To find the resultant force acting on the particle at point O, we need to determine the vector sum of the forces $F_1$, $F_2$, and $F_3$.

Step 1: Decompose the forces into their components.

1. Force $F_1 = 80.0 \, \text{N}$, directed at an angle of $30^\circ$ below the positive x-axis.

   • $F_{1x} = F_1 \cos(30^\circ) = 80 \times \frac{\sqrt{3}}{2} = 40\sqrt{3} \approx 69.28 \, \text{N}$
   • $F_{1y} = F_1 \sin(30^\circ) = 80 \times \frac{1}{2} = 40 \, \text{N}$ (downward, so it's negative in the y-axis).

2. Force $F_2 = 60.0 \, \text{N}$, directed along the positive y-axis.

   • $F_{2x} = 0$
   • $F_{2y} = 60 \, \text{N}$

3. Force $F_3 = 40.0 \, \text{N}$, directed along the negative x-axis.

   • $F_{3x} = -40 \, \text{N}$
   • $F_{3y} = 0$

Step 2: Sum of components.

1. Sum of x-components: $F_{\text{resultant}, x} = F_{1x} + F_{2x} + F_{3x} = 40\sqrt{3} + 0 - 40 \approx 29.28 \, \text{N}$

2. Sum of y-components: $F_{\text{resultant}, y} = F_{1y} + F_{2y} + F_{3y} = -40 + 60 + 0 = 20 \, \text{N}$

Step 3: Calculate the magnitude of the resultant force.

$F_{\text{resultant}} = \sqrt{(F_{\text{resultant}, x})^2 + (F_{\text{resultant}, y})^2}$ $F_{\text{resultant}} = \sqrt{(29.28)^2 + (20)^2} \approx \sqrt{856.0384 + 400} \approx \sqrt{1256.0384} \approx 35.45 \, \text{N}$

Step 4: Calculate the direction of the resultant force.

$\theta = \tan^{-1} \left( \frac{F_{\text{resultant}, y}}{F_{\text{resultant}, x}} \right)$ $\theta = \tan^{-1} \left( \frac{20}{29.28} \right) \approx 34.56^\circ \text{ above the positive x-axis}$

Result

• Magnitude of the resultant force: $35.45 \, \text{N}$
• Direction: $34.56^\circ$ above the positive x-axis.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do we find the resultant force if there are more than three forces acting?
2. What happens to the resultant force direction if one of the forces is reversed?
3. How do we calculate the components if the forces are not perpendicular?
4. Can we use graphical methods to find the resultant force? If yes, how?
5. How would friction or other resistive forces affect the resultant force calculation?

Tip: Always resolve forces into components first, then combine them to simplify complex problems.