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A sonometer wire of density rho and radi...

A sonometer wire of density `rho` and radius r is held between two bridges at a distance L apart . Tension in the wire is T. then the fundamental frequency of the wire will be

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Consider a wire of linear density m stretched between two rigid supports a distance L apart. Let T be tension in the string. In the fundamental mode of its vibration, only one loop is formed with the antinode at the centre and two nodes at its ends.

`:. L=(lambda)/(2)" " :. lambda=2L" "`.....(1)
The speed of the transverse wave on the stretched wire is
`v=nlambda=sqrt(T//m)" "`.....(2)
where n is the frequency of vibration.
`:.` In the fundametal maode,
`n=(1)/(lambda)sqrt((T)/(m))=(1)/(2L)sqrt((T)/(m))" "`....(3)
Now,under the tension T, the longtudinal stress in the wire is `T//4`, where A is its area of cross section. If the elognation produced is l, the longitudinal strain in the wire is `l//L`.
Then, Young's modulus of the material is
`Y=("stress")/("strain")=(T//A)/(l//L)=(TL)/(Al)`
`:. T=(YAl)/(L)" "`...(4)
Also, if `rho` is the density of the material, the linear density of the wire,
`m=Arho" "`....(5)
Substituting the expressions for T and from Eqs, (4) and (5) in Eq. (3),
`n=(1)/(2L)sqrt((YAl//L)/(Arho))=(1)/(2L) sqrt((Yl)/(rhoL))`
which is the required expression.
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