An annular ring with inner and outer radii `R_1` and `R_2` is rolling wihtout 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/F_2` is

Options

A. 1
B. `R_1/R_2`
C. `R_2/R_2`
D. `(R_1/R_2)^2`

Answer

To solve the problem, we need to analyze the forces acting on two particles located at the inner radius \( R_1 \) and the outer radius \( R_2 \) of the annular ring that is rolling without slipping.

### Step-by-Step Solution:

1. **Understanding the System**:
- We have an annular ring with inner radius \( R_1 \) and outer radius \( R_2 \).
- The ring is rolling without slipping with a uniform angular speed \( \omega \).

2. **Identifying Forces**:
- For a particle at the inner radius \( R_1 \), the force experienced is \( F_1 \).
- For a particle at the outer radius \( R_2 \), the force experienced is \( F_2 \).

3. **Centripetal Force**:
- Both particles experience centripetal force due to their circular motion.
- The centripetal force \( F \) acting on a particle is given by the formula:
\[ F = m \cdot a_c \]
where \( a_c \) is the centripetal acceleration.

4. **Centripetal Acceleration**:
- The centripetal acceleration for a particle moving in a circle of radius \( r \) with angular speed \( \omega \) is given by:
\[ a_c = r \cdot \omega^2 \]
- Therefore, for the inner particle at radius \( R_1 \):
\[ F_1 = m \cdot R_1 \cdot \omega^2 \]
- For the outer particle at radius \( R_2 \):
\[ F_2 = m \cdot R_2 \cdot \omega^2 \]

5. **Finding the Ratio of Forces**:
- To find the ratio of the forces \( \frac{F_1}{F_2} \):
\[ \frac{F_1}{F_2} = \frac{m \cdot R_1 \cdot \omega^2}{m \cdot R_2 \cdot \omega^2} \]
- The mass \( m \) and angular speed \( \omega^2 \) cancel out:
\[ \frac{F_1}{F_2} = \frac{R_1}{R_2} \]

6. **Final Result**:
- Thus, the ratio of the forces experienced by the two particles is:
\[ \frac{F_1}{F_2} = \frac{R_1}{R_2} \]