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To solve this problem, we need to consider the equilibrium conditions for the system shown. The beam is in equilibrium, meaning the sum of all forces and the sum of all moments acting on the beam must be zero.

### Given Information:
- The weight of the 5 kg mass acting downwards at a distance of 8 m from point B.
  - Force due to this weight = \( W = mg = 5 \times 9.81 = 49.05 \, \text{N} \)
- There is a 65 N force applied at an angle of 100° from the horizontal at point A.
- A 21 N force is acting vertically downward 12 m from point B.
- \( x \) is the unknown distance from point B where the reaction force \( R_f \) acts.

### Steps to Solve:
We will apply the following equilibrium conditions:

1. **Sum of vertical forces = 0**:
   The vertical forces acting on the beam are:
   - The 65 N force at point A has a vertical component \( 65 \sin(100^\circ) \).
   - The weight of the 5 kg mass, 49.05 N.
   - The 21 N force acting downward.
   - The vertical reaction force \( R_f \).

   Summing these forces, we get:
   \[
   R_f + 65 \sin(100^\circ) = 49.05 + 21
   \]

2. **Sum of moments about point B = 0**:
   Taking moments about point B eliminates the need to consider \( R_f \), as its moment arm is zero. We need to account for the moment contributions from:
   - The 5 kg mass (49.05 N) acting at a distance of 8 m.
   - The 65 N force applied at point A, with its moment arm of 12 m and an effective vertical force component \( 65 \sin(100^\circ) \).
   - The 21 N force acting at a distance of 12 m.

   Writing the moment equation about B:
   \[
   49.05 \times 8 + 21 \times 12 = R_f \times x + 65 \sin(100^\circ) \times 12
   \]

Let's solve these two equations systematically to find \( x \) and \( R_f \). I'll begin the calculations now.The calculations yield the following results:

- The reaction force \( R_f \) at point B is approximately **6.04 N**.
- The distance \( x \) is calculated to be **-20.5 meters**, which indicates that the current setup or assumptions may need revision. A negative distance suggests a possible error in the forces' direction, assumptions, or in their placement in the system.

Would you like to explore this further, check assumptions, or correct specific parameters?

Here are 5 related questions to consider:
1. How do changes in the angle of the applied force affect the equilibrium?
2. What happens if the applied force at point A is horizontal instead of inclined?
3. Can the beam still remain in equilibrium if additional weights are added?
4. How would the moments change if point B was the pivot instead of A?
5. What impact would friction have on the reaction forces in this system?

**Tip**: Always ensure the sum of forces and moments are both zero when dealing with equilibrium conditions.

Calculate the value of x and reaction force Rf for the condition given in the figure. Assume the object is in equilibrium.

Solve for the reaction force (Rf) and distance (x) in an equilibrium beam with applied forces and moments. Includes a detailed step-by-step explanation using trigonometry and equilibrium conditions. Suitable for undergraduate engineering or physics students.

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