An aluminium wire is clamped at each end and under zero tension at room temperature. Reducing the tempreture, which results in a decrease in the wire's equilibrium length, increase the tension in the wire. What strain `((DeltaL//L))` rejults in a transverse wave speed of `100 m//s`? Take the cross-seectional area of the wire to be `5.00xx10^(-6)m^(2)`, the density to be `2.70xx10^(3)` kg//m^(3)`, abd young's modulus to be `7.00xx10^(10)N//m^(2)`.

Explanation

We expect some small fraction like `10^(-3)` as an answer. The situation might remind you of fitting a hot iron rim to a wagon wheel or a shrink-fit plastic seal to a medicine bottle.
We must review the relationship of strain to stress, and algebraically combine it with the equation for the speed of a string wave.
the expression for the elastic modulus
`Y=(F//A)/(DeltaL//L)`
becomes an equation for strain
`(DeltaL)/(L)=(F//A)/(Y)`
`we substitute into the equation for the wave speed
`v=sqrt(T//mu)=sqrt(F//mu)`
where tension T means the same as streching force F.
`mu=(m)/(L) =(p(AL))/(L)=pa`
we substitute and, solve, `v^(2)=(F)/(mu)=(1)/(p)(F/A) or (F/A)=pv^(2)`
`and substitute (ii) and (i) to obtain
`(DeltaL)/(L)=(pv^(2)/(y)=((2.70xx10^(3) kg//m^(3))(100m//s)^(2))/(7.00xx10^(10) N//m)^(2))=3.86xx10^(-4)`
The young's modulus equations describes streching that does not really happen, being causes and cancelled out simultaneously bu changes in tension and tempreture. this wire could be used as a thermometer. pluck it, pour liquid nitrogen onto it, pluck it again, and listen to the frequency go up.