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 frequinces 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

D

Explanation

The 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 meaning.
`therefore (upsilon_(A))/(upsilon_(B))=sqrt((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)=4orupsilon_(A)=4upsilon_(B)`