https://r2.erweima.ai/mathbot/image/1724056106170jpck6yuj.jpg

The image shows a diagram of two forces, \( f_1 = 10 \, \text{N} \) and \( f_2 = 5 \, \text{N} \), each acting at an angle of \( 30^\circ \) relative to the x-axis.

To solve this, we can calculate the resultant force by resolving each force into its horizontal and vertical components and then summing these components.

### Resolving Forces:
1. **For \( f_1 = 10 \, \text{N} \):**
   - Horizontal component \( f_{1x} = f_1 \cos(30^\circ) \)
   - Vertical component \( f_{1y} = f_1 \sin(30^\circ) \)

2. **For \( f_2 = 5 \, \text{N} \):**
   - Horizontal component \( f_{2x} = f_2 \cos(30^\circ) \)
   - Vertical component \( f_{2y} = f_2 \sin(30^\circ) \)

### Components Calculation:
- \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \) and \( \sin(30^\circ) = \frac{1}{2} \)

1. **For \( f_1 \):**
   - \( f_{1x} = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3} \, \text{N} \)
   - \( f_{1y} = 10 \times \frac{1}{2} = 5 \, \text{N} \)

2. **For \( f_2 \):**
   - \( f_{2x} = 5 \times \frac{\sqrt{3}}{2} = \frac{5\sqrt{3}}{2} \, \text{N} \)
   - \( f_{2y} = 5 \times \frac{1}{2} = 2.5 \, \text{N} \)

### Summing Components:
- **Horizontal components (X-direction):**
  - \( f_x = f_{1x} + f_{2x} = 5\sqrt{3} + \frac{5\sqrt{3}}{2} \)
  - \( f_x = \frac{10\sqrt{3}}{2} + \frac{5\sqrt{3}}{2} = \frac{15\sqrt{3}}{2} \approx 12.99 \, \text{N} \)

- **Vertical components (Y-direction):**
  - \( f_y = f_{1y} + f_{2y} = 5 + 2.5 = 7.5 \, \text{N} \)

### Resultant Force Calculation:
The magnitude of the resultant force \( R \) is given by:
\[
R = \sqrt{f_x^2 + f_y^2}
\]
Substituting the values:
\[
R = \sqrt{\left(\frac{15\sqrt{3}}{2}\right)^2 + 7.5^2}
\]
\[
R = \sqrt{12.99^2 + 7.5^2} \approx \sqrt{168.92 + 56.25} \approx \sqrt{225.17} \approx 15.01 \, \text{N}
\]

### Direction of the Resultant Force:
The angle \( \theta \) relative to the x-axis is given by:
\[
\theta = \tan^{-1}\left(\frac{f_y}{f_x}\right)
\]
\[
\theta = \tan^{-1}\left(\frac{7.5}{12.99}\right) \approx \tan^{-1}(0.577) \approx 30^\circ
\]

### Final Answer:
- **Magnitude of the resultant force** \( R \) is approximately \( 15.01 \, \text{N} \).
- **Direction of the resultant force** is approximately \( 30^\circ \) relative to the positive x-axis.

Would you like more details or have any questions?

---

**Related Questions:**
1. How would the resultant force change if both angles were increased to \( 45^\circ \)?
2. What is the effect of increasing the force \( f_2 \) to 10 N on the resultant force?
3. How do you calculate the resultant force if the forces are not coplanar?
4. How do the horizontal and vertical components contribute to the net force?
5. What would happen to the direction of the resultant force if \( f_1 \) and \( f_2 \) were equal?

**Tip:** Understanding how to resolve forces into components is crucial when dealing with vector addition problems, especially in physics and engineering contexts.

Learn how to solve a vector addition problem involving forces of 10 N and 5 N, each at 30 degrees to the x-axis. Calculate the resultant force magnitude and direction using trigonometric formulas. Suitable for high school students studying physics and engineering.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation