Two equal, straight strips of two different metals are fastened together parallel to each other, a small fired distance d apart to form a bimetallic strip. Find the radius of curvature of the bimetallic strip when it is heated from `0^(@)C" to "t^(@)C`.

Explanation

Let A and B be two straight strips fastened parallel to each other at `0^(0)`C . When heated to `t^(0)C` the strips of different metals expand differently. So, the bimetallic strip bends. The radius of curvature of strip A is `r_(1)` and that of B is `r_(2)`.
If `l_(0)` is the original length of the metal strips at `0^(0)C and l_(1) , l_(2)` are the lengths of strips A , B respectively at `t^(0)C ,` then `l_(1) = l_(0) (1 + alpha_(1) t)` and `l_(2) = l_(0) (1 + alpha_(2) t)` .... (i)
Where `alpha_(1) and alpha_(2)` are the coefficient of linear expansion of strips A and B respectively.
their common centre of curvature O and `r_(1),r_(2)` are the radii of curvature of strips A, B respectively, then
`l_(1) = r_(1) phi and l_(2) = r_(2) phi` ............ (ii)
` (##AKS_NEO_CAO_PHY_XI_V01_PMH_C12_SLV_021_S01.png" width="80%">
Substituting `l_(1) and l_(2) ` from equation (i)
`l_(0) (1 + alpha_(1) t) = r_(1) phi and l_(0) (1 + alpha t) = r_(2) phi)`
`(r_(1) - r_(2) ) phi = l_(0) (alpha_(1) - alpha_(2) )t `
`phi = (l_(0) (alpha_(1) - alpha_(2))t)/((r_(1) - r_(2))) = (l_(0) (alpha_(1) alpha_(2))t)/(d) = (therefore r_(1) - r_(2) = d)`
`l_(0) ( 1 + alpha_(1) t) = r_(1) phi and l_(0) ( 1 + alpha_(2) t ) = r_(2) phi`
` (##AKS_NEO_CAO_PHY_XI_V01_PMH_C12_SLV_021_S02.png" width="80%">
Adding
`l_(0) (1 + alpha_(1) t + l + alpha_(2) t) = (r_(1) + r_(2) ) phi`
`l_(0) (2 + (alpha_(1) + alpha_(2))t) = (r_(1) + r_(2)) (l_(0) (alpha_(1) - alpha_(2))t)/(d)`
` 2 = (r_(1) + r_(2)) ((alpha_(1) - alpha_(2))t)/(d)`
`because (alpha_(1) + alpha_(2))`t is neglected when compared with 2 )
`(r_(1) + r_(2))/(2) = (d)/((alpha_(1) - alpha_(2))t)`
`(r_(1) + r_(2))/(2) ` is the average radius of curvature of the bimetallic strip when heated. It is denoted by r.
`r = (d)/((alpha_(1) - alpha_(2))t)`
When the bimetallic strip is cooled by `1^(0)` C , it bends in opposite direction with same average radius of curvature.