JEE MAIN - Physics (2005 - No. 63)
An annular ring with inner and outer radii $${R_1}$$ and $${R_2}$$ is rolling without slipping with a uniform angular speed. The ratio of the forces experienced by the two particles situated on the inner and outer parts of the ring, $${{{F_1}} \over {{F_2}}}\,$$ is
$${\left( {{{{R_1}} \over {{R_2}}}} \right)^2}$$
$${{{{R_2}} \over {{R_1}}}}$$
$${{{{R_1}} \over {{R_2}}}}$$
$$1$$
Explanation
Let the mass of each particle is m.
Then force experienced by each particle, $$F = m{\omega ^2}R$$
$$\therefore$$ $${{{F_1}} \over {{F_2}}} = {{m{\omega ^2}{R_1}} \over {m{\omega ^2}{R_2}}}$$
$$ \Rightarrow $$ $${{{F_1}} \over {{F_2}}} = {{{R_1}} \over {{R_2}}}$$
Then force experienced by each particle, $$F = m{\omega ^2}R$$
$$\therefore$$ $${{{F_1}} \over {{F_2}}} = {{m{\omega ^2}{R_1}} \over {m{\omega ^2}{R_2}}}$$
$$ \Rightarrow $$ $${{{F_1}} \over {{F_2}}} = {{{R_1}} \over {{R_2}}}$$
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