A Wire of density ` 9 g * cm^(-3)` is elongated by 0 . 05 cm when stretched between two clamps 100 cm apart. Find out the lowest frequency of transverse vibration in the wire. Given, Young's modulus of the material of the wire ` = 9 xx 10^(11) dyn * cm^(-2)` .

Explanation

Length of the wire, L = 100 cm , elongation, l = 0.05 cm, density of the material, `rho = 9 g * cm^(-3)` , Young.s modulus of the material ,
` Y = 9 xx 10^(11) dyn * cm^(-2)` .
Let ` alpha = ` area of cross section of the wire
` :.` Mass per unit length, ` m = alpha rho`
Longitudinal stress `= ("tension")/("area of cross-section") = (T)/(alpha)` ,
longitudinal strain `= ("elongatio")/("original length") = (l)/(L)*`
`:. Y = ("longitudinal stress")/("longitudinal strain") = (T//alpha)/(l//L) = (TL)/(alpha l)`
or, ` T = (y alpha l)/(L)`
The lowest frequency of transverse vibration = fundamental frequency,
`n = (1)/(2L) sqrt((T)/(m)) = (1)/(2L) sqrt((Yalphal)/(L)*(1)/(alpha rho))=(1)/(2 xx 100) sqrt(((9 xx 10^(11))xx 0 . 05)/(100 xx 9))`
` = 35 . 36 ` Hz .