This problem involves determining the resultant of three forces acting on a plate and finding the point where this resultant force crosses the plate (essentially, the center of mass of the forces). To achieve this, we need to find the x- and y-coordinates of the point where the resultant crosses the plate by using the principle of moments (or torque).

Let's break down the steps:

Forces given:

1. Force $F_A = 35 \, \text{kN}$ at point A (making a 40° angle from the x-axis).
2. Force $F_B = 25 \, \text{kN}$ at point B (along the x-axis).
3. Force $F_C = 40 \, \text{kN}$ at point C (making a 30° angle from the y-axis).

Coordinates:

• The plate is circular with a radius of 4 m.
• Points A, B, and C are located on the circumference of the plate.

We will proceed with the following steps:

Step 1: Resolve each force into its x- and y-components

Forces at angles need to be split into x- and y-components:

• Force at $A$ (35 kN): $F_{Ax} = 35 \sin(40^\circ)$ $F_{Ay} = 35 \cos(40^\circ)$

• Force at $B$ (25 kN): $F_{Bx} = 25 \, \text{kN} \quad (\text{acts directly along the x-axis, no y-component})$ $F_{By} = 0$

• Force at $C$ (40 kN): $F_{Cx} = 40 \cos(30^\circ)$ $F_{Cy} = 40 \sin(30^\circ)$

Step 2: Calculate the moments (torque)

To find the point where the resultant force crosses the plate, we apply the principle of moments. We need to calculate the moment of each force about the origin (point $O$).

Step 3: Solve for the resultant's x- and y-coordinates

We use the following formulas to find the coordinates of the point where the resultant force acts:

$x_r = \frac{\sum (F_i \cdot x_i)}{\sum F_i}$ $y_r = \frac{\sum (F_i \cdot y_i)}{\sum F_i}$

Now, I will calculate these components using the given data.The x- and y-coordinates of the point where the resultant of the three forces crosses the plate are approximately:

• $x_r = -0.69 \, \text{m}$
• $y_r = 1.70 \, \text{m}$

This indicates that the resultant force crosses the plate slightly to the left of the origin on the x-axis and above the origin on the y-axis.

Let me know if you'd like more details or further clarification.

Here are 5 follow-up questions related to this concept:

1. How do we calculate the moment of a force about a point?
2. Why do we resolve forces into components when calculating the resultant force?
3. What would happen to the resultant force if one of the forces increased significantly?
4. How can we determine the magnitude and direction of the resultant force itself?
5. What is the physical significance of the point where the resultant crosses the plate?

Tip: Always ensure that your angles are in the correct units (degrees vs radians) when performing trigonometric calculations.