https://file.aiquickdraw.com/mathbot/file/1725313912206pjpnowcq.jpg

Determine the magnitude of the resultant  FR=F1+F2+F3
  of the three forces by first finding the resultant  F′=F2+F3
  and then forming  FR=F′+F1
 .
Express your answer to three significant figures and include the appropriate units.

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.

Learn how to calculate the magnitude of the resultant force \( F_R = F_1 + F_2 + F_3 \) using vector addition and component methods. Step-by-step solution provided with forces \( F_1 = 30 \, \text{N}, F_2 = 20 \, \text{N}, F_3 = 50 \, \text{N} \). Result expressed to three significant figures. Suitable for high school physics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation