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)

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=vλ.The frequency does not change as the masses are changed.Thus, f=p2LTpμ and also f=p+12LTp+1μEquating 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=n2LTnμ=n2Lmgμ⇒m=4L2f2μn2g          …(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