A uniform string is vibrating with a fundamental frequency 'f. The new frequency, if radius and length both are doubled would be……..

Options

A. 2f
B. 3f
C. `f/4`
D. `f/3`

Answer

C

Explanation

Linear density of mass for a given string,
`mu = (M)/(L) = (V rho )/(L) =( AL rho )/(L ) = A rho = jpi r ^(2) rho `
Now fundamental frequency is,
`f - (1)/(2L ) sqrt((T)/(mu)) = (1)/(2L ) sqrt ((T)/( pi r ^(2) rho ))`
` therefore f = (1)/(2Lr ) sqrt ((T)/(pi rho )) implies f prop (1)/(Lr )`
`therefore (f _(1))/( f.) = (L.r.)/(Lr) implies (f)/(f.) = ((2L) (2r))/( Lr ) implies f._(1) = (f)/(4)`