To calculate the resultant forces in the $x$ and $y$ directions, we'll break down each force into its respective components.

Step 1: Identify the Forces and Angles

From the diagram:

1. Force 1 (3 N) at $30^\circ$ above the negative $x$-axis.
2. Force 2 (2 N) at $30^\circ$ below the negative $x$-axis.
3. Force 3 (5 N) along the negative $y$-axis.
4. Force 4 (4 N) at $45^\circ$ above the positive $x$-axis.
5. Force 5 (8 N) along the positive $x$-axis.

Step 2: Resolve the Forces into Components

1. Force 1 (3 N at $150^\circ$):

   • $F_{1x} = 3 \cos(150^\circ)$
   • $F_{1y} = 3 \sin(150^\circ)$

2. Force 2 (2 N at $210^\circ$):

   • $F_{2x} = 2 \cos(210^\circ)$
   • $F_{2y} = 2 \sin(210^\circ)$

3. Force 3 (5 N along negative $y$-axis):

   • $F_{3x} = 0$
   • $F_{3y} = -5 \, \text{N}$

4. Force 4 (4 N at $45^\circ$):

   • $F_{4x} = 4 \cos(45^\circ)$
   • $F_{4y} = 4 \sin(45^\circ)$

5. Force 5 (8 N along positive $x$-axis):

   • $F_{5x} = 8 \, \text{N}$
   • $F_{5y} = 0 \, \text{N}$

Step 3: Calculate the Components

Now, let's calculate these components and sum them to find the resultant forces in the $x$ and $y$ directions.### Resultant Forces:

1. Resultant Force in the $x$-direction:

   • $F_x \approx 6.50 \, \text{N}$ (positive direction)

2. Resultant Force in the $y$-direction:

   • $F_y \approx -1.67 \, \text{N}$ (negative direction)

These components give us the net force acting on the object in the $x$ and $y$ directions.

Would you like more details or have any questions?

Related Questions:

1. How would the net force change if one of the forces increased or decreased?
2. What would be the effect on the net force if one of the forces was removed?
3. How can the resultant vector be graphically represented on the coordinate plane?
4. What is the magnitude and direction of the resultant vector?
5. How would the forces be analyzed if they acted in three dimensions?

Tip: Always check the direction (sign) of your components to ensure they align with the physical setup of the problem.