What is the resultant of three coplanar forces: 300 N at `0^(@)`, 400 N at `30^(@)` and 400 N at `150^(@)` ?

Options

A. 500 N
B. 700 N
C. 1100 N
D. 300 N

Answer

To find the resultant of the three coplanar forces, we will break down each force into its horizontal (x) and vertical (y) components and then sum these components to find the resultant force.

### Step 1: Identify the forces and their angles
- **Force 1 (F1)**: 300 N at 0° (along the positive x-axis)
- **Force 2 (F2)**: 400 N at 30°
- **Force 3 (F3)**: 400 N at 150°

### Step 2: Calculate the components of each force

**Force 1 (F1 = 300 N at 0°):**
- \( F_{1x} = 300 \cos(0°) = 300 \)
- \( F_{1y} = 300 \sin(0°) = 0 \)

**Force 2 (F2 = 400 N at 30°):**
- \( F_{2x} = 400 \cos(30°) = 400 \times \frac{\sqrt{3}}{2} = 200\sqrt{3} \)
- \( F_{2y} = 400 \sin(30°) = 400 \times \frac{1}{2} = 200 \)

**Force 3 (F3 = 400 N at 150°):**
- \( F_{3x} = 400 \cos(150°) = 400 \times -\frac{\sqrt{3}}{2} = -200\sqrt{3} \)
- \( F_{3y} = 400 \sin(150°) = 400 \times \frac{1}{2} = 200 \)

### Step 3: Sum the components

**Total horizontal component (Rx):**
\[ R_x = F_{1x} + F_{2x} + F_{3x} = 300 + 200\sqrt{3} - 200\sqrt{3} = 300 \]

**Total vertical component (Ry):**
\[ R_y = F_{1y} + F_{2y} + F_{3y} = 0 + 200 + 200 = 400 \]

### Step 4: Calculate the resultant force (R)

The resultant force can be calculated using the Pythagorean theorem:
\[ R = \sqrt{R_x^2 + R_y^2} = \sqrt{300^2 + 400^2} \]
\[ R = \sqrt{90000 + 160000} = \sqrt{250000} = 500 \, \text{N} \]

### Final Result
The resultant of the three coplanar forces is **500 N**.

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