Question

A metal string is fixed between rigid supports. It is initially at negligible tensin. Its Young modulus is `Y`, density `rho` and coefficient of thermal expansion is `alpha`. If it is now cooled through a temperature `=t`, transverse waves will move along it with speed
A. `Ysqrt(alphat)/rho`
B. `alphatsqrt(Y/rho)`
C. `sqrt((Yalphat)/rho)`
D. `tsqrt((Yalpha)/rho)`

Answer 1

c) From, `Y = (F//A)/(DeltaL//L)=(T.L)/(ADeltaL)`
where, `F=T="tension"`.
On cooling, `DeltaL = alphaL(Deltatheta)= alphaLt`
`thereforeY = (TL)/(AalphaLt)`
`T=YAalphaT`
Also, mass per unit length of string
`m=Arho`
As wave velocity = `=sqrt(T/M)` `therefore` v=`sqrt((YAalphat)/(Arho)) = sqrt((Yalphat)/rho)`