Two strings A and B of lengths, `L_(A)=80 cm and L_(B)=x cm` respectively are used separately in a sonometer. The ratio of their densities `(d_(A)//d_(B))` is 0.81. The diameter of B is one -half that of A. If the strings have the same tension and fundamental frequency the value of x is

Options

A. 33
B. 102
C. 144
D. 130

Answer

C

Explanation

Given, `(T_(A))/(T_(B))=1, (L_(A))/(L_(B))=(80)/(x)`
`(D_(A))/(D_(B))=(2)/(1),(d_(A))/(d_(B))=(0.81)/(1)`
Let `mu_(1) and mu_(2)` be the linear densities
`therefore (mu_(A))/(mu_(B))=((D_(A))/(D_(B)))^(2)xx(d_(A))/(d_(B))=((2)/(1))^(2)xx0.81`
`=4xx0.81=3.24`
`therefore (v_(1))/(v_(2))=(L_(B))/(L_(A))xxsqrt((T_(A))/(T_(B))xx(mu_(B))/(mu_(A)))`
`1=(x)/(80)xxsqrt(1xx(2)/(3.24))`
`therefore x=80xx sqrt(3.24)=144`