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(C) The fundamental frequency of a vibrating string is given by $f = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$,where $T$ is the tension and $\mu$ is the line

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$1: 1$

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(C) The fundamental frequency of a vibrating string is given by $f = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$,where $T$ is the tension and $\mu$ is the linear mass density.Linear mass density $\mu = 	ext{Area} 	imes 	ext{Density} = (\pi R^2) ho$.Substituting $\mu$ into the frequency formula: $f = \frac{1}{2L} \sqrt{\frac{T}{\pi R^2 ho}} = \frac{1}{2LR} \sqrt{\frac{T}{\pi ho}}$.Given that the material is the same,$ho$ is constant. Since $T$ is also constant,we have $f \propto \frac{1}{LR}$.For the first string: $L_1 = L$,$R_1 = 2r$. Thus,$f_1 \propto \frac{1}{L \cdot 2r} = \frac{1}{2Lr}$.For the second string: $L_2 = 2L$,$R_2 = r$. Thus,$f_2 \propto \frac{1}{2L \cdot r} = \frac{1}{2Lr}$.The ratio of their frequencies is $\frac{f_1}{f_2} = \frac{1/2Lr}{1/2Lr} = 1:1$.

Two strings of the same material having lengths $L$ and $2L$,and radii $2r$ and $r$ respectively,are vibrating in the fundamental mode. The tension applied to both strings is the same. The ratio of their respective fundamental frequencies is:

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