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 :

Options

A. `2piSY alphaDeltaT`
B. `SY alphaDeltaT`
C. `piSY alphaDeltaT`
D. `2SY alphaDeltaT`

Answer

D

Explanation

(d) `Y=(F//S)/(DeltaL//L)impliesDeltaL=(FL)/(SY)`
`:. LalphaDeltaT=(FL)/(SY) [:' DeltaL=LalphaDeltaT]`
`:. F=SYalphaDeltaT`
`:.` The ring is pressing the wheel from both sides,
`:. F_(net)=2F=2YSalphaDeltaT`