To find the resultant force from the given diagram, we need to calculate the sum of all forces in both the $x$- and $y$-directions. Let’s break down the forces step by step.

Given Information:

• $F_1 = 30 \, \text{lbf}$ at an angle of $60^\circ$ from the negative $x$-axis
• $F_2 = 18 \, \text{lbf}$ along the positive $x$-axis
• $F_3 = 5 \, \text{lbf}$ along the negative $y$-axis
• $F_4 = 10 \, \text{lbf}$ along the negative $y$-axis

Breaking down the forces into components:

Force $F_1$:

• $F_1$ makes a $60^\circ$ angle with the negative $x$-axis.
• $F_{1x} = -F_1 \cos(60^\circ) = -30 \cos(60^\circ)$
• $F_{1y} = F_1 \sin(60^\circ) = 30 \sin(60^\circ)$

Force $F_2$:

• $F_2$ is along the positive $x$-axis.
• $F_{2x} = 18 \, \text{lbf}$
• $F_{2y} = 0$

Force $F_3$:

• $F_3$ is along the negative $y$-axis.
• $F_{3x} = 0$
• $F_{3y} = -5 \, \text{lbf}$

Force $F_4$:

• $F_4$ is along the negative $y$-axis.
• $F_{4x} = 0$
• $F_{4y} = -10 \, \text{lbf}$

Summing up the components:

Sum of $x$-components:

$\Sigma F_x = F_{1x} + F_{2x} + F_{3x} + F_{4x}$

Sum of $y$-components:

$\Sigma F_y = F_{1y} + F_{2y} + F_{3y} + F_{4y}$

Now, let's calculate these values and find the magnitude of the resultant force $F_R$ and its direction.### Results:

1. Sum of $x$-components: $\Sigma F_x = 3.0000 \, \text{lbf}$
2. Sum of $y$-components: $\Sigma F_y = 10.9808 \, \text{lbf}$
3. Resultant force: $F_R = 11.3832 \, \text{lbf}$
4. Direction of the resultant: $\theta_R = 74.7194^\circ$ (from the positive $x$-axis)

The resultant force has a magnitude of 11.3832 lbf and is directed at an angle of 74.72° from the positive $x$-axis.

Would you like more details or have any questions?

Here are 5 related questions:

1. How would the resultant force change if $F_2$ increased?
2. What would happen to the resultant direction if $F_1$ were doubled?
3. How can we decompose forces into components for other angles?
4. How does the system behave if $F_3$ acts at a different angle?
5. What is the effect of force direction changes on the resultant magnitude?

Tip: Always check that the angle reference for forces aligns with the coordinate axes when resolving components.