A copper and a tungsten plate having a thickness `d` 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 `alpha_(Cu) and alpha_(W)`.

Options

A. `delta([1+(alpha_c+alpha_t)T])/([2(alpha_c-alpha_t)T])`
B. `delta([1+(alpha_c+alpha_t)T])/([(alpha_c-alpha_t)T])`
C. `delta([2+(alpha_c+alpha_t)T])/([(alpha_c-alpha_t)T])`
D. `delta([2+(alpha_c+alpha_t)T])/([2(alpha_c-alpha_t)T])`

Answer

D

Explanation

The average length of copper plate at a temparature `T is l_(C) = l_(0) (1+alpha_(C)T)`. Where `l_(0)` is the length of copper plate at `0^(@)C` . The length of the tungsten plate `l_(t) =l_90)(1+alpha_(t)T)`

we shall assume that the edges of plates are displaced during deformation and that an increase the plate thickness due to heating can be neglected from figure we have length of copper value plate: `l=(C) = phi(R + delta //2)` and length of tunfsten plate `l_(t) = phi (1+alpha_(c)T)`
`phi(R-delta//2) = l_(0)(1+alpha_(t)T)`
To eliminate the unknown quantities, `f` and `l_(0)` divide the equation (i) by (ii) term wise:
`rArr ((R+delta//2))/((R-delta//2)) = ((1+alpha_(C)T))/((1+alpha_(t)T))`
`rArr R = delta([2+alpha_(c)+alpha_(t)T])/([2(alpha_(c)-alpha_(t))T])`.