String A has a length L, radius of cross-section r, density of material p and is under tension T. String b has all these quantities double those of string A. If `v_(a)` and `v_(b)` are the corresponding fundamentals frequencies of the vibrating string, then

Options

A. `upsilon_(A)=2upsilon_(B)`
B. `upsilon_(A)=4upsilon_(B)`
C. `upsilon_(B)=4upsilon_(A)`
D. `upsilon_(A)=upsilon_(B)`

Answer

B

Explanation

Fundamental frequency of a string is
`upsilon=1/(2L)sqrt(T/(pir^(2)rho))=1/(2Lr)sqrt(T/(pirho))`
where the symbols have their usual meanings.
`therefore(upsilon_(A))/(upsilon_(B))=(L_(B)/L_(A))(r_(B)/r_(A))(T_(A)/T_(B)rho_(B)/(rho_(A)))^(1//2)`
Substituting the given values, we get
`(upsilon_(A))/(upsilon_(B))=((2L)/L)((2r)/r)(T/(2T)(2rho)/(rho))^(1//2)=4,upsilon_(A)=4upsilon_(B)`