https://file.aiquickdraw.com/mathbot/file/1730333633573859bgh19.jpg

To solve for the net forces in the x and y directions, let’s analyze each force in the diagram and decompose them into their horizontal (x) and vertical (y) components.

### Given Forces:
1. **Normal force, \( N = 38.3 \, \text{N} \)**
   - Direction: Upwards at an angle of \(40^\circ\) to the vertical.
2. **Weight, \( W = 50 \, \text{N} \)**
   - Direction: Vertically downwards.
3. **Force of friction, \( f = 15.2 \, \text{N} \)**
   - Direction: Parallel to the surface, angled downwards to the left at \(40^\circ\) from the horizontal.

### Step 1: Resolve Each Force into Components
#### Normal Force \( N \)
Since \(N\) is directed at an angle of \(40^\circ\) above the horizontal:
- \( N_x = N \sin(40^\circ) = 38.3 \sin(40^\circ) \)
- \( N_y = N \cos(40^\circ) = 38.3 \cos(40^\circ) \)

#### Weight \( W \)
Since \(W\) is vertical:
- \( W_x = 0 \)
- \( W_y = -50 \, \text{N} \) (downward direction is taken as negative)

#### Friction Force \( f \)
Since \(f\) is directed at \(40^\circ\) below the horizontal:
- \( f_x = -f \cos(40^\circ) = -15.2 \cos(40^\circ) \)
- \( f_y = -f \sin(40^\circ) = -15.2 \sin(40^\circ) \)

### Step 2: Calculate Components
Using approximate values for trigonometric functions:
- \( \sin(40^\circ) \approx 0.6428 \)
- \( \cos(40^\circ) \approx 0.7660 \)

#### Normal Force Components
- \( N_x = 38.3 \times 0.6428 \approx 24.61 \, \text{N} \)
- \( N_y = 38.3 \times 0.7660 \approx 29.35 \, \text{N} \)

#### Friction Force Components
- \( f_x = -15.2 \times 0.7660 \approx -11.65 \, \text{N} \)
- \( f_y = -15.2 \times 0.6428 \approx -9.78 \, \text{N} \)

### Step 3: Sum of Forces in Each Direction
#### Net Force in the x-direction, \( F_x \)
\[
F_x = N_x + f_x = 24.61 - 11.65 \approx 12.96 \, \text{N}
\]

#### Net Force in the y-direction, \( F_y \)
\[
F_y = N_y + W_y + f_y = 29.35 - 50 - 9.78 \approx -30.43 \, \text{N}
\]

### Final Answer
- **Net force in the x-direction, \( F_x \approx 12.96 \, \text{N} \)**
- **Net force in the y-direction, \( F_y \approx -30.43 \, \text{N} \)**

Would you like more details on any step, or do you have any questions?

### Additional Questions to Explore:
1. How would the net forces change if the angle of the frictional force changed?
2. What would happen to the net force in the y-direction if the weight increased?
3. How would adding an additional horizontal force affect the x-direction force balance?
4. How can we calculate the resultant force magnitude and direction from \( F_x \) and \( F_y \)?
5. What role does friction play in balancing forces on an inclined plane?

### Tip:
When resolving forces on inclined planes, always set up a coordinate system aligned with the plane to simplify component calculations.

Using the free body diagram below, calculate the net forces in the x and y direction.

Learn how to calculate the net forces in x and y directions using a free body diagram with forces including normal force, weight, and friction. This example explores vector decomposition and force resolution for high school physics students.

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