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 each force into its components along the x-axis and y-axis, and then sum these components to find the resultant vector.

### Step 1: Identify the Forces and Their Angles
- Force 1: \( F_1 = 300 \, \text{N} \) at \( 0^\circ \)
- Force 2: \( F_2 = 400 \, \text{N} \) at \( 30^\circ \)
- Force 3: \( F_3 = 400 \, \text{N} \) at \( 150^\circ \)

### Step 2: Calculate the Components of Each Force
**For Force 1:**
- \( F_{1x} = F_1 \cos(0^\circ) = 300 \cos(0) = 300 \, \text{N} \)
- \( F_{1y} = F_1 \sin(0^\circ) = 300 \sin(0) = 0 \, \text{N} \)

**For Force 2:**
- \( F_{2x} = F_2 \cos(30^\circ) = 400 \cos(30) = 400 \times \frac{\sqrt{3}}{2} = 200\sqrt{3} \, \text{N} \)
- \( F_{2y} = F_2 \sin(30^\circ) = 400 \sin(30) = 400 \times \frac{1}{2} = 200 \, \text{N} \)

**For Force 3:**
- \( F_{3x} = F_3 \cos(150^\circ) = 400 \cos(150) = 400 \times -\frac{\sqrt{3}}{2} = -200\sqrt{3} \, \text{N} \)
- \( F_{3y} = F_3 \sin(150^\circ) = 400 \sin(150) = 400 \times \frac{1}{2} = 200 \, \text{N} \)

### Step 3: Sum the Components
**Total x-component:**
\[ R_x = F_{1x} + F_{2x} + F_{3x} = 300 + 200\sqrt{3} - 200\sqrt{3} = 300 \, \text{N} \]

**Total y-component:**
\[ R_y = F_{1y} + F_{2y} + F_{3y} = 0 + 200 + 200 = 400 \, \text{N} \]

### Step 4: Calculate the Resultant Force
The magnitude of the resultant force \( R \) can be found using the Pythagorean theorem:
\[ R = \sqrt{R_x^2 + R_y^2} = \sqrt{300^2 + 400^2} = \sqrt{90000 + 160000} = \sqrt{250000} = 500 \, \text{N} \]

### Step 5: Calculate the Angle of the Resultant
The angle \( \theta \) of the resultant with respect to the x-axis can be calculated using:
\[ \theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{400}{300}\right) = \tan^{-1}\left(\frac{4}{3}\right) \]

### Final Result
The resultant of the three coplanar forces is \( 500 \, \text{N} \) at an angle \( \theta = \tan^{-1}\left(\frac{4}{3}\right) \).

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