1. **Problem statement:** A pulley has two rims with radii $R_1$ (inner) and $R_2$ (outer). Two blocks with masses $m_1$ and $m_2$ hang from strings wrapped around these rims. The system is in rotational equilibrium. We need to find the correct equation relating $R_1$, $R_2$, $m_1$, and $m_2$. 2. **Concept:** For rotational equilibrium, the net torque on the pulley must be zero. Torque $\tau$ is given by: $$\tau = r \times F$$ where $r$ is the radius and $F$ is the force (weight of the block here). 3. **Forces:** The force due to each block is its weight: $$F_1 = m_1 g, \quad F_2 = m_2 g$$ where $g$ is acceleration due to gravity. 4. **Torque balance:** Since the pulley is in equilibrium, the clockwise torque equals the counterclockwise torque: $$R_1 F_1 = R_2 F_2$$ Substitute forces: $$R_1 m_1 g = R_2 m_2 g$$ 5. **Simplify:** Cancel $g$ from both sides: $$\cancel{g} R_1 m_1 = \cancel{g} R_2 m_2$$ So, $$R_1 m_1 = R_2 m_2$$ 6. **Conclusion:** The correct equation is option B: $$R_1 m_1 = R_2 m_2$$