Two strings of same material having lengths as 'L', '2L', and radii '2r', 'r' respectively, are vibrating in…
Two strings of same material having lengths as 'L', '2L', and radii '2r', 'r' respectively, are vibrating in the fundamental mode. Tension applied to both the
strings is same. The ratio of their respective fundamental frequencies is
$4: 3$
$1: 2$
$1: 1$
$3: 4$
Solution
We are comparing the fundamental frequencies of two strings of different lengths and radii, but made of the same material, with the same tension applied.
The fundamental frequency of a string is given by:
$f=\frac{1}{2 L} \sqrt{\frac{T}{\mu}}$
Where:
- $L$ : Length of the string
- $T$ : Tension applied
- $\mu$ : Linear mass density ( $\mu=\rho A=\rho \pi r^2$, with $\rho$ being the material density and $r$ the radius)