question_answer

An annular ring with inner and outer radii \[{{R}_{1}}\] and \[{{R}_{2}}\]is rolling without slippling 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, \[\frac{{{F}_{1}}}{{{F}_{2}}}\]is

A)  1            

B)                         \[\left( \frac{{{R}_{1}}}{{{R}_{2}}} \right)\]

C)  \[\frac{{{R}_{2}}}{{{R}_{1}}}\]                   

D)         \[{{\left( \frac{{{R}_{1}}}{{{R}_{2}}} \right)}^{2}}\]

Correct Answer: B

Solution :

Let panicle \[A\] is situated on the inner part and \[B\] on the outer part of the ring. As the ring is moving with uniform angular speed therefore, both particles will feel centrifugal force. \[\therefore \]  \[\frac{{{F}_{1}}}{{{F}_{2}}}=\frac{{{F}_{A}}}{{{F}_{B}}}=\frac{m{{\omega }^{2}}{{R}_{1}}}{m{{\omega }^{2}}{{R}_{2}}}\] \[\Rightarrow \]               \[\frac{{{F}_{1}}}{{{F}_{2}}}=\frac{{{R}_{1}}}{{{R}_{2}}}\]