A copper and a tungsten plate having a thickness d each are riveted together so that at $0^0$C they form a flat bimetallic plate. Find the average radius of the curvature of this plate at temperature T. The coefficient of linear expansion for Copper and tungsten are ${\mathrm{\alpha }}_{\mathrm{Cu}} \mathrm{and} {\mathrm{\alpha }}_{\mathrm{w}}$.

detailed solution

Correct option is D

The average length of copper plate at a temperature T is lC = l0(1+αCT) where l0 is the length of copper plate at 0°C.The length of the tungsten plate is lt = l0(1+αtT)We shall assume that the edges of plates are not displaced during deformation and that an increase in the plate thickness due to heating can be neglected.From figure we have length of copper plate: lC= ϕ(R+δ2) and length of tungsten plate lt = ϕ(R-δ2))Consequently, ϕ(R+δ2) = l0(1+αcT)------- (i)ϕ(R-δ2) = l0(1+αtT)------(ii)To eliminate the unknown quantities, f and l0 we divide the equation (i) by (ii) term wise:⇒(R+δ2)(R-δ2)  = (1+αCT)(1+αtT)⇒R = δ[2+(αc+αt)T][2(αc-αt)T]