Q. Two stretched strings of same material are vibrating under same tension in fundamental mode. The ratio of their frequencies is 1 : 2 and ratio of the length of the vibrating segments is 1 : 4. Then the ratio of the radii of the strings is
A) 2 : 1
B) 4 : 1
C) 3 : 2
D) 8 : 1
Explanation
(d) Frequency $f = \frac{1}{2L}\sqrt{\frac{T}{M}} = \frac{1}{2L}\sqrt{\frac{T}{\pi r^{2}(1)\rho}}$
$= \frac{1}{2rL}\sqrt{\frac{T}{\pi\rho}} \Rightarrow \frac{f_{1}}{f_{2}} = \left( \frac{r_{2}}{r_{1}} \right)\left( \frac{L_{2}}{L_{1}} \right) \Rightarrow \frac{1}{2} = \left( \frac{r_{2}}{r_{1}} \right)\left( \frac{4}{1} \right)$
$\Rightarrow \frac{r_{2}}{r_{1}} = \frac{1}{8} \Rightarrow \frac{r_{1}}{r_{2}} = \frac{8}{1}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.