Question

The frequency of a vibrating wire is n.If it's tension is doubled , density is halved and diameter is doubled then the new frequency will be​

Answer 1

Given that,

The frequency of a vibrating wire is n. Its tension is doubled, density is halved and its diameter is doubled.

To find,

The new frequency of the vibrating wire.

Solution,

The frequency of a vibrating wire is given by the formula as follows :

$n=\dfrac{1}{2l}\sqrt{\dfrac{T}{M}}$

l is length of wire

M is mass of wire per unit its length

Mass in terms of density is given by :

$M=\dfrac{V\times d}{l}$, V is volume and d is density

or

$M=\dfrac{A\times l\times d}{l}\\\\M=A\times d\\\\M=\pi r^2 d$

So, frequency is given by :

$n=\dfrac{1}{2l}\sqrt{\dfrac{T}{\pi r^2d}}\\\\n=\dfrac{1}{2l r}\sqrt{\dfrac{T}{\pi d}}$

Now, T' = 2T, r' = 2r and d' = d/2

New frequency is given by :

$n=\dfrac{1}{2 r'}\sqrt{\dfrac{T'}{\pi d'}}\\\\n'=\dfrac{1}{2l (2r)}\sqrt{\dfrac{2T}{\pi (d/2)}}\\\\n'=\dfrac{1}{2 r}\sqrt{\dfrac{T}{\pi d}}\\\\n'=n$

So, the new frequency remains the same.