The problem asks to determine the magnitude of the moment about point $B$ due to the force $F_3 = 766 \, \text{N}$, neglecting the force $F_1$.

Step-by-step approach:

1. Identify the components of force $F_3$:

   • $F_3$ makes an angle of $30^\circ$ with the horizontal.
   • The components of $F_3$ are:
     • Horizontal component: $F_{3x} = F_3 \cdot \cos(30^\circ)$
     • Vertical component: $F_{3y} = F_3 \cdot \sin(30^\circ)$

2. Moment arm of each component: The moment arm is the perpendicular distance from point $B$ to the line of action of the force component.

   • For the horizontal component $F_{3x}$, the moment arm is $0.05 \, \text{m}$ (vertical distance from point $B$ to the line of action).
   • For the vertical component $F_{3y}$, the moment arm is the horizontal distance from point $B$ to the line of action of $F_{3y}$, which is $0.5 \, \text{m}$.

3. Moment calculation: The moment $M_B$ at point $B$ is given by:

   $M_B = F_{3x} \cdot d_{y} + F_{3y} \cdot d_{x}$ where:

   • $d_y = 0.05 \, \text{m}$ (moment arm for $F_{3x}$),
   • $d_x = 0.5 \, \text{m}$ (moment arm for $F_{3y}$).

   Now, let's compute:

   • $F_{3x} = 766 \, \text{N} \cdot \cos(30^\circ) = 766 \cdot 0.866 = 663.96 \, \text{N}$
   • $F_{3y} = 766 \, \text{N} \cdot \sin(30^\circ) = 766 \cdot 0.5 = 383 \, \text{N}$

   The moment at point $B$ is:

   $M_B = (663.96 \, \text{N}) \cdot (0.05 \, \text{m}) + (383 \, \text{N}) \cdot (0.5 \, \text{m})$ $M_B = 33.198 \, \text{N} \cdot \text{m} + 191.5 \, \text{N} \cdot \text{m}$ $M_B = 224.698 \, \text{N} \cdot \text{m}$

Final Answer:

The magnitude of the moment about point $B$ due to the force $F_3$ is approximately 224.7 N·m.

Would you like more details or have any further questions?

Here are some related questions for you to explore:

1. How does the angle of the applied force affect the moment?
2. What happens if the length from point $B$ to the point of application of $F_3$ increases?
3. How would the moment change if the force $F_3$ were applied vertically?
4. Can you explain the physical meaning of the moment and its direction?
5. How does the moment arm influence the effectiveness of the applied force?

Tip: Always remember that moments depend on both the force magnitude and the perpendicular distance from the point of rotation to the line of action of the force!