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

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

AIIMSAIIMS 2010Waves

Solution:

Fundamental frequency of a vibrating string is expressed as
$f =\frac{1}{2L}\sqrt{\frac{T}{\mu}}=\frac{1}{LD}\sqrt{\frac{T}{\pi\rho}}$
where, D = diameter of string
$\rho$ = density of the material of string
As length L and radius are doubled, the new frequency
$f ' =\frac{1}{\left(2L\right)\left(2D\right)}\sqrt{\frac{T}{\pi\rho}}=\frac{1}{4} f$