object can be hung from a string(with a linear mass density In the arrangement shown below, an object can be hung from a string(with a linear mass density µ = 0.00200 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 =2.50 m. When the mass m of the object is either 9.0 kg or 16.0 kg, standing waves are observed; however, no standing waves are observed with any mass between these values. 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
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 itsfundamental 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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