A wooden wheel of radius R is made of two semicircular parts (see figure), 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 `2 pi R`. To fit the ring on the whell, it is heated so that its temperature rises by `Delta T` and it just steps over the wheel. As it cools down to surrounding temperature, it presses the semicirular parts together. If te doefficient of linear expansion of the metal is `alpha`, and its Youngs' modulus is dY, the force that on part of the wheel applies on the other part is :

Options

A. `2SYalpha DeltaT`
B. `2piSYalphaDeltaT`
C. `SYalphaDeltaT`
D. `piSYalphaDeltaT`

Answer

A

Explanation

`Y = ("stress")/("strain") implies "stress" = Y xx "strain"`
`(F)/(S) = Y(alphaDeltaT) implies F = YSalphaDeltaT`
Therefore force by one part on other part = `2F = 2SYalphaDeltaT`