1. **Problem Statement:** Determine the support reactions at point A for the beam with a collar that can slide vertically, subjected to a 900 N downward force at the center, a 500 N·m clockwise moment at the center, and geometry as described. 2. **Free Body Diagram and Known Data:** - Vertical force at center: $F = 900$ N downward - Moment at center: $M = 500$ N·m clockwise - Lengths: $1.5$ m from A to center, $1.5$ m from center to bend, $1$ m from bend to B - Beam angle at bend: $45^\circ$ - Collar at A can slide vertically, so vertical reaction $A_y$ exists, but horizontal reaction $A_x$ is zero (no horizontal restraint). 3. **Reactions to find:** - Vertical reaction at A: $A_y$ - Horizontal reaction at A: $A_x = 0$ (since collar slides vertically) 4. **Equilibrium Equations:** - Sum of vertical forces: $$\sum F_y = 0 = A_y - 900$$ - Sum of horizontal forces: $$\sum F_x = 0 = A_x = 0$$ - Sum of moments about A: $$\sum M_A = 0$$ 5. **Calculate moment arm for the 900 N force:** - The 900 N force acts at center, $1.5$ m from A along the beam. - The beam is horizontal from A to center, so moment arm is $1.5$ m vertically. 6. **Calculate moment arm for the 500 N·m moment:** - Moment is applied at center, so it directly adds to moment equilibrium. 7. **Calculate moment from the 900 N force about A:** $$M_{900} = 900 \times 1.5 = 1350 \text{ N}\cdot\text{m (clockwise)}$$ 8. **Sum moments about A:** Taking clockwise as positive, $$\sum M_A = 1350 + 500 - M_{A_y} = 0$$ 9. **Moment due to vertical reaction $A_y$ at A:** - Since $A_y$ acts at A, its moment arm is zero, so it produces no moment. 10. **Check for other moments:** - No other forces produce moments about A. 11. **Solve for $A_y$ from vertical forces:** $$A_y - 900 = 0 \Rightarrow A_y = 900 \text{ N (upward)}$$ 12. **Check moment equilibrium:** $$1350 + 500 = 1850 \neq 0$$ This indicates the beam is not in equilibrium with only $A_y$ reaction. 13. **Consider horizontal reaction $A_x$:** - The collar can slide vertically, so $A_x$ can exist. - The beam is angled at $45^\circ$ from the center to B, so the force at B can create horizontal reaction at A. 14. **Resolve forces at B:** - Since no external force at B, reaction at B is zero. 15. **Reconsider moment arm for $A_y$:** - The beam is horizontal from A to center, so $A_y$ does not create moment about A. 16. **Moment equilibrium must be balanced by a horizontal reaction at A:** - Let $A_x$ be the horizontal reaction at A. - The vertical shaft allows vertical sliding, so $A_x$ can exist. 17. **Calculate moment arm of $A_x$ about A:** - $A_x$ acts horizontally at A, so no moment arm about A. 18. **Moment equilibrium must be balanced by a moment reaction at A:** - Since collar is fixed to member, it can resist moment. 19. **Let $M_A$ be the moment reaction at A:** - Sum moments about A: $$1350 + 500 - M_A = 0 \Rightarrow M_A = 1850 \text{ N}\cdot\text{m (counterclockwise)}$$ 20. **Summary of reactions:** - $A_x = 0$ (no horizontal force) - $A_y = 900$ N upward - $M_A = 1850$ N·m counterclockwise **Final answer:** $$A_x = 0, \quad A_y = 900 \text{ N (upward)}, \quad M_A = 1850 \text{ N}\cdot\text{m (counterclockwise)}$$