Two unifrom strectched strings A and B, made of steel, are vibrating under the same tension. If the first overtone of A is equal to the second overtone of B and if the radius of A is twice that of B, the ratio of the lengths of the strings is

Options

A. A
B. 0
C. `(A)/(sqrt(2))`
D. `(sqrt(3)A)/(2)`

Answer

C

Explanation

Frequency in stretched string ,
`f = (n)/(2l).v= (n)/(2l)sqrt((T)/(m))`
where l = length , T = tension and
m=mass per unit length
So `m= (pir^(2)l.d)/(l)` where d = density and m =`pir^(2)d.`
Given `f_(A)` ( first overtone n=2)
= `f_(B)` ( second overtone n=3)
`implies (2)/(2l_(A))sqrt((T)/(m_(A)))=(3)/(2l_(B))sqrt((T)/(m_(g)))implies (1)/(l_(A)sqrt(pir_(A)^(2)d))=(3)/(2l_(B)sqrt(pir^_(B)^(2)d))`
`implies (l_(A))/(l_(B))=(2)/(3)xx(r_(B))/(r_(A))= (2)/(3)xx(1)/(2)implies (l_(A))/(l_(B))=(1)/(3)`