To solve for the magnitude of the resultant force $F_R = F_1 + F_2 + F_3$, we'll first find the resultant of forces $F_2$ and $F_3$ (denoted as $F'$), and then add this to $F_1$.

Step 1: Decompose Forces $F_2$ and $F_3$

1. Force $F_3$ (50 N) is already along the $x$-axis:

   • $F_{3x} = 50 \, \text{N}$
   • $F_{3y} = 0 \, \text{N}$

2. Force $F_2$ (20 N) makes a 20° angle with the negative $x$-axis:

   • $F_{2x} = -20 \cos(20^\circ) \, \text{N}$
   • $F_{2y} = 20 \sin(20^\circ) \, \text{N}$

Step 2: Calculate $F' = F_2 + F_3$

Now, let's sum the components:

• $F'_x = F_{2x} + F_{3x}$
• $F'_y = F_{2y} + F_{3y}$

Substitute the values:

• $F_{2x} = -20 \cos(20^\circ) \approx -18.79 \, \text{N}$
• $F_{2y} = 20 \sin(20^\circ) \approx 6.84 \, \text{N}$

So:

• $F'_x = -18.79 + 50 = 31.21 \, \text{N}$
• $F'_y = 6.84 + 0 = 6.84 \, \text{N}$

Step 3: Add $F_1$ to $F'$

Force $F_1$ (30 N) is at an angle whose components are based on the 3-4-5 triangle:

• $F_{1x} = -\frac{4}{5} \times 30 = -24 \, \text{N}$
• $F_{1y} = \frac{3}{5} \times 30 = 18 \, \text{N}$

Now, add these components to the components of $F'$:

• $F_{Rx} = F'_x + F_{1x} = 31.21 + (-24) = 7.21 \, \text{N}$
• $F_{Ry} = F'_y + F_{1y} = 6.84 + 18 = 24.84 \, \text{N}$

Step 4: Find the Magnitude of $F_R$

Finally, the magnitude of the resultant force $F_R$ is: $F_R = \sqrt{F_{Rx}^2 + F_{Ry}^2} = \sqrt{(7.21)^2 + (24.84)^2} \approx \sqrt{51.98 + 617.73} \approx \sqrt{669.71} \approx 25.9 \, \text{N}$

Answer:

The magnitude of the resultant force $F_R$ is 25.9 N (to three significant figures).

Would you like details or have any questions? Here are some related questions to explore:

1. How would the result change if the angle of $F_2$ was different?
2. What is the direction of the resultant force $F_R$?
3. How can you use vector addition graphically to verify this result?
4. What would happen if $F_1$ was directed along the positive $y$-axis instead?
5. How can you calculate the work done by the resultant force $F_R$ if it moves an object along a straight path?

Tip: When working with vector forces, always break them down into components before performing any calculations. This simplifies the math and helps avoid errors.