A flexible metallic wire at temperature `0^(@)C` has young's modulus Y, length `l` coefficient of thermal expansion `alpha`, volume mass density `rho`. It is clamped between two rigid supports at a distance `l` from each other. If it cools down temperature `Deltatheta` such that a transverse wave can travel through it with speed v. Then

Options

A. `Deltatheta=(rhov^(2))/(2Yalpha)`
B. `Deltatheta=(rhov^(2))/(Yalpha)`
C. frequency of wire if it vibrates with n loops at `-Deltatheta^(@)C` is equal to `(n)/(2l)sqrt((YalphaDeltatheta)/(rho))`
D. Frequency of wire if it vibrates with n loops at `-Deltatheta^(@)C` is given by `(n)/(l)sqrt((YalphaDeltatheta)/(rho))`

Answer

A, C

Explanation

strain `=(Deltal)/(l)=("stress")/(Y)=(T//A)/(Y)`
Relative decrease in length `(Deltal)/(l)=alphaDeltatheta`
speed of transverse wave in wire `=sqrt((T)/(mu))`
`becauseV=sqrt((YalphaDeltatheta)/(rho))(mu//A=rho)`
`impliesDeltatheta=(rhoV^(2))/(Yalpha)`
`f=(nv)/(2l)=(n)/(2l)sqrt((YalphaDeltatheta)/(rho))`