An object of mass m is hung from a thin taut string (with$$ linear mass density μ$$=2$$ $$g/m) that passes over a light pulley as shown in figure. The string is connected to a vibrator which can be vibrated with constant frequency f, and the length of the string between vibrator and the pulley is L $$=$$ $$2.0$$ m. The standing waves are formed when the mass m of the object is either $$16.0$$ kg or $$25.0$$ kg. It is observed that no standing waves are formed with any mass between these values, however. What is the largest object mass (in$$ $$kg) for which standing waves could be observed? (Use$$ $$g=9.8$$ $$m/s2)

Question

An object of mass m is hung from a thin taut string (with$$ linear mass density μ$$=2$$ $$g/m) that passes over a light pulley as shown in figure. The string is connected to a vibrator which can be vibrated with constant frequency f, and the length of the string between vibrator and the pulley is L $$=$$ $$2.0$$ m. The standing waves are formed when the mass m of the object is either $$16.0$$ kg or $$25.0$$ kg. It is observed that no standing waves are formed with any mass between these values, however. What is the largest object mass (in$$ $$kg) for which standing waves could be observed? (Use$$ $$g=9.8$$ $$m/s2)

Infinity Learn · Accepted Answer

Let p be the number of loops in the standing wave resulting from the $$25.0$$ kg mass. Then p $$+$$ $$1$$ is the number of loops for the standing wave resulting from the $$16.0$$ kg mass.The wavelength for standing waves in similar end condition, λ$$=2Ln$$ and the frequency is $$f=v$$λ.The frequency does not change as the masses are changed.Thus, $$f=p2LTp$$μ and also $$f=p+12LTp+1$$μEquating the expressions for f, we $$havep+1p=TpTp+1=25.0$$×$$g16.0$$×$$g=54Therefore$$, $$4p$$ $$+$$ $$4$$ $$=$$ $$5p$$, or p $$=$$ $$4$$. Using either expression for f, we find $$f=42$$×$$2.025.0$$×$$9.80.00200=350$$ HzThe tension in the string for n loops formed, $$Tn=mgHence$$ frequency, $$f=n2LTn$$μ$$=n2Lmg$$μ⇒$$m=4L2f2$$μ$$n2g$$          …$$(i)For$$ largest mass m, the number of loops should be least, n $$=$$ $$1Hence$$, $$m=4(2.0)2(350)2$$×$$0.00200(1)2$$×$$9.80=400$$ kg

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