1. **Problem Statement:** A horizontal beam of length $L$ is supported at two points: a fixed support at $A$ (left) and a roller support at $B$ (right). The beam carries a uniformly distributed load $w$ (force per unit length) acting downwards along its entire length. 2. **Objective:** Determine the reactions at supports $A$ and $B$ due to the uniformly distributed load $w$. 3. **Relevant Formulas and Rules:** - The total load on the beam is $W = wL$. - The load acts at the centroid of the distribution, which is at the midpoint of the beam, i.e., at $L/2$ from either end. - For equilibrium: - Sum of vertical forces must be zero: $R_A + R_B = W$ - Sum of moments about any point must be zero. 4. **Calculations:** - Taking moments about point $A$ (counterclockwise positive): $$\sum M_A = 0 = R_B \times L - wL \times \frac{L}{2}$$ $$R_B L = wL \times \frac{L}{2}$$ $$R_B = \frac{wL^2}{2L} = \frac{wL}{2}$$ - Using vertical force equilibrium: $$R_A + R_B = wL$$ $$R_A = wL - R_B = wL - \frac{wL}{2} = \frac{wL}{2}$$ 5. **Interpretation:** Both supports $A$ and $B$ share the load equally, each supporting half of the total uniformly distributed load. **Final answer:** $$R_A = \frac{wL}{2}, \quad R_B = \frac{wL}{2}$$