A wooden wheel of radius R is made of two semicircular part . The two parts are held together by a ring made of a metal strip of cross sectional area S and length L. L is slightly less than `2piR`. To fit the ring on the wheel, it is heated so that its temperature rises by `DeltaT`and it just steps over the wheel. As it cools down to surrounding temperature, it process the semicircle parts together. If the coefficient of linear expansion of the metal is `alpha`, and it Young's modulus is Y, the force that one part of the wheel applies on the other part is :

Explanation

increase in length
`DeltaL = alphaLDeltaT rArr (DeltaL)/(L) = alphaDeltaT`
The thermal stress developed is
`(T)/(S) = Y(DeltaL)/(L) = YalphaDeltaT, T = SYalphaDeltaT`
From the `FBD` of one part of the wheel, `F = 2T`
Where `F` is the force applied by one part of the wheel on other part, `F = 2SYalphaDeltaT`