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Question

 A copper and a tungsten plate having thickness δ each are riveted together so that at 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 αCu and αW
  1. δ[ 1+(αCu+αW)T ][ 2(αCuαW)T ]      
  2. δ[1+(αCu+αW)T][(αCuαW)T]     
  3. δ[ 2+(αCu+αW)T ][ (αCuαW)T ]     
  4. δ[ 2+(αCu+αW)T ][ 2(αCuαW)T ]      


Solution

The correct option is D δ[ 2+(αCu+αW)T ][ 2(αCuαW)T ]      
The average length of copper plate at a Temperature T is lCu=lo(1+αCuT), where lo is the length of copper plate at 0 C. The length of the tungsten plate is lW=lo(1+αWT) 



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 he neglected.
From figure, we have length of copper plate: lCu=ϕ(R+δ/2) and length of tungsten plate lW=ϕ(Rδ/2)
Consequently, ϕ(R+δ/2)=lo(1+αCuT)(i)
ϕ(Rδ/2)=lo(1+αWT)(ii)
To eliminate the unknown quantities ϕ and lo, we divide the equation (i) by (ii) term wise:
 
(R+δ/2)(Rδ/2)=(1+αCuT)(1+αWT)

R=δ[ 2+(αCu+αW)T ][ 2(αCuαW)T ]    ...(use componendo - dividendo)

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