To determine the magnitude and direction of the resultant force from the system of forces shown in your image, we can use vector addition.

Given Information:

There are several forces acting at different angles:

• 4 kN at an angle of $15^\circ$
• 7 kN at an angle of $90^\circ$
• 3 kN at an angle of $180^\circ$
• 8 kN at an angle of $255^\circ$

Steps to Solve:

1. Resolve each force into its horizontal ($F_x$) and vertical ($F_y$) components. The general formula for resolving a force $F$ at an angle $\theta$ is: $F_x = F \cos(\theta), \quad F_y = F \sin(\theta)$

   • For the 4 kN force at $15^\circ$: $F_x = 4 \cos(15^\circ), \quad F_y = 4 \sin(15^\circ)$
   • For the 7 kN force at $90^\circ$: $F_x = 7 \cos(90^\circ), \quad F_y = 7 \sin(90^\circ)$
   • For the 3 kN force at $180^\circ$: $F_x = 3 \cos(180^\circ), \quad F_y = 3 \sin(180^\circ)$
   • For the 8 kN force at $255^\circ$: $F_x = 8 \cos(255^\circ), \quad F_y = 8 \sin(255^\circ)$

2. Sum all the horizontal and vertical components: $R_x = \sum F_x, \quad R_y = \sum F_y$ where $R_x$ is the total horizontal component and $R_y$ is the total vertical component.

3. Calculate the magnitude of the resultant force using the Pythagorean theorem: $R = \sqrt{R_x^2 + R_y^2}$

4. Find the direction (angle) of the resultant force using the inverse tangent: $\theta_R = \tan^{-1}\left(\frac{R_y}{R_x}\right)$

Let me compute these values.The resultant force of the system is calculated as follows:

• Horizontal component ($R_x$): $-1.21 \, \text{kN}$
• Vertical component ($R_y$): $0.31 \, \text{kN}$
• Magnitude of the resultant force: $1.25 \, \text{kN}$
• Direction of the resultant force: $165.69^\circ$ (measured counterclockwise from the positive x-axis)

Would you like more details on any step or further clarification?

Here are five questions to expand on this problem:

1. How do you calculate the components of forces at specific angles?
2. Why is the angle of the resultant measured from the positive x-axis?
3. What is the significance of resolving forces into horizontal and vertical components?
4. How would the solution change if one of the forces was negative?
5. Can vector addition be applied to forces acting in three dimensions?

Tip: Always ensure angles are measured correctly relative to the positive x-axis for consistent calculations.