1. Interpret the situation

An annular ring is rolling without slipping with uniform angular speed $\omega$.

We need the ratio of the forces experienced by:

• a particle on the inner circle of radius $R_1$
• a particle on the outer circle of radius $R_2$

Since the ring is rotating with uniform angular speed, each particle moves in a circle about the center of the ring.

So the required force is the centripetal force:

$F = m\omega^2 r$

where $r$ is the distance of the particle from the center.

2. Force on particle at inner radius $R_1$

If the mass of each particle is $m$, then

$F_1 = m\omega^2 R_1$

3. Force on particle at outer radius $R_2$

Similarly,

$F_2 = m\omega^2 R_2$

4. Take the ratio

$\frac{F_1}{F_2} = \frac{m\omega^2 R_1}{m\omega^2 R_2} = \frac{R_1}{R_2}$

5. Match with options

$\frac{F_1}{F_2} = \frac{R_1}{R_2}$

So the correct option is:

C. $\displaystyle \frac{R_1}{R_2}$

6. Comparison with stored correct answer

Stored correct answer: C

Our derived answer: C

They match.