Replace the system of forces and couple by a single force couple system at A. Refer figure.

Steps to Solve

1. Resolve Forces into Components

Calculate the horizontal and vertical components of the applied forces. The forces are:

• 200 N acting horizontally to the right.
• 100 N acting at a 75-degree angle.

For the 100 N force:

• Horizontal component: $$ F_{x} = 100 \cos(75^\circ) $$
• Vertical component: $$ F_{y} = 100 \sin(75^\circ) $$

2. Calculate Resultant Force

Sum the horizontal and vertical forces to get the resultant force:

• Total Horizontal Force: $$ R_{x} = 200 + F_{x} $$
• Total Vertical Force: $$ R_{y} = F_{y} $$

The magnitude of the resultant force $R$ can be calculated using: $$ R = \sqrt{R_{x}^2 + R_{y}^2} $$

3. Determine Angle of Resultant Force

Calculate the direction (angle $\theta$) of the resultant force: $$ \theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) $$

4. Calculate Moments About Point A

Calculate the moments produced by each force about point A. Positive moments are counterclockwise and negative moments are clockwise. The moments are:

• For the 200 N force: $$ M_{200} = 200 \times 2 $$
• For the 100 N force (4 m from A): $$ M_{100} = 100 \sin(75^\circ) \times 4 $$

Combine these moments and include the couple (80 Nm counterclockwise) present in the system: $$ M = M_{200} + M_{100} - 80 $$

5. Calculate Final Couple at A

Determine the net moment at point A. If there is a clockwise moment, it will be represented positively: $$ \text{Net Moment} = \text{Moment from forces and couples} $$