This problem requires finding the reaction forces at points $A$ and $B$ using the principles of static equilibrium. Let's solve it step by step.

Given:

• Force $X = 42 \, \text{lb}$ (acting horizontally to the left)
• Force $Y = 50 \, \text{lb}$ (acting vertically downward)
• Distance from force application to point $A$ (horizontal) = 4 inches = 0.33 feet
• Total length between points $A$ and $B$ = 20 feet

Assumptions:

• The beam is in static equilibrium, so the sum of forces in both the horizontal and vertical directions, and the sum of moments, must all be zero.

Step 1: Sum of Moments about Point $A$

Take the sum of moments about point $A$, which eliminates the reaction at $A$. Moments are calculated using the perpendicular distance from the point to the line of action of the force.

• The horizontal force $X$ creates a counterclockwise moment: $M_X = X \times 10 \, \text{ft} = 42 \, \text{lb} \times 10 \, \text{ft} = 420 \, \text{lb-ft}$

• The vertical force $Y$ creates a clockwise moment: $M_Y = Y \times 4/12 \, \text{ft} = 50 \, \text{lb} \times 0.33 \, \text{ft} = 16.67 \, \text{lb-ft}$

• The reaction force at $B$, $B_y$, also creates a clockwise moment about point $A$: $M_B = B_y \times 20 \, \text{ft}$

For equilibrium, the sum of moments about point $A$ is zero: $M_X - M_Y - M_B = 0$ Substitute the values: $420 \, \text{lb-ft} - 16.67 \, \text{lb-ft} - B_y \times 20 \, \text{ft} = 0$ Solving for $B_y$: $B_y = \frac{420 - 16.67}{20} = \frac{403.33}{20} = 20.17 \, \text{lb}$

Step 2: Sum of Vertical Forces

For vertical forces, the sum of the forces in the vertical direction must be zero: $A_y + B_y - Y = 0$ Substitute the known value of $B_y$: $A_y + 20.17 \, \text{lb} - 50 \, \text{lb} = 0$ Solving for $A_y$: $A_y = 50 - 20.17 = 29.83 \, \text{lb}$

Step 3: Sum of Horizontal Forces

The sum of horizontal forces must also be zero: $A_x - X = 0$ Thus, $A_x = X = 42 \, \text{lb}$.

Final Reactions:

• Reaction at $B$ (vertical component) is $B_y = 20.17 \, \text{lb}$.
• Reaction at $A$:
  • Vertical component: $A_y = 29.83 \, \text{lb}$
  • Horizontal component: $A_x = 42 \, \text{lb}$

Would you like more detailed explanations on any part, or do you have other questions? Here are some related questions:

1. What are the conditions for static equilibrium?
2. How do you calculate the moment of a force?
3. What is the importance of reaction forces in structural analysis?
4. How would the reaction forces change if the force $Y$ were applied closer to point $A$?
5. How can this analysis be extended to a three-dimensional structure?

Tip: Always check your unit consistency, especially when dealing with inches and feet, as unit errors can lead to incorrect solutions.