A wire is stertched with negligible tension at 30^(@)Cbetween two rigid supports. Find the velocity of the transverse wave in the wire at 20^(@)C . Given, the coefficient of linear expansion, the Young's modulus and the density of the material of the wire ofalpha = 18 xx 10^(-6@) C^(-1), Y = 12 xx 10^(11) dyn * cm^(-2)andrho = 6 g* cm^(-3)respectively.

A wire is stertched with negligible tension at 30^(@)Cbetween two rigi

1.	A wire is stertched with negligible tension at 30^(@)Cbetween two rigid supports. Find the velocity of the transverse wave in the wire at 20^(@)C . Given, the coefficient of linear expansion, the Young's modulus and the density of the material of the wire ofalpha = 18 xx 10^(-6@) C^(-1), Y = 12 xx 10^(11) dyn * cm^(-2)andrho = 6 g* cm^(-3)respectively.
Answer» Solution :Decrease in temperature, `theta = 30 - 20 = 10^(@) C ` ,due to this decrement, the wire will TEND to contract. As aresult, a themal stress will be developed. This thermal stress ` Y alpha theta` , So the tension in the wire, `Y = Y alpha theta A ` , where a = area of CROSS section of the wire. Again, mass per unit LENGTH of the wire, ` m = A * 1* RHO = Arho` `:.`The velocity of transverse wave in the wire is ` V = sqrt((T)/(m)) sqrt((Y alpha theta A)/(A rho))` ` sqrt((Yalpha theta)/(rho)) = sqrt(((12 xx 10^(11)) xx (18 xx 10^(-6)) xx 10)/(6))` ` = 6000 cm* s^(-1)`