At `27^(@)`C a copper wire is held taut between two rigid supports, with negligible tension. If the temperature is now reduced to `12^(@)C`, find the speed of transverse wave in the wire.
For copper coefficient of linear expansion `= 1.7 xx 10^(-5).^(@)C^(-1)` Young's modulus `= 1.4 xx 10^(11) N//m^(2)`
Density `= 9 xx 10^(3) kg//m^(3)`

Explanation

Change in length of copper wire due to change in temperature from `27^(@)`C to `12^(@)`C is
`Delta l = alpha l Delta T`
`= 1.7 xx 10^(-5) xx l xx (27 - 12)`
`= l xx 25.5 xx 10^(-5)`m
Now, Young.s modulus is
`Y = (F)/(A).(l)/(Delta l)`
`therefore` Tension produced in wire, `F = (Y A Delta l)/(l)`
`= (1.4 xx 10^(11) xx A xx l xx 25.5 xx 10^(-5))/(l)`
`= 35.7 xx 10^(6) xx AN`
Mass per unit length is
`m = A xx 1 xx rho = A xx 9 xx 10^(3)` kg/m
`therefore` Speed of transverse wave
`upsilon = sqrt((T)/(m)) = sqrt((35.7 xx 10^(6) xx A)/(A xx 9 xx 10^(3)))`
= 62.9 m/s