question_answer

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, \[\frac{{{F}_{1}}}{{{F}_{2}}}\] is

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

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

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

D) 1

Correct Answer: C

Solution :

[c] \[{{a}_{1}}=\frac{v_{1}^{2}}{{{R}_{1}}}=\frac{{{\omega }^{2}}R_{1}^{2}}{{{R}_{1}}}={{\omega }^{2}}{{R}_{1}}~~~\] \[{{a}_{2}}=\frac{v_{2}^{2}}{{{R}_{2}}}={{\omega }^{2}}{{R}_{2}}~\] Taking particle   masses equal    \[\frac{{{F}_{1}}}{{{F}_{2}}}=\frac{m{{a}_{1}}}{m{{a}_{2}}}=\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{R}_{1}}}{{{R}_{2}}}\]