Two metal rods are fixed end to end between two rigid supports as shown in figure. Each rod is of length `l` and area of cross-section is `A`. When the systeam is heated up,datermine the condition when the juncation between rods does not shift ? (`Y_(1)` and `Y_(2)` are Young's modulus of materials of rods, `alpha_(1)` and `alpha_(2)` are coefficients of linear expansion)

Explanation

Since, each rod is prevented from expension so, they are under compression and mechanical strain. The strain in each rod in zero.
`(e_(1))/(l) = alpha_(1)DeltaT - (F)/(AY_(1)), (e_(2))/(l) = alphaDeltaT - (F)/(AY_(2)) , (e_(1))/(l) = (e_(2))/(l) = 0`
`alpha_(1)DeltaT - (F)/(AY_(1)) = 0` and `alpha_(2)DeltaT - (F)/(AY_(2)) = 0`
`alpha_(1)DeltaT = (F)/(AY_(1))`..........(1) and `alpha_(2)DeltaT = (F)/(AY_(2))`.......(2)
Dividing `(1)` by `(2)`, we get `alpha_(1)Y_(1) = alpha_(2)Y_(2)`