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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 wheel, 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 passes the semicircular parts together. If the coefficient of linear expansion of the metal is $$\alpha$$, and its Young's modulus is $$Y$$, the force that one part of the wheel applies on the other part is:

A
$$2SY\alpha \Delta T$$
B
$$2\ \pi\ SY\alpha \Delta T$$
C
$$SY\alpha \Delta T$$
D
$$\pi\ SY\alpha \Delta T$$
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Solution
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Correct option is C. $$SY\alpha \Delta T$$

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