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(D) 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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(D) 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 $\mu = 	ext{mass per unit length} = ho 	imes 	ext{Area of cross-section} = ho \pi r^2$,where $ho$ is the density of the material.Given: $L_1 = L$,$r_1 = 2r$,$L_2 = 2L$,$r_2 = r$,and $T_1 = T_2 = T$.For the first string: $f_1 = \frac{1}{2L_1} \sqrt{\frac{T}{ho \pi r_1^2}} = \frac{1}{2L} \sqrt{\frac{T}{ho \pi (2r)^2}} = \frac{1}{4L} \sqrt{\frac{T}{ho \pi r^2}}$.For the second string: $f_2 = \frac{1}{2L_2} \sqrt{\frac{T}{ho \pi r_2^2}} = \frac{1}{2(2L)} \sqrt{\frac{T}{ho \pi r^2}} = \frac{1}{4L} \sqrt{\frac{T}{ho \pi r^2}}$.Taking the ratio: $\frac{f_1}{f_2} = \frac{\frac{1}{4L} \sqrt{\frac{T}{ho \pi r^2}}}{\frac{1}{4L} \sqrt{\frac{T}{ho \pi r^2}}} = 1$.

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