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Two equal, straight strips of two different metals are fastened together parallel to each other, a small.fixed distanced apart to form a bimetallic strip. Find the radius of curvature of the bimetallic strip when it is heated from 0^(0)C to t^(0) C. |
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Answer» Solution :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. |
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