The image presents a problem asking to determine the resultant of a system of concurrent forces in a two-dimensional plane, with the force magnitudes and angles given. Here's the breakdown:

Given:

• $F_1 = 150 \, \text{N}$ at an angle of 30° above the positive $x$-axis.
• $F_2 = 200 \, \text{N}$ at an angle of 60° above the negative $x$-axis.
• $F_3 = 80 \, \text{N}$ at an angle of 45° below the negative $x$-axis.
• $F_4 = 180 \, \text{N}$ along the negative $y$-axis.

Steps to solve:

1. Resolve each force into its components:

   • $F_1$ (150 N, 30°):
     • $F_{1x} = 150 \cos(30°)$
     • $F_{1y} = 150 \sin(30°)$
   • $F_2$ (200 N, 60°):
     • $F_{2x} = -200 \cos(60°)$ (negative since it's along the negative $x$-axis)
     • $F_{2y} = 200 \sin(60°)$
   • $F_3$ (80 N, 45°):
     • $F_{3x} = -80 \cos(45°)$ (negative since it's along the negative $x$-axis)
     • $F_{3y} = -80 \sin(45°)$ (negative since it's below the $x$-axis)
   • $F_4$ (180 N, along negative $y$-axis):
     • $F_{4x} = 0$ (no $x$-component)
     • $F_{4y} = -180$ (entirely along negative $y$-axis)

2. Sum up all the components:

   • $F_x = F_{1x} + F_{2x} + F_{3x} + F_{4x}$
   • $F_y = F_{1y} + F_{2y} + F_{3y} + F_{4y}$

3. Find the magnitude of the resultant force:

   • $R = \sqrt{F_x^2 + F_y^2}$

4. Find the angle of the resultant force:

   • $\theta = \tan^{-1}\left(\frac{F_y}{F_x}\right)$

I'll compute the values of the components and determine the resultant.The components and resultant for the system of forces are as follows:

• $F_x = -26.67 \, \text{N}$ (negative, meaning it's toward the negative $x$-axis)
• $F_y = 11.64 \, \text{N}$ (positive, meaning it's toward the positive $y$-axis)

The magnitude of the resultant force is:

• $R = 29.09 \, \text{N}$

The angle of the resultant force is:

• $\theta_R = 156.42^\circ$ (measured counterclockwise from the positive $x$-axis)

This angle indicates that the resultant force is directed primarily in the second quadrant (between the negative $x$-axis and positive $y$-axis).

Would you like further clarification on any steps, or more detailed explanation?

Here are some related questions for deeper exploration:

1. How do the angles of forces affect their resultant direction?
2. How would changing the magnitude of $F_1$ alter the resultant?
3. What happens if we add another force along the positive $y$-axis?
4. How does the angle calculation change if we resolve forces using different reference axes?
5. What are practical applications of determining the resultant force in engineering?

Tip: To simplify vector problems, always break forces into their components first before summing them up.