Question

Two strings of the same material and same length are given equal tension. If they are vibrating with fundamental frequencies \(1600\) Hz and \(900\) Hz, then the ratio of their respective radii is

• A. \(16:9\)
• B. \(4:3\)
• C. \(81:256\)
• D. \(3:4\)
• E. \(9:16\)

Hint:

For strings of same material: \[ f\propto \frac{1}{r} \] when length and tension are fixed.

Answer 1

Answer: (E)

For a stretched string: \[ f=\frac{1}{2L}\sqrt{\frac{T}{\mu}} \] Since same material and same length: \[ \mu \propto A \propto r^2 \] So, \[ f\propto \frac{1}{r} \] Therefore, \[ \frac{r_1}{r_2}=\frac{f_2}{f_1} =\frac{900}{1600} =\frac{9}{16} \] Hence, \[ \boxed{(E)\ 9:16} \]